COSMIC-TIME GENERATIVE PRESENTISM (CTGP) A Generative Spacetime Ontology Compatible with General Relativity ========================================================================= =========================== ABSTRACT Cosmic-Time Generative Presentism (CTGP) proposes a dynamic ontology of spacetime in which reality exists only at an advancing present boundary, generated through continuous causal-temporal progression in which each present state emerges from the lawful evolution of immediately preceding states. Earlier states no longer exist as independently real regions of spacetime; the specific physical organizations constituting those earlier states cease to exist as present realities, while their causal consequences continue propagating through later physical structures. Present systems therefore inherit transformed physical continuities, causal records, and distributed consequences of prior states, rather than preserving those earlier states themselves. Unlike static block-universe interpretations of relativity, CTGP treats spacetime as a continuously generated structure emerging through causal propagation governed by physical law. The framework preserves relativistic causal structure while identifying cosmological time, which on large scales is the proper time of the cosmic matter flow, as the natural parameter governing the progression of physical reality. In this model, each present state of the universe arises as the causal continuation of prior states, while the past remains accessible only through physical information encoded in present systems — radiation fields, gravitational structures, and cosmological relics such as the cosmic microwave background. In the emergent geometric regime, CTGP identifies the generative present with the level sets of the cosmological time function: the maximal proper time elapsed since the initial singularity. By a theorem of Andersson, Galloway, and Howard, these level sets are Cauchy surfaces whenever cosmological time is regular, so the present remains globally well defined even where the flow of matter develops caustics inside collapsed structure. The framework therefore applies insofar as the actual universe is globally hyperbolic with regular cosmological time. Identifying these level sets with the present is an interpretive posit rather than a theorem, and the present can be located from its past alone, without reference to the future. The continuum generation parameter σ is a monotone relabeling of cosmological time, which introduces no new field and can optionally be expressed through a constrained action that leaves the Einstein equations unchanged. The paper then determines which physical realizations of the generation parameter are consistent. A natural hypothesis — that generation proceeds faster where matter is denser — fails: an exact density-weighted law drives the generation flow out of alignment with matter within a light-crossing time of any structure, and every stable soft completion forces a uniform generation rate inside bound structures, suppressing any density-weighted cosmological effect below one part in 10⁴⁰. No local density-dependent generation law in the classes analyzed survives these constraints: the generation rate may vary with cosmic epoch but not across space, and CTGP predicts no intrinsic cosmological axis. We further show that presentist and eternalist readings of identical physics are observationally equivalent, so the ontology cannot be verified by measurement at fixed physics. It remains empirically exposed: presentism requires the fundamental laws to be generable from the present state, with no dependence on future boundary data and no global consistency conditions
requiring the whole history at once — a structural commitment that a confirmed chronology violation, interventionist retrocausal dependence, or fundamental final-state condition would refute. A unique final state that simply results from deterministic evolution does not count; only a final condition that must be imposed as an independent input to fix earlier evolution would. Current physics is consistent with that commitment but does not establish it. Because it entails a unique factual past, CTGP also requires single-outcome realism in quantum theory, which excludes standard Everettian ontology. CTGP traces the causal, record-forming, and experiential arrows of time to the direction of generation, relates the thermodynamic arrow to it without reducing it, answers the principal objections — relativity of simultaneity, Lorentz invariance, diffeomorphism invariance, strongcurvature foliation, point presentism, the absence of any intrinsic marker of the present, and underdetermination — and constrains where physically instantiated processes associated with experience may occur, without claiming to determine the nature of consciousness. Keywords: presentism; generative presentism; philosophy of time; general relativity; cosmological time function; foliation; causality; initial-value formulation; observational equivalence; cosmic microwave background; decoherence; quantum foundations; qualia READER’S GUIDE Three layers of claim run through this paper and should be kept distinct when assessing it. Layer 1 is established physics that CTGP adopts without modification: the initial-value formulation of general relativity, global hyperbolicity where applicable, cosmological proper time, relativistic causal structure, conservation laws, and the observed matter-frame and CMB structure. Layer 2 is CTGP’s ontological interpretation, which is the framework’s distinctive philosophical claim: only the current generative state exists, the past possesses causal but not ontological persistence, and the future is genuinely ungenerated. Layer 3 comprises physical hypotheses about how the generation parameter is realized in the continuum: the form of the generation law, any coupling it has to matter, and any observable consequences. Layer 1 is not at issue. Layer 2 is argued on grounds of explanatory economy and carries one structural empirical commitment, that physical law be generable from the present. Layer 3 is where physical hypotheses can fail, and §11 shows that one natural family of them does. Objections directed at one layer should not be taken to bear on another. The paper is organized in four parts. Part I (Interpretive Framework, §§1–5) sets out the philosophical case: the motivation, the position of CTGP among presentism, eternalism, and the growing block, the phenomenological constraint, the comparison with block-universe and growingblock models, and the five core postulates. Part II (Formal Framework, §§6–10) develops the mathematics: the generative present as the level sets of cosmological time, the generation parameter and its admissible forms, general relativity as initial-value generation, the ADM decomposition, the action principle and cosmological foliation, and compatibility with quantum theory and with a pre-geometric ontology. Part III (Physical Realization and Empirical Standing, §§11–12) determines which generation laws are dynamically admissible and states precisely what observation can and cannot establish about the ontology. Part IV (Evaluation, §§13–17) addresses objections, treats cosmological observables as causal records, situates CTGP relative to causal set theory and growingblock models, and collects the formal results. Appendices A–E contain ADM derivations, causalstructure theorems, a vulnerability assessment, the extension of CTGP to conscious experience, and a worked example (Supernova 1987A). Readers primarily interested in the physics case may proceed directly to §6, §9, §11, and §12. The
discussion of conscious experience is confined to Appendix D and is not presupposed by any argument in the main text. ========================================================================= =========================== PART I. Interpretive Framework ========================================================================= =========================== ---------------------------------------------------------------------------------------------------1. Introduction ---------------------------------------------------------------------------------------------------CTGP begins from two distinct sources of friction. The first is the well-known tension between (i) the mathematical convenience of four-dimensional spacetime descriptions and (ii) the phenomenology of temporal passage and the asymmetry between past and future. The second, and less often foregrounded, is the explanatory gap between block universe ontology and the very existence of dynamic experience: if the universe is simply a completed four-dimensional structure, it is unclear how or why any part of that structure would constitute or generate the felt succession of 'nows' that characterizes conscious experience. CTGP’s core claim is not that relativity is false, but that the ontology suggested by a static ‘block’ is not forced by the formalism. GR’s initial-value structure is entirely compatible with a generative reading: evolution equations are not merely descriptive of a pre-existing block, but lawlike rules governing a continuously advancing generative flow, whose state at any moment admits hypersurface descriptions in the emergent-geometric limit. More precisely, minimal generative laws govern the continuous propagation of the present boundary Σ_σ along the generation flow vector n^μ, while effective low-energy laws (GR, QFT) govern the detailed dynamics realized upon generated hypersurfaces. References to the “σ field” denote the effective continuum representation of the primitive generative ordering parameter σ (§10.4). The generative reading of general relativity advanced here does not conflict with quantum mechanics or current approaches to quantum gravity. Relativistic quantum field theory preserves relativistic causal structure at all experimentally tested scales, while decoherence explains the emergence of classical behavior without requiring fundamental retrocausality or the abandonment of causal ordering. Several interpretations of quantum mechanics remain compatible with a causally ordered ontology (§10.1), and among quantum-gravity approaches, causal set theory is a natural companion (§15.2). CTGP articulates this reading precisely and suggests that it satisfies all local relativistic constraints while sustaining an open-future ontology. The framework therefore pursues three interlocking goals: (a) preserving local relativistic constraints, (b) articulating a global ontology in which only the present physical state exists, while causal consequences of prior states continue propagating through transformed matter-energy configurations, inherited structure, and causal records within later states — without implying that such records exhaust all truths about the past, and without implying that physical existence
vanishes except for surviving traces, since CTGP locates non-persistence specifically at the level of the earlier state’s organization rather than at the level of physical continuity itself — and the future remains ungenerated, and (c) grounding the phenomenology of temporal experience — including conscious qualia — in the generative structure of the present hypersurface. These goals are addressed in order: the philosophical case first, the mathematics second, and the implications for mind and experience third. CTGP constrains where and when outcomes become ontologically definite — at the advancing present boundary — without requiring a further selector beyond lawful physical dynamics; conscious agency, where present, contributes as an emergent causal process within those dynamics rather than as an extra-physical source of outcome selection. CTGP introduces no modification to Einstein’s equations and is empirically equivalent to GR at the level of local dynamics. A note on CTGP’s fundamental ontological character: CTGP is a fundamentally continuous framework. Its core mathematical structures — the timelike gradient ∇_μσ, the generation flow vector n^μ, smooth scalar-field representations, continuous propagation equations, smooth Cauchy foliation, and ADM evolution — are all continuously defined. Causal-set models and other discrete approaches to quantum gravity are compatible with CTGP as possible discrete realizations of its generative-ordering principles, but they are not its ontological foundation. CTGP’s fundamental picture is a continuously advancing generative flow, not a sequence of discrete frames. 1.1 Empirical Posture ~~~~~~~~~~~~~~~~~~~~~ CTGP is a thesis about what exists: the present, not the past or the future. Two readings of the same physics that differ only in this respect make identical predictions for every measurement (§12.1), so the framework does not claim that any instrument can detect the nonexistence of the past. Its empirical exposure lies elsewhere: in the requirement that physical law be generable from the present, which can be refuted (§12.2), and in the consistency of its continuum realization, which the paper tests (§11). ---------------------------------------------------------------------------------------------------2. Philosophical Background: Locating CTGP Among Presentism, Eternalism, and the Growing Block ---------------------------------------------------------------------------------------------------2.1 The Standard Taxonomy ~~~~~~~~~~~~~~~~~~~~~~~~~ Three major ontological positions on time are standardly distinguished in the philosophy of physics literature. Eternalism (or the 'block universe' view) holds that past, present, and future are all equally real; the universe is a completed four-dimensional manifold, and 'now' is merely an indexical — like 'here' — with no special ontological status. Presentism holds that only the present exists; past and future are not real. The growing block view, associated with C. D. Broad (Broad 1923) and developed in a relativistic setting by Ellis and Rothman (Ellis & Rothman 2010), holds that past and present are real but the future is not: reality grows as new events are generated. A fourth option, the moving spotlight view, keeps the whole block but lets presentness move through it; Skow (Skow 2015) argues that it is the block universe’s strongest rival, although he ultimately defends the block.
CTGP occupies a distinctive form of cosmologically grounded presentism, referred to throughout as generative presentism. Unlike eternalism, CTGP denies the co-existence of past, present, and future. Unlike growing-block theories, CTGP does not hold that past stages continue to exist; M(σ) is a formal history, not an accumulating region of spacetime. Earlier states existed when present but no longer exist. The present alone possesses ontological existence, while causal consequences of prior states persist through records, memories, radiation fields, material structures, and other inherited physical encodings. This is the framework's full position on the taxonomy, and it is not re-argued below: §4.3 and §15.1 address only what CTGP inherits from prior growing-block models, not whether it is one. 2.2 The Explanatory Challenge for the Block Universe ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The block universe faces a philosophical challenge that is not merely intuitive but structural. In a truly static four-dimensional block, every event — including every purported moment of 'experiencing the present' — is simply a fixed point in the manifold. The block does not evolve or generate; it simply is. The verbs standardly used to describe temporal experience — 'moving through,' 'arriving at,' 'leaving behind,' 'anticipating' — all presuppose change occurring over time. But change occurring over time is precisely what the block universe denies at the ontological level. One standard response is the worldline perspective: from any point along a worldline, past events are causally connected and future events are not yet accessible. This gives a local asymmetry, but it is a descriptive asymmetry, not an ontological one. The worldline runs through a pre-existing structure; nothing is generated. The worldline perspective provides an account of why an observer would represent things as temporally ordered, but it does not explain why there is any experience at all — why the mathematical structure would be accompanied by, or would constitute, the felt succession of 'nows.' Structure alone does not equal experience. The point can be stated without assuming its conclusion. A complete description of a succession is not necessarily a succession that is actually occurring. A four-dimensional block can contain every state of a process and every relation among those states — that B follows A, that the organism at B remembers A — and the eternalist rightly holds that this is what a temporally extended process is, with no moving spotlight required. CTGP concedes all of that. What it disputes is that temporal relations, however complete, amount to temporal becoming: a score is not a performance, and a trajectory drawn on a spacetime diagram does not move through the diagram. CTGP therefore treats succession as a primitive feature of reality rather than as a relation within a completed structure. The present state is generated from its causal predecessor; the predecessor ceases to be the present state; the successor does not yet exist. The disagreement is thus precise: whether a complete temporal structure suffices for becoming, or whether becoming is an additional ontological feature that the ontology must itself contain. CTGP takes this challenge as a positive constraint on ontology: an adequate ontology of time must not merely describe the temporal ordering of events but must sustain the conditions under which genuine temporal becoming — and the experience grounded in it — is possible. The generative presentist structure of CTGP is its response. 2.3 What CTGP Adds to Prior Becoming-Theories ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Existing theories of becoming in the literature — notably Ellis and Rothman's 'crystallizing block universe' (Ellis & Rothman 2010), together with the presentist appeals to cosmic time defended by Craig (Craig 2001) and Zimmerman (Zimmerman 2011) — establish the basic architecture but leave open several questions that CTGP addresses explicitly. First, they do not provide a physically grounded growth parameter defined without coordinates for realistic, inhomogeneous spacetimes: cosmic-time presentists usually take the present from the homogeneous slices of idealized Friedmann models or from the rest frame of the cosmic background radiation. CTGP identifies the growth parameter with cosmological time, a geometric invariant fixed by the causal and metric structure of the spacetime (§6.1). Second, they do not engage the phenomenology of conscious experience as a constraint on the ontology; CTGP treats the present edge Σ_σ as a natural locus for the ongoing physical processes associated with experience and develops the implications for philosophy of mind. Third, they do not articulate the physically instantiated criterion that distinguishes ontological generation from computational simulation. CTGP fills all three gaps. In the emergent regime, σ is fixed by the causal and metric structure already present in general relativity and introduces no new propagating degrees of freedom in the configuration λ = 0 adopted in §6.9. ---------------------------------------------------------------------------------------------------3. Phenomenology of Temporal Experience ---------------------------------------------------------------------------------------------------3.1 The Phenomenological Constraint ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Any adequate theory of time must satisfy what we call the phenomenological constraint: it must explain, rather than explain away, the felt asymmetry between past and future — the sense that one is moving through time from a fixed past into an open future, that the present is where things are 'happening,' and that the past is closed while the future is not. This is not merely an intuitive or folk-psychological datum; it is a structural feature of experience that any reductive or eliminative account must engage. CTGP argues that the block universe does not meet this constraint in the sense sought here — not because it gives the wrong answer, but because its ontology supplies no becoming for the phenomenology to track. On the block view, all events are equally real and equally fixed. The subjective sense of temporal flow becomes at best an illusion — a feature of how observers represent their situation from inside the block — and at worst an inexplicable brute fact about certain classes of mathematical structures. The eternalist can reply that thermodynamics, memory formation, and the architecture of cognition explain why observers have this phenomenology; CTGP’s counter is that such mechanisms explain why an observer has the experience of passage, not whether passage is ontologically real. That is the philosophical disagreement at issue. 3.2 Flow, Asymmetry, and the Generated Present ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP's generative ontology provides the required apparatus. Reality advances as a continuously flowing generative process: the generation flow vector n^μ propagates the present boundary Σ_σ forward along a smooth, timelike field rather than advancing it frame-by-frame. The differential notation Σ_σ → Σ_{σ+dσ} represents an infinitesimal increment of this continuous flow, not a
discrete jump between fixed stages. This flow is asymmetric by design: M(σ) formally represents the generated history of the universe up to the present flow parameter σ, while only the present state possesses ontological existence, and Σ_σ is the advancing edge at which new structure continuously emerges. M(σ) should accordingly be read as a formal representation of causal ancestry and generative ordering, not as a collection of simultaneously existing historical states; the symbol organizes the lawful sequence of generation, it does not store the generated stages as co-present entities. The experienced sense of temporal flow corresponds to this actual, continuously unfolding generative process — not a sequence of discrete snapshots, but a river whose front perpetually advances. Crucially, this is not a mere relabeling of the block universe in generative vocabulary. The formal structure of M(σ) differs from the block in that future hypersurfaces are not part of the domain at any stage σ — they do not exist to be described, even in principle. The open future is not a gap in our knowledge of a pre-existing structure; it is an ontological absence. This is the precise sense in which CTGP sustains an 'open future': the future is genuinely ungenerated, not merely unknown. 3.3 Multiple Registers of the Argument ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The phenomenological case for CTGP can be made in at least three registers. At the intuitive level, temporal experience has a directionality — a before-and-after — that is not symmetrically available in both directions. The past is experienced as fixed and determinate; the future is experienced as open and indeterminate. This asymmetry is not accounted for by the block universe, where all directions in the manifold are equally determinate. At the analogical level, the distinction between a finished DVD (block universe) and a live production (CTGP) captures the ontological point: the DVD encodes a complete structure that is merely revealed as playback proceeds, while a live performance is actually created as it unfolds. At the formal level, CTGP’s M(σ) structure makes the asymmetry mathematically precise: the present is ontologically real, the past is historically real but no longer existent, and the future is excluded from realization. 3.4 Alignment of Phenomenological and Physical Arrows ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP offers a single ontological account of three phenomena that are typically treated as independent explananda: experiential temporal flow, the thermodynamic arrow of time, and the directional structure of causal propagation. In standard physics, these are often traced to separate origins — subjective flow to neuroscience or philosophy of mind, thermodynamic asymmetry to lowentropy initial conditions, and causal directionality to the light-cone structure of relativity. CTGP provides a single underlying account: all three arise from the continuously advancing generative flow represented by M(σ), whose present boundary Σ_σ advances in one direction only along the generation flow vector n^μ. The following arrows all point in the same direction within CTGP, and their alignment is not coincidental. The causal and memory-formation arrows point in the direction of generation by construction; the thermodynamic and cosmological-expansion arrows are related to that direction but not reduced to it, since each also depends on contingent features of our universe’s initial conditions and matter content. The thermodynamic arrow — the increase of entropy from past to future
— runs in the direction in which stages are generated, since each new hypersurface is generated from its predecessor and never the reverse. Generation supplies the orientation, not the gradient: as in standard cosmology, the increase itself requires a low-entropy initial condition, which CTGP must assume exactly as eternalism does. What the generative flow contributes is that this increase cannot run counter to the direction in which stages are produced. The causal arrow — the asymmetry between cause and effect, between light-cone past and future — is the direction of generation itself; where an action formulation is used, it is represented formally by restricting the action’s domain to M(σ), which excludes ungenerated future regions. The cosmological expansion arrow — the directional growth of the universe under FLRW dynamics — provides the physical substrate for σ itself, whose gradient, in FLRW, aligns with the matter-frame congruence. And the memory formation arrow — the capacity of present systems to inherit and encode physical consequences of past events but not future ones, some of which remain sufficiently organized to function as recoverable records — is explained by the persistence of causal continuity within M(σ): information generated by earlier states propagates forward through physical structure, while future hypersurfaces do not yet exist to leave consequences in the present. This alignment strengthens the overall explanatory power argument for CTGP. On CTGP’s assessment, the block interpretation explains these arrows through distinct structures — a low-entropy boundary condition, Lorentzian causal structure, and the architecture of cognition — rather than through a single ontological mechanism by which their convergence would follow. CTGP traces their convergence to a single underlying source: the one-directional succession of generated stages, ordered by cosmological time and formally represented by M(σ). An ontology that provides an account of several arrows of time through a single generative structure has, on this assessment, greater explanatory economy than one that accounts for them separately; whether unification of this kind is an explanatory virtue is part of the philosophical dispute. The economy at issue is explanatory unification: one ontological mechanism underwriting arrows that otherwise require separate explanations. It is not overall ontological parsimony. CTGP’s account of past truth carries its own cost in primitive past-tensed properties (§5.2.1), and the two must be weighed against each other rather than counted on the same side. ---------------------------------------------------------------------------------------------------4. Comparison with Block Universe and Growing-Block Models ---------------------------------------------------------------------------------------------------4.1 The Block Universe Position ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The block universe (eternalism) holds that the complete four-dimensional Lorentzian manifold is the fundamental ontological structure. Every event — every point (x, t) in spacetime — is equally real, regardless of whether it is in the past, present, or future relative to any observer. The GR field equations and the associated mathematical formalism are naturally read on this view as describing an already-complete structure, not generating one. The block universe has significant virtues: it is mathematically parsimonious, it accommodates the relativity of simultaneity without requiring a preferred foliation, and it avoids questions about the mechanism of temporal 'passage.' Terminological note: CTGP is classed throughout as generative presentism rather than as a growing block. It shares the growing block's commitment to directed generation but not its ontology — earlier stages are represented within the formal history M(σ) as completed causal structure while
possessing no ontological existence, and this is a difference in what exists, not in what is drawn. [FIGURE: Figure 1] Figure 1. Ontological structure of the block universe (eternalism) and of CTGP (generative presentism). In the block universe all slices exist equally and “now” is an indexical with no special ontological status. In CTGP the double line marks the active present edge Σ_σ; single lines mark earlier stages, which belong to the formal history M(σ) but no longer exist; the future is ungenerated. The two readings agree on every observation (§12.1); they differ in what exists, and presentism additionally requires the laws of nature to be generable from the present (§12.2). 4.2 The Livestream vs. Recording Distinction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP's central claim against the block universe is best expressed through the recording-versuslivestream distinction. A finished recording (DVD, film) encodes a complete temporal sequence; the sequence is revealed as playback proceeds, but every frame is already determined before playback begins. The block universe is structurally analogous: all events are determined; their 'unfolding' for any observer is merely representational. A live production, by contrast, is actually being created as it proceeds; the next scene is not pre-existing but is genuinely generated by the performers, crew, and circumstances at the moment of production. CTGP claims the universe is more like a live production than a finished recording. This is not merely a metaphor. The formal claim is that the GR initial-value structure, read generatively, produces Σ_{σ+Δσ} from Σ_σ by lawful dynamics without consulting any pre-existing future state. The 'next slice' is not selected from an already-complete manifold but is genuinely produced. CTGP's M(σ) formalism makes this precise: the domain formally represents the generated history leading to the present state, not a restriction over a pre-existing complete block. 4.3 What CTGP Inherits from Growing-Block Models, and Where It Departs ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ellis and Rothman (2010)'s 'crystallizing block universe' provides the closest prior art. They argue that spacetime crystallizes from an uncertain quantum future into a determinate past, using quantum indeterminacy as the mechanism for genuine becoming. CTGP shares with growing-block theories a commitment to objective temporal becoming but differs in several respects. First, CTGP does not require quantum indeterminacy as the engine of becoming; the generation parameter σ is defined by cosmological time, a classical geometric invariant, and is compatible with multiple quantum interpretations. Second, CTGP provides an explicit formal treatment of σ-reparameterization of ADM evolution, which grounds the growth parameter in the variational structure of the Einstein–matter system without requiring quantum indeterminacy as the mechanism of generation, while remaining compatible with objective-collapse, consistent-histories, and epistemic interpretations alike. The moving spotlight view occupies a different position: the present moves through a complete fourdimensional manifold. Skow (Skow 2015) argues that it is the strongest rival to the block universe, although he ultimately defends the block. CTGP rejects the moving spotlight framing because it presupposes the block and adds a moving 'now' to it — a double ontology that CTGP finds unmotivated. CTGP's ontological economy is greater: there is no pre-existing block; there is only the continually generated present state together with its inherited causal structure, formally represented by M(σ).
4.4 The Mathematical Formalism Is Interpretation-Neutral ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ An important clarification: the four-dimensional mathematical formalism of GR is neutral between the block universe and growing-block interpretations. The field equations G_{μν} = 8πG T_{μν} do not, by themselves, determine whether the solution represents a pre-existing structure or a generated one. CTGP's claim is interpretive, not that the mathematics is wrong. What CTGP adds is a principled reason — grounded in the initial-value structure and the phenomenological constraint — to prefer the generative reading over the static one. 4.5 Ontological Underdetermination and Explanatory Preference ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The observation that GR is interpretation-neutral (§4.4) raises the question of how to adjudicate between ontologies that are empirically equivalent at the level of local predictions. Physics routinely faces this situation: quantum mechanics is compatible with Everettian, Bohmian, and collapse interpretations of the same Schrödinger equation; statistical mechanics admits both Boltzmann and Gibbs readings of the Liouville equation; and GR admits both block and growingspacetime readings of the Einstein equation. Empirical equivalence does not imply ontological parity. The standard philosophical move is to apply explanatory criteria: among empirically equivalent ontologies, preference should be given to the one that provides an account of an additional observed phenomenon without introducing new physical laws (see, e.g., Maudlin 2012; Rickles 2016). CTGP formalizes this as the Phenomenological Completeness Principle: among empirically equivalent ontologies, preference should be given to the one that provides an account of additional observed phenomena without introducing new physical laws. The block universe and CTGP make identical local predictions. They differ in how they treat one datum: the phenomenology of temporal experience — the felt asymmetry between past and future, the sense of genuinely unfolding succession. The block universe typically explains this datum at the level of observer cognition and information processing rather than at the level of fundamental spacetime ontology. CTGP differs by treating the asymmetry of temporal experience as reflecting an objective asymmetry in the ontological structure of reality itself. CTGP provides an account of it: temporal experience is grounded in the continuously advancing generative flow at Σ_σ — the smooth propagation of the present boundary along n^μ, not a discrete sequence of frames. CTGP therefore functions not as a competing physical model but as an interpretive completion of relativistic spacetime — an explanatory expansion that accounts at the ontological level for a datum the block universe explains through other structures, while remaining consistent with all existing physics. The Phenomenological Completeness Principle, stated more formally, runs as follows. Given two empirically equivalent formalisms, the one whose fundamental ontology provides a causal-explanatory structure for an additional, well-attested phenomenon is to be preferred, provided it does so without introducing new physical laws or contradicting existing ones. On CTGP’s assessment, it satisfies this criterion: it grounds the phenomenological arrow of time in the ontological arrow of generation, which an action formulation can represent formally by restricting its domain (§9.1). On this view, the felt direction of temporal experience is not an illusion generated by observer psychology; it is the immediate experiential correlate of the asymmetric structure of generation,
which M(σ) represents formally — a profile that always includes the past and never the future. The block universe generally interprets the same phenomenology as arising from the informationprocessing structure of observers embedded within a static spacetime manifold. A static fourdimensional block has no interior direction; every internal relation between events is given all at once. CTGP’s succession of stages runs in one direction only, and the felt direction of experience tracks that succession. Block-universe theorists explain the arrows through distinct structures: a low-entropy boundary condition for the thermodynamic arrow, Lorentzian causal structure for the causal arrow, and the architecture of cognition for temporal experience. On CTGP’s assessment, this leaves their correlation without a single ontological source, whereas CTGP derives them from one. Whether a unified account is preferable to a distributed one is the philosophical question at issue, not a result either side can prove. The two ontologies are also economical in different respects: presentism posits fewer existing things, eternalism less structure, since it needs neither a distinguished foliation nor primitive succession. Which economy should weigh more is itself part of the dispute. ---------------------------------------------------------------------------------------------------5. The Core Postulates of CTGP ---------------------------------------------------------------------------------------------------5.1 Postulates 1–3: The Present, Causal Continuity, and the Cosmological Parameter ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~ Postulate 1 (Ontological Primacy of the Present): At any stage of cosmic evolution, only the current hypersurface of reality—the present state of the universe—possesses ontological existence as presently real. The present is not, however, an isolated instant: it exists as part of a continuous causal-temporal progression whereby each state emerges through the lawful evolution of immediately preceding states and in turn contributes to the emergence of subsequent states. Earlier states do not persist as independently existing regions of spacetime. Their physical consequences continue within later states through ongoing causal continuity, of which only a subset remains organized as information-bearing records accessible in present physical systems. Stated without formalism, the picture is as follows. Write P_n for the entirety of what exists at generative stage n. Existence at that stage is P_n and nothing besides: P_{n−1} has ceased to exist as a state and P_{n+1} has not yet been generated. Generation carries P_n to P_{n+1}, at which point existence is P_{n+1}. The present is accordingly not a distinguished region within a larger existing whole but the whole of what exists. Two clarifications prevent the obvious misreadings. First, that P_{n−1} has ceased to exist as a state does not mean it left nothing behind: its causal consequences are incorporated into P_n, and a subset of them remain organised as records (Postulate 4, §14). Ceasing to exist and becoming irrelevant are different things. Second, the future is unrealised rather than hidden: P_{n+1} is not an existing region we are unable to observe but a set of potential successors of which one is generated. Postulate 2 (Causal Continuity): The present state of the universe arises as the causal continuation of immediately prior physical states. Every event in the present is linked through chains of physical interaction to earlier events within its past light cone. In the emergent-geometric regime, this causal ordering is represented by relativistic spacetime geometry; more fundamentally, causal ordering is the primitive structure from which spacetime geometry is reconstructed (Causal Reconstruction Principle, §10.3). Generative ordering is therefore prior to metric structure, not
derived from it. Postulate 3 (Cosmological Temporal Parameter): The progression of the present boundary is parameterized by cosmological time. In the big-bang-origin formulation used here, this is the maximal proper time elapsed since the initial singularity (§6.1); a surface-relative generalization for bouncing spacetimes is given in §6.10.1. In the emergent geometric regime, the generation parameter σ is a monotone relabeling of this function (Appendix A.3). This provides a physically grounded temporal ordering of successive states of reality and resolves objections based on the relativity of simultaneity. In FLRW solutions, cosmological time is the familiar cosmic time t, equal to the proper time along the comoving matter congruence. That is how the parameter is realized and measured, not what defines it (§6.3). Prior to geometric emergence, σ functions as a pregeometric ordering parameter without presupposing proper time. Its use introduces no preferred frame beyond what is already present in standard cosmology. Terminology. Throughout this framework, the matter frame denotes the local rest frame of the cosmological matter distribution: u^μ is the unit timelike eigenvector of T_{μν}, and ρ_c := T_{μν} u^μ u^ν is the energy density measured in that frame. The matter-frame congruence is the family of worldlines tangent to u^μ. In FLRW and in single-stream regions it coincides with the maximizing geodesics that define cosmological time (§6.3). It is not used to define co-presence, and no postulate requires a physical system to belong to it in order to occupy a generative stage. The distinction is developed in §9.4.1. 5.2 Postulate 4: Persistence Through Causal Continuity and Records ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Postulate 4 (Persistence Through Causal Continuity and Records): Although earlier states of the universe no longer exist as independent regions of spacetime, the physical consequences of those states continue through later states via lawful, continuous transformations of matter, energy, fields, and information-bearing structures. Certain aspects of those consequences remain organized as causal records that encode information about past events, while other aspects become distributed, transformed, or effectively unrecoverable. Such records include radiation fields, particle distributions, gravitational structures, and cosmological relics, the most extensive of which are the cosmic microwave background and the neutral hydrogen observed through the 21 cm line (§14). The signal exists now; the emitting matter existed earlier; the earlier state itself no longer exists. What remains is not the earlier organization but the continuing propagation of its physical consequences, one subset of which functions as a causal record. 5.2.1 Persistence and the Truth of the Past While CTGP holds that earlier states do not persist as independently existing regions of spacetime, the causal consequences of those states continue propagating within later states through ongoing physical evolution. Some of those consequences remain organized as causal encodings and records accessible in the present. This should not be taken to imply that present encodings exhaust all truths about the past. CTGP distinguishes between: (i) ontological persistence, which is limited to present physical structures and their encoded
causal records, and (ii) truth about past events, which may extend beyond what is currently encoded. On this view, past events were fully real at their corresponding present stage and remain determinately true as events that occurred, even if aspects of those events are no longer physically encoded or recoverable in the present state of the universe. Present encodings therefore function as partial physical persistence and evidential traces, not as the total grounding of all past truths. The grounding objection. A standard objection to presentism asks what makes past-tense statements true if the past does not exist. CTGP adopts a Lucretian answer (Bigelow 1996). The truthmaker for a past-tense truth is the present world itself, which instantiates past-tensed properties: the present universe has the property *having been, at stage σ′, such that p*. These properties are primitive. They are not grounded in any further fact, whether a present record or an existing past stage, since no past stage exists to ground them. Nor are they records. Records are categorical physical structures that carry evidence about the past and can be degraded or erased (§14.6). Past-tensed properties belong to the present state as a whole and, once acquired, are retained at every later stage (Definition 17.20). This is the precise sense in which the present lawfully descends from earlier stages: descent is not a relation to something that still exists but a property the present has. Since past-tensed properties are properties of Pₙ itself, this account adds no independently existing past object or historical stage to the present ontology. This also fixes the status of M(σ). M(σ) is not the truthmaker. It is the formal representation of the present world’s past-tensed profile, laid out as if its stages co-existed so that the ordinary semantics of tense can be applied. The ground is the present, and the history is its representation. CTGP therefore faces neither horn of the usual dilemma: it does not reify M(σ), which would make it a growing block, and it does not leave past truths ungrounded. The formal semantics (Definition 17.20) indexes earlier stages in the metalanguage; a clause such as “W_σ′ ⊨ φ” is read as tensed — φ was the case at σ′ — and carries no commitment to existing past stages. The cost should be stated. Past-tensed properties do not supervene on the present’s categorical physical configuration, since two present states could agree in every categorical respect while differing in their pasts. This is the familiar objection that Lucretian properties “cheat” (Sider 2001). CTGP accepts this cost as the formal counterpart of a claim it makes on independent grounds: records are evidence rather than grounds, and truths about the past outrun what is encoded. If past truths supervened on present categorical structure, then losing a record would change what happened, and CTGP denies that (Proposition 17.22). This cost also bears on the explanatory-economy argument of §3.4: the economy claimed there is unification of the arrows of time under one mechanism, not parsimony of ontology, and the primitive properties posited here count against the latter. CTGP therefore does not eliminate past facts; it relocates them into the present, as irreducible pasttensed properties of what exists now. The ontology is leaner than eternalism in what exists, not in what is fundamental. 5.2.2 Conservation, Organization, and Recoverability CTGP distinguishes four separate concepts that are often conflated:
(i) Conservation of physical existence. Matter, energy, fields, and physical processes continue through lawful transformations. In general relativity this continuity is underwritten by local conservation laws, ∇μTμν = 0; global conserved quantities follow, by Noether’s theorem, only where the spacetime has corresponding symmetries. CTGP does not claim that physical existence disappears when a state ceases to be present. (ii) Conservation of information. Information may persist in transformed or distributed forms even when no localized or practically recoverable record remains. Questions regarding the ultimate conservation of information depend upon the relevant physical domain and remain distinct from questions of record formation. (iii) Conservation of organization. Particular arrangements, structures, configurations, and correlations may cease to exist even while the physical constituents involved continue in transformed states. CTGP identifies the loss of present existence primarily with the loss of the earlier state’s specific organization, not with the annihilation of all physical continuity. (iv) Recoverability of knowledge. The existence of physical consequences does not imply that those consequences remain sufficient to reconstruct prior states. Ontological persistence and epistemic accessibility are distinct. Something may continue to exist in transformed or distributed form while becoming effectively unrecoverable as a coherent record. Accordingly, CTGP’s claim that earlier states no longer exist refers to the non-persistence of those states as present ontological organizations, not to the disappearance of all physical continuity arising from them. 5.3 Postulate 5 and the Generative Minimality Principle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Postulate 5 (Hierarchical Law Emergence): Reality is governed at all stages by minimal generative laws sufficient for the lawful progression of the present boundary Σ_σ. These laws admit a hierarchy of levels of description: the primitive generative ordering is fundamental; causal ordering is a fundamental-emergent hybrid; classical spacetime dynamics (GR) and relativistic quantum field theory are effective descriptions valid in their domains; and thermodynamic regularities are emergent. Within the emergent geometric regime, the generative ordering is represented by cosmological time and adds no dynamics to general relativity, which is used without modification; the “σ-generation law” names whatever pre-geometric physics realizes the ordering, not a modification of GR. Higherlevel effective laws — including classical spacetime dynamics, relativistic causal structure, and low-energy field behavior — emerge progressively as physical conditions permit their stable realization. CTGP is therefore never lawless at any stage of cosmic evolution; the Planck epoch is governed by minimal generative structure even when the smooth Lorentzian manifold does not yet exist as a continuum approximation. Together these postulates define a model in which spacetime is not a static four-dimensional block but a dynamically generated causal structure whose present boundary advances through cosmological time. The five postulates jointly entail the Generative Minimality Principle: the minimal lawful structure required for the existence of physical succession is the σ field and its constraint equations. All additional physical laws are effective or emergent within this hierarchy.
5.4 Continuous Causal Becoming and Conservation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP does not interpret temporal succession as a sequence of disconnected states appearing and disappearing independently. Rather, each present state emerges through continuous causal evolution from immediately preceding states. Temporal progression is therefore understood as an uninterrupted process of physical becoming, in which later states arise from, inherit, and transform prior states through lawful causal continuity. Earlier states cease to exist as present realities, yet the transition from one state to the next is not a discontinuous replacement but a continuous evolution of physical reality itself. Accordingly, CTGP views time as an ongoing causal-temporal progression in which reality continuously becomes what it next is through lawful physical transformation. Present physical records are instantiated in finite substrates and are therefore constrained by entropy production, decoherence, noise, and storage limitations. Record formation depends on amplification and stabilization of correlations; record persistence depends on resistance to thermodynamic degradation; and record erasure, when it occurs via logically irreversible operations, incurs a minimum thermodynamic cost as described by Landauer’s principle. CTGP therefore entails that the universe’s accessible causal archive is finite and dynamically evolving, without implying that all past truths are stored or that failure of record formation carries a Landauer cost. CTGP’s conception of continuous causal-temporal evolution is consistent with, and clarified by, the geometric understanding of conservation laws expressed through Noether’s theorem. A natural question for any presentist framework is what grounds physical continuity if earlier states do not persist. The structure of modern physics provides part of the answer. Conserved quantities arise from symmetries of physical law rather than from the persistence of particular physical configurations: time-translation symmetry yields energy conservation, spatial-translation symmetry momentum conservation, and rotational symmetry angular-momentum conservation. In general relativity, however, these global conservation laws hold only where the spacetime possesses the corresponding symmetries; an expanding universe, for example, conserves no global energy in general. What holds generally is local conservation, ∇μTμν = 0, and that local lawful continuity is what CTGP requires. Matter distributions, field configurations, and organizational structures may evolve, or cease to exist in their earlier form, while still participating in lawful transformations constrained by local conservation. On this reading, continuous causal evolution within M(σ) is not arbitrary succession but lawful evolution constrained by invariant structures inherited from spacetime and dynamical symmetries; CTGP does not invoke Noether’s theorem as a proof of its ontology, but as independent physical support for its continuity claims. Conservation laws govern physical quantities, not particular arrangements. The continued conservation of energy, momentum, angular momentum, and other invariant quantities does not imply persistence of earlier physical organizations; conserved quantities remain embedded within continuously evolving physical processes whose specific configurations change through time. CTGP therefore interprets conservation as evidence of lawful continuity across temporal evolution rather than as evidence that earlier states remain ontologically present. This refines item (i) of the conservation/organization/recoverability distinction above: physical existence continues through symmetry-constrained lawful transformation, while the earlier state’s specific organization — item (iii) — is precisely what does not persist. Conservation, continuity, and presentism are accordingly
not in tension: the persistence of causal influence reflects lawful transformation governed by conserved structure, not static preservation of earlier organizations. ========================================================================= =========================== PART II. Formal Framework ========================================================================= =========================== ---------------------------------------------------------------------------------------------------6. Mathematical Framework ---------------------------------------------------------------------------------------------------The continuum formalism of CTGP employs standard Lorentzian geometry (M, g_{μν}) and standard matter fields as an effective low-energy description valid in the regime where classical geometry has emerged (see §10.3). At the fundamental level, CTGP remains agnostic regarding the microscopic realization of its pre-geometric generative structure. The novelty is not in modifying field equations but in restricting ontological commitment to the present state while using M(σ) as a formal representation of generated history. 6.1 The Generated Domain and the Generative Present ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Let Σ_σ denote a spacelike hypersurface in a foliation. The generated domain at stage σ is M(σ) = ⋃_{σ′ < σ} Σ_{σ′} The stages Σ_σ are not chosen freely. In the emergent geometric regime, CTGP identifies them with the level sets of a geometric invariant, the cosmological time function. Definition (Generative present). For each event p, let the cosmological time be τ(p) = sup { L(γ) : γ a past-directed causal curve starting at p }, where L denotes Lorentzian length. Cosmological time is *regular* if τ(p) < ∞ for every p and τ → 0 along every past-inextendible causal curve. CTGP identifies the generative present with the level sets Σ_τ = {τ = const}. The continuum generation parameter is σ = F(τ) for any strictly increasing function F; only the level sets are physical, and the choice of F is a labeling convention (Appendix A.3a). By a theorem of Andersson, Galloway, and Howard (Theorem 17.4), regular cosmological time is a continuous time function whose level sets are Cauchy surfaces; τ is locally Lipschitz, and every event lies on a timelike geodesic from the initial singularity whose length equals τ(p). Wherever τ is differentiable, −∇^μτ is the future-directed unit tangent of that maximizing geodesic, so g^{μν}∂_μτ∂_ντ = −1. Three consequences follow. In FLRW with a big-bang singularity, τ is cosmic time t, and the generative present agrees with the cosmological foliation of §9.3. Where the family of maximizing geodesics develops caustics, as it does inside collapsed structure, τ loses differentiability: its level sets acquire corners but remain Cauchy surfaces, so the present stays
globally well defined through structure formation. At such points the level sets are achronal topological hypersurfaces rather than smooth spacelike ones; since τ is locally Lipschitz it is differentiable almost everywhere, and statements that use the unit normal of the foliation hold wherever it is differentiable. And τ is the maximal single-valued continuation of the unit-norm solution of g^{μν}∂_μσ∂_νσ = −1: before caustics the two coincide, and afterward the smooth-field description breaks down while the present defined by τ does not (§11.4). CTGP is formulated for globally hyperbolic spacetimes. This should be read as an entailment of the framework rather than as a restriction adopted for convenience: since generation is the successive production of Cauchy hypersurfaces, a solution admitting no such foliation is not a spacetime that lacks a present but a solution that could not be a generated history at all. CTGP therefore entails that physically realized spacetime is globally hyperbolic, and classifies Cauchy-horizon and closedtimelike-curve solutions as unrealized idealizations. The entailment carries falsifiable content through what it excludes: a confirmed observation of closed timelike curves, or of any phenomenon requiring a Cauchy-horizon-crossing region to be physically realized, would refute the framework (§9.4.2; §13). Eternalism, by contrast, permits spacetimes containing closed timelike curves and laws that allow them. The two views therefore do not make the same claim about global hyperbolicity, even though both are consistent with the observations to date: CTGP takes a risk that eternalism does not, which is the same asymmetry that underlies the generability constraint (§12.2). Locating the present without the future. Cosmological time is defined on a spacetime (M, g), and the theorem that its level sets are Cauchy surfaces concerns the whole manifold. It might therefore seem that CTGP needs the completed block to say which surface is present. It does not. By definition, τ(p) is the supremum of the lengths of past-directed causal curves from p, so its value at p depends only on the causal past J⁻(p); the regularity conditions are likewise past-directed. Whether an event lies on Σ_σ is therefore fixed by its past alone, and no future region enters. What makes the lengths of those past curves well defined, although the past does not exist, is what grounds every past truth (§5.2): τ is itself a past-tensed magnitude of the present, fixed by the present’s having been preceded by causal histories of those lengths. Global statements about the full manifold belong to the mathematical representation of the history as it would be generated; they show that the construction is well behaved, and they do not ground which slice is present. Clarification on process and representation. Neither Σ_σ nor the union M(σ) should be read as implying that reality consists of a sequence of discrete, ontologically fundamental temporal slices. Σ_σ is a mathematical representation of a continuously evolving physical reality, not a freestanding entity that reality simply “is.” The hypersurfaces function analogously to the instantaneous configurations used in classical mechanics or field theory: useful analytical cross-sections of a continuous process, not independently existing states stacked one against the next. Correspondingly, M(σ) = ⋃_{σ′<σ} Σ_{σ′} is a mathematical reconstruction of generated history and should not be interpreted as an ontological accumulation of independently existing temporal slices; it represents a single continuously evolving causal-temporal process viewed through a foliation-based description, not a growing pile of spacetime sheets. Reality itself is the uninterrupted lawful evolution connecting these representations — the succession of hypersurfaces describes one continuous process of becoming, not a series of separately existing states. Geometric interpretation of the growth hypersurface. In the emergent geometric regime, each hypersurface Σ_σ is defined as a level set of the scalar field σ(x): Σ_σ = {x ∈ M | σ(x) = const}. Because σ admits representation as a scalar field in this regime, this construction is covariant under diffeomorphisms. The hypersurfaces therefore represent a physical ordering of generative
events rather than a coordinate-dependent slicing, and no preferred coordinate frame is introduced; the foliation is fixed by the level sets of cosmological time, of which σ is a relabeling, and involves no additional field dynamics. To ensure causal consistency, the gradient ∇_μσ is required to remain timelike: ∇_μσ ∇^μσ < 0. This condition — guaranteed wherever τ is differentiable, since ∇_μτ is then a unit timelike covector — ensures that the generative ordering corresponds to a physically admissible temporal direction and that Σ_σ is everywhere spacelike. CTGP does not claim that general relativity singles out a unique foliation in every spacetime. It claims that cosmological time, defined from the causal and metric structure alone, supplies a physically distinguished foliation wherever it is regular, and that in our universe this foliation approximately coincides, on large scales, with the rest frame of the cosmic matter distribution. The construction does not presuppose the idealized fundamental observers of standard cosmology: the maximizing geodesics of cosmological time play the role those observers would play, and they are fixed by the geometry wherever cosmological time is regular, however lumpy the matter distribution. This encodes CTGP’s generative temporal asymmetry: future hypersurfaces Σ_{σ′>σ} are excluded from M(σ). Figure 2 illustrates this structure. [FIGURE: Figure 2] Figure 2. The generated domain M(σ). The present edge Σ_σ is the active boundary of generation; earlier stages Σ_{σ′} (σ′ < σ) constitute the formal history M(σ), and future stages are not part of the domain. Lines are schematic level sets of σ, not independently existing slices. 6.2 Causal-Theoretic Foundation of the Generation Parameter ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP introduces a scalar parameter σ intended to index the growth of the generated spacetime domain M(σ). A natural concern is that any such parameter appears ad hoc: why should the ordering of ontological generation be determined by a particular scalar rather than another curvature invariant or entropy measure? A deeper answer emerges from the causal structure of relativistic spacetime itself. The starting point is a result due to Hawking, King, and McCarthy (1976), which suggests that in suitably well-behaved relativistic spacetimes the chronological relation I+(p) — the set of events reachable from p by future-directed timelike curves — determines the conformal structure of the spacetime metric. Once the chronological ordering is known, the topology and light-cone structure can be reconstructed uniquely up to an overall conformal factor. Malament (1977) subsequently strengthened this result by indicating that the causal ordering relation alone suffices to determine the conformal metric structure under mild conditions. The philosophical significance is direct: temporal precedence in relativistic spacetime is not a coordinate artifact but is already encoded in the causal structure itself. CTGP therefore does not introduce σ as an external time parameter imposed upon GR. Rather, σ is understood as a physically selected refinement of the causal ordering already implicit in globally hyperbolic spacetime. Most scalars — Ricci scalar, Weyl invariants, Kretschmann scalar — cannot serve this role because they do not increase monotonically along every future-directed causal curve and do not define Cauchy foliations; they fail the most basic requirement of a temporal ordering parameter. From the causal ordering, a continuous global time function can be constructed via Geroch’s volumetime method. Let J−(p) and J+(p) denote the causal past and future of an event p, and let μ be an
invariant measure on spacetime. Quantities such as t−(p) = μ(J−(p)) increase strictly along every future-directed causal curve in any distinguishing spacetime and, in globally hyperbolic spacetimes, define a continuous time function (Geroch 1970), yielding a continuous causal ordering parameter. This establishes that relativistic spacetime already contains the ordering CTGP needs, independently of any interpretive choice. However, a continuous causal ordering is not yet sufficient for CTGP, which requires smooth hypersurfaces Σ_σ suitable for Cauchy evolution. This refinement is provided by the Bernal–Sánchez theorems. In any globally hyperbolic spacetime (M, g), there exists a smooth temporal function τ whose gradient is everywhere timelike, and whose level sets Σ_τ = const are smooth spacelike Cauchy hypersurfaces, with M ≅ ℝ × Σ globally (Bernal and Sánchez 2003, 2005). The existence of σ as a smooth temporal function whose level sets are spacelike Cauchy hypersurfaces is therefore not an additional physical postulate but a consequence of the causal geometry of globally hyperbolic spacetimes. The logical structure is: causal ordering exists (HKM, Malament) → a continuous time function exists (Geroch) → a smooth Cauchy temporal function exists (Bernal–Sánchez) → CTGP interprets σ as a physically calibrated member of this class. In the pre-geometric regime, Σ_σ denotes a generation stage within the causal ordering induced by σ. Only in the emergent geometric regime does Σ_σ admit representation as a smooth spacelike hypersurface. Two further lines of motivation indicate that the cosmological-time foliation is not an arbitrary representative of this class. First, in FLRW cosmologies the level sets of cosmological time coincide with the rest frame of the matter flow, along which entropy production supplies a natural thermodynamic clock. This is a suggestive coincidence in the symmetric case, not a derivation of σ from entropy production. Second, in a discrete quantum-gravity setting, σ admits a possible realization within causal-set approaches, where growth dynamics suggest a natural analogue of spacetime generation (Rideout and Sorkin, 2000): one possible interpretation is that σ corresponds, in an appropriate continuum approximation, to an averaged measure of generated causal structure. CTGP does not presently provide a rigorous derivation of this correspondence and therefore treats it as a motivating analogy rather than a proven result. Importantly, causal-set approaches represent one possible discrete realization of CTGP’s more general continuous generative-flow ontology; CTGP’s continuum structure is the generation flow ∇_μσ, and discrete models provide compatible but nonfoundational instantiations. These two independent lines of motivation suggest possible deeper foundations for σ, but neither presently constitutes a derivation of the CTGP formalism. 6.3 The Generation Parameter as Cosmological Time ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The causal-structure results of HKM, Malament, Geroch, and Bernal–Sánchez jointly establish three facts: that relativistic spacetime possesses an objective causal ordering not reducible to coordinate choice; that this ordering can be represented by a continuous global time function; and that in globally hyperbolic spacetimes it can be smoothed into a temporal function whose level sets are spacelike Cauchy hypersurfaces. Geometry alone determines the existence of admissible temporal orderings, though not yet their physical interpretation. CTGP resolves that remaining freedom through cosmological time, which is fixed by the causal and metric structure alone. In FLRW its level sets are orthogonal to the comoving matter congruence; beyond FLRW the alignment is approximate (see *Alignment with matter* below). The matter frame is where the present is realized,
not what defines it. These reconstruction results are additionally strengthened by the pre-geometric ontology of §10.3: since causal structure determines most geometric information up to conformal factors, CTGP’s commitment to fundamental causal ordering naturally supports the claim that geometry is emergent from causality rather than prior to it. The HKM and Malament theorems are not merely mathematical scaffolding for CTGP; under the pre-geometric reading, they become expressions of the ontological priority of causal ordering over metric structure. The reconstruction direction — from causal order to geometry, never the reverse — is precisely the direction CTGP’s Causal Reconstruction Principle requires. In the emergent regime the continuum generation parameter is σ = F(τ) with F strictly increasing. Wherever τ is differentiable, ∇_μσ = F′(τ)∇_μτ, and σ satisfies the norm condition ∇_μσ ∇^μσ = −f(σ), f(σ) ≡ F′(F⁻¹(σ))² > 0. The canonical choice F = identity gives σ = τ and f = 1: σ is cosmological proper time, as Postulate 3 states. Other choices relabel the stages without changing them. In FLRW, for example, σ(t) = ∫ρ̄(t′) dt′ labels each stage by the cumulative mean energy density of the background; because ρ̄ depends only on cosmic time, this is a relabeling, not a new physical structure. Alignment with matter. In cosmological spacetimes the maximizing geodesics that define τ are the comoving free-fall worldlines. In FLRW they coincide exactly with the matter congruence, whose fourvelocity u^μ is the timelike eigenvector of T_{μν}, so that ∇_μσ ∝ u_μ. Beyond FLRW, alignment holds with the free-fall congruence of pressureless matter in single-stream regions at linear order. It is not claimed for pressure-supported components — the baryon–photon plasma and radiation do not follow the same geodesics — or within multistream regions of collapsed structure, where τ remains well defined but the matter flow is not single-valued. Admissibility of the generation law. The dependence of the generation rate on σ alone is not a convention. Laws in which the rate responds to local matter density — for instance f = ρ_c², with ρ_c := T_{μν}u^μu^ν the matter-frame energy density — are dynamically inconsistent with alignment and, in the presence of structure, with a timelike foliation (§11). A generation rate may vary with cosmic epoch through F, but not across space at fixed σ. 6.4 Causality as a Fundamental Structural Principle ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A common objection to present-edge ontologies is that quantum mechanics or a future theory of quantum gravity may ultimately require the abandonment of causality. CTGP regards this conclusion as interpretive rather than established. Within CTGP, causality is a structural feature of the generated domain M(σ). Every event contained within the present possesses a causal past that is likewise contained within M(σ). This follows from the causal structure of globally hyperbolic spacetimes (Theorem 17.1) and does not depend on whether the underlying microphysics is deterministic or indeterministic. Current quantum theory does not require the conclusion that causal ordering is absent. Bell
experiments constrain local hidden-variable theories but do not uniquely determine the metaphysical status of causality. Multiple interpretations of quantum mechanics remain compatible with relativistic causal structure and with all presently available observations. CTGP is therefore fully compatible with standard relativistic quantum field theory in its experimentally tested domain. At the quantum-gravity level, approaches that treat causal order as primitive, such as causal set theory, are natural companions (§15.2); CTGP’s smooth σ-field may be read as the continuum description of a deeper causal ordering whose directionality is preserved even where continuum spacetime ceases to apply. CTGP also distinguishes between causal continuity and deterministic predictability. A process may be fundamentally stochastic while still occurring within a lawful causal framework. The absence of a known determining cause for a particular outcome does not imply the absence of prerequisite physical structures, lawful constraints, or causal continuity with prior generated states. CTGP does not require strict determinism; it requires only that present reality emerges from prior generated reality through lawful physical processes, and that no confirmed observation presently requires the existence of causal influence entirely disconnected from physical state evolution. CTGP distinguishes between uncaused outcomes and unconditioned existence. An event may lack a prior determining cause while still requiring a pre-existing physical framework, lawful constraints, and generated domain within which the event occurs. No confirmed observation presently requires the existence of events that occur independently of all prerequisite physical structure. Retrocausality is a speculative interpretive option rather than a settled feature of quantum physics, and it would conflict with CTGP’s postulate that the future is ontologically absent; CTGP does not require it. Singularities — black hole interiors and the Big Bang — mark the breakdown of the smooth continuum description, not the breakdown of causal ordering itself. One assumption deserves explicit statement. CTGP’s primitive ordering is definite: of two causally related stages, one precedes the other. Quantum processes with indefinite causal order, such as the quantum switch (Chiribella et al. 2013), which has been realized in photonic experiments (Procopio et al. 2015), superpose the order in which operations act on a target system. Such processes are described by a single definite evolution of the total system, including the control degree of freedom, and are in that sense compatible with a definite generative ordering at the level of the whole. Whether quantum gravity requires indefinite causal order at the level of spacetime itself is an open question; if it did, CTGP’s primitive ordering would require generalization. 6.5 Timelike Character of the Spacetime Generation Field ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Wherever τ is differentiable, g^{μν}∂_μτ∂_ντ = −1, so ∇_μσ = F′(τ)∇_μτ is everywhere timelike. This is not an additional assumption: it follows from the definition of cosmological time as maximal elapsed proper time. Because a timelike gradient defines a local causal orientation — a preferred arrow at each point consistent with the light-cone structure — the generation parameter behaves as a cosmic time field fixed by the spacetime’s own causal and metric structure rather than by any externally imposed coordinate. Two consequences follow. First, spacetime generation has a locally defined forward direction: the
timelike gradient of σ selects, at every event, which causal direction corresponds to ontological growth, in a fully covariant manner consistent with the causal structure of general relativity. Second, for the canonical choice σ = τ, generation advances at unit rate in the proper time of the generation flow everywhere on each stage; any variation of the rate is a function of cosmic epoch only. The generation rate is therefore not sourced by local matter density, and §11 shows that a density-sourced rate cannot be made consistent with a timelike, matter-aligned foliation. 6.6 The Generation Flow Vector ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The timelike gradient ∇_μσ is a covector. Raising its index and normalizing gives the futuredirected unit vector field that we designate the generation flow vector (signature −+++): n^μ = −∇^μσ / √f(σ) The generation flow vector n^μ represents, at each spacetime event, the local direction along which generation proceeds. It is the analog of the matter four-velocity in cosmological applications: where matter flow aligns with the generation flow (§6.3), n^μ and u^μ coincide. In more general regions, n^μ provides the independently defined generation direction, defined through vacuum regions as well, since cosmological time does not depend on the presence of matter. The generation flow vector is an effective continuum construct defined only after the emergence of Lorentzian geometry and should not be interpreted as a fundamental entity of the pre-geometric regime. The growth edge of reality propagates along this field. More precisely, the present hypersurface Σ_σ advances in the direction of n^μ: each new level set of σ is generated by the dynamics operating along the flow defined by the generation flow vector. This connects the philosophy-of-time concept of “ontological generation” to a covariant structure physicists recognize from fluid dynamics and congruence theory. The generation flow vector is not an independently postulated field; it is fully determined by the spacetime generation field σ and therefore by the causal and metric structure of the spacetime. Because ∇_μσ ∝ ∇_μτ and the normalized flow has vanishing acceleration, the generation flow lines are timelike geodesics (Proposition 11.1). It introduces no new degrees of freedom beyond those already present in the CTGP variational structure. 6.6.1 Evolution of the Generative Field Along the generation flow, dσ/dτ = F′(τ); for σ = τ, σ advances at unit rate along every generator. The propagation equation ∇_μ(λ∇^μσ) = ½λf′(σ), obtained from the action of §9.1, is linear and homogeneous in λ and reduces to a conservation law for the current λ∇^μσ when f is constant: given σ, it transports the multiplier λ along the flow from its value on an initial surface (§6.9). 6.7 Constraint-Preserving Selection of the Present Foliation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Identifying the present with a foliation also requires that each leaf be admissible initial data. This requirement is supplied by the initial-value structure of general relativity. For a spacelike hypersurface Σ_σ to represent a physically admissible present, the induced metric h_{ij}, extrinsic curvature K_{ij}, and matter data on that hypersurface must satisfy the Hamiltonian constraint
R^{(3)} + K² − K_{ij}K^{ij} = 16πGρ and the momentum constraints ∇_j(K^j_i − δ^j_i K) = 8πGj_i. These equations are consistency conditions on any admissible “instant” of the universe; a hypersurface whose induced data fail to satisfy them cannot serve as valid initial data for Einstein–matter evolution. CTGP therefore refines its selection principle: the present hypersurfaces Σ_σ are those members of the admissible temporal foliation for which the induced data satisfy the ADM constraints and whose succession is generated by the Einstein–matter evolution equations without appeal to future boundary conditions. In this sense, σ labels not merely a monotone scalar ordering but a constraintpreserving family of generated presents. The parameter is not introduced as a freely chosen cosmic clock and only afterward tied to matter; rather, it indexes the sequence of hypersurfaces on which the relativistic initial-value problem is well posed. Cosmological time then selects one member of this class without further choice. Because its level sets are Cauchy surfaces of a solution of the Einstein–matter equations, their induced data satisfy the constraints automatically. This three-stage structure — causal ordering, Cauchy foliation, and constraint-preserving selection — grounds σ at three independent levels. The three most common objections are thereby answered: (i) “σ is arbitrary” fails because σ’s existence is guaranteed by GR causal theorems, and cosmological time fixes it uniquely up to relabeling; (ii) “why not any scalar” fails because most scalars are not monotone on all future-directed causal curves and do not define Cauchy foliations; (iii) “growth is gauge” fails because the underlying ordering derives from the causal structure of spacetime itself, not from a choice of coordinates. 6.8 Ontological vs. Dynamical Status of σ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A potential source of confusion about CTGP’s theoretical commitments concerns the status of σ: is it a new dynamical field, a derived scalar, and does it carry independent energy? This subsection addresses these questions explicitly. Under the pre-geometric interpretation developed in §10.3, σ plays three distinct roles that should be explicitly distinguished: (1) as a generation parameter, it labels the sequence of causal stages that constitute the generated history M(σ), a formal representation rather than an accumulating region; (2) as an ordering structure, it induces the causal ordering whose continuum limit admits foliation by generated hypersurfaces; and (3) as a minimal lawful substrate, the primitive generative ordering represented by σ constitutes the minimal lawful structure required for the succession of physical states. In the emergent geometric regime, this primitive ordering admits representation as the σ field, but the field representation is not fundamental. This three-role structure elevates σ beyond a bookkeeping parameter. The deepest ontological commitment of CTGP is not that a fundamental scalar field generates reality, but that primitive generative ordering and its associated causal relations constitute the minimal lawful structure required for the succession of physical states and for the emergence of effective physical laws. At the level of continuum dynamics, σ is fixed by the geometry: cosmological time is a functional of the metric, so σ carries no initial data of its own. The action of §9.1 encodes σ through a Lagrange multiplier λ enforcing the norm condition ∇_μσ ∇^μσ = −f(σ). The multiplier sector is not inert in general: as in mimetic gravity, λ contributes a pressureless, dust-like stress-energy whose amplitude is set by initial data (§6.9). With λ = 0 on an initial surface, it vanishes identically
and the Einstein equations are unmodified. This is the configuration CTGP adopts, because the Layer 2 present is defined by cosmological time without reference to λ. This status is precisely analogous to several well-understood scalars in GR and thermodynamics. Proper time τ along a worldline is a derived scalar fixed by the metric and the worldline’s tangent vector; it introduces no new degrees of freedom and is entirely determined by the existing dynamical content of GR. Entropy production scalars in relativistic thermodynamics are similarly secondary: they are constructed from the stress-energy tensor and thermodynamic state variables, track the causal direction of physical processes, and do not add new fields to the theory. York time in canonical gravity — the trace of the extrinsic curvature K, used as an intrinsic time variable in certain foliations — provides perhaps the closest structural analog: it is extracted from the existing ADM variables, increases monotonically in expanding cosmologies, and defines a foliation without being an independent physical degree of freedom. σ plays an exactly analogous role: it is a physically calibrated member of the class of smooth Cauchy temporal functions guaranteed by the Bernal–Sánchez theorems, selected by maximal elapsed proper time rather than stipulated as an independent field. The concern that σ “secretly introduces a new field that modifies GR” therefore misreads the formalism. A new field modifies GR by adding independent degrees of freedom whose dynamics can diverge from those of the Einstein–matter system; scalar–tensor theories (Brans–Dicke), f(R) gravity, and quintessence models all do this. σ does not: it is fixed by the metric through cosmological time, and with λ = 0 the constraint sector carries no stress-energy. CTGP’s ontological claim — that σ indexes genuine spacetime generation — is interpretive, not dynamical. The physics is unmodified; the ontological reading of the physics is what CTGP proposes. In summary, with λ = 0 (§6.9) the continuum generation parameter introduces no new propagating degrees of freedom and modifies no gravitational dynamics. Physical hypotheses that would give σ dynamics of its own — density-dependent generation rates and soft completions of the norm condition — are analyzed in §11; none survives the consistency requirements while producing an observable effect. Covariance and compatibility with General Relativity. All quantities introduced in the CTGP framework are constructed from the metric, its causal structure, and scalar invariants. Consequently, the theory remains invariant under arbitrary spacetime diffeomorphisms. The Einstein field equations G_{μν} = 8πG T_{μν} are unchanged (§9.1), and the generative dynamics operate only at the level of the ontological realization of spacetime events rather than modifying the local gravitational field equations. In this sense CTGP extends the interpretation of spacetime evolution without altering the empirically verified structure of General Relativity. Within the emergent geometric regime, σ is represented by cosmological time, a geometric scalar, and therefore transforms covariantly under diffeomorphisms. Fundamentally, however, σ is interpreted as a pregeometric generative ordering parameter rather than as a scalar field defined on a pre-existing manifold. This covariance is guaranteed by construction and does not require additional assumptions. 6.9 Status of the Lagrange Multiplier λ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The action of §9.1 contains a Lagrange multiplier λ that enforces the norm condition ∇_μσ ∇^μσ = −f(σ).
Variation with respect to σ gives the propagation equation ∇_μ(λ ∇^μσ) = ½ λ f′(σ), which is linear and homogeneous in λ and reduces to a conservation law for the current λ∇^μσ when f is constant. Given σ, it transports λ along the generation flow from its value on an initial surface. The multiplier is physically meaningful. Variation with respect to the metric shows that, on the constraint surface, λ contributes the stress-energy T^{(σ)}_{μν} = 2λ∇_μσ∇_νσ (Proposition 17.7): a pressureless fluid moving along the generation flow, with energy density proportional to λf. This is the structure of mimetic gravity (Chamseddine and Mukhanov 2013), in which such a component mimics cold dark matter; like any pressureless fluid, it develops caustics where its flow lines cross. CTGP sets λ = 0 on an initial surface. The propagation equation then keeps λ = 0 everywhere, the constraint sector contributes no stress-energy, and the Einstein–matter system is exactly that of general relativity. This choice reflects the division of labor in the framework: the Layer 2 present is defined by cosmological time (§6.1) and needs no dynamical support from λ. A nonzero λ would be a Layer 3 hypothesis — a dark-matter-like component carried by the generation flow — degenerate at the background level with ordinary cold dark matter and subject to the caustic problem discussed in §11.4. CTGP therefore adopts none of the phenomenology associated with mimetic gravity: no dustlike component, no modification of the expansion history, and no new propagating degree of freedom. The resemblance is structural only, and it is confined to the form of the constraint. 6.10 Non-Arbitrariness of the Generative Present ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Theorem (Uniqueness of the Cosmological-Time Foliation). Let (M, g) be a spacetime with regular cosmological time τ. Then the foliation by the level sets of τ is uniquely determined: its leaves are the level sets of τ, which are Cauchy surfaces (Theorem 17.4), and every admissible generation parameter has the form σ = F(τ) with F strictly increasing. No choice of coordinates, observers, or matter calibration enters, because τ is defined from the causal and metric structure alone and is preserved by isometries (Proposition 17.8). The “arbitrary scalar” objection therefore does not apply. Of the many Cauchy time functions whose existence the Bernal–Sánchez theorems guarantee, the generative present is the unique one defined by maximal elapsed proper time, up to relabeling of its values. The logical chain is: causal geometry → cosmological time → a distinguished foliation → CTGP’s interpretation of that foliation as ontological co-presence. The first three links are results of general relativity; the last is CTGP’s interpretive thesis, not a theorem. Geometric naturalness does not by itself confer ontological privilege: the geometry makes the identification non-arbitrary, sparing the presentist a choice among admissible foliations by fiat, but the identification itself is a primitive posit of the ontology and one of its costs (§16). The theorem’s scope is set by its hypothesis. The present formulation is a big-bang-origin formulation: regular cosmological time requires a past boundary at finite proper time, and past-eternal, bouncing, cyclic, and emergent cosmologies lie outside its scope. §6.10.1 extends the construction to universes with a single uniform bounce; the remaining
cases would require a further generalization. This limitation concerns the continuum representative, not the ontology. Cosmological time is CTGP’s realization of the primitive generative ordering in the emergent geometric regime (§10.3), not the ultimate source of that ordering; a generalized construction would replace it while leaving the ordering itself intact. 6.10.1 Extension Through a Bounce Cosmological time fails in a bouncing universe without a beginning for a precise reason. (A bounce whose contracting phase itself began at an earlier singularity still has regular cosmological time, measured from that singularity.) If the contracting phase extends indefinitely into the past, causal curves continue through the bounce instead of ending at a past boundary, so τ is infinite everywhere and both regularity conditions of Theorem 17.4 fail. The construction can nevertheless be extended when the spacetime supplies a distinguished Cauchy surface to play the role of the initial singularity. Definition (Surface-relative time). Let S be a spacelike Cauchy surface. For an event p, let τ_S(p) = ± sup { L(γ) : γ a causal curve between S and p }, τ_S = 0 on S, with the positive sign when p lies to the future of S and the negative sign when it lies to the past. The sign is fixed by the spacetime’s time orientation, not by an additional convention. On each side of S, τ_S is cosmological time with S in place of the initial singularity. The one-sided analogues of the regularity conditions of Theorem 17.4 — τ_S finite, and τ_S → 0 along every causal curve approaching S — are not additional physical assumptions: in a globally hyperbolic spacetime they follow from S being a Cauchy surface, since every past-inextendible causal curve on the future side reaches S and the region between S and any event is compact. The argument of Andersson, Galloway, and Howard therefore carries over on each side, and the level sets of τ_S are Cauchy surfaces (Proposition 17.23). The settled case. In an exact FLRW universe with a single bounce, the surface at the moment of the bounce — where the scale factor stops decreasing and begins increasing, ȧ = 0 — has zero mean curvature, is unique, and is a Cauchy surface. Taking S to be this surface, τ_S is cosmic time measured from the bounce, and the generative present is well defined through the bounce with no averaging prescription. The stages before the bounce are simply earlier stages, generated and ceased like any past stage. Such a universe has no first stage; CTGP’s ontology and its tense semantics (Definition 17.20) do not require the sequence of past stages to be finite. Scope. Three limits fix the scope of the extension. First, S is an input rather than an output: the construction presupposes a distinguished Cauchy surface instead of deriving one from the causal structure alone, as cosmological time does in the big-bang case. Second, in a non-uniform bounce the regions of zero mean curvature need not join into a single surface, and whether a unique maximal Cauchy surface exists is open; the existence and uniqueness theorems for such surfaces assume conditions — compact Cauchy surfaces, symmetries, or energy conditions — that a bounce typically violates. Third, a cyclic universe has many surfaces of zero mean curvature, and an emergent universe with an Einstein-static past has zero mean curvature on every slice, so neither supplies a distinguished S. These cases remain outside the present continuum representative. As before, the limitation concerns the representative, not the ontology: replacing τ by τ_S where S exists is a change of representative, not of the Layer 2 claim.
6.11 Admissible Generation Laws ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The kinematic structure of CTGP — a Cauchy foliation by the level sets of cosmological time, aligned with matter in cosmology — rests on established results: the causal-structure theorems of Appendix B, the cosmological time function, and the observed CMB rest frame. The generation law is the one element that could in principle carry new physics: the function f that fixes how fast σ advances. This freedom is narrower than it appears. Any generation law in which f carries spatial matter dependence gives the generation flow an acceleration away from free fall, a_μ = −D_μ ln √f (Proposition 11.1). That acceleration destroys alignment with matter and, in the presence of structure, the timelike character of the foliation; stable soft completions fare no better (§11.3). Within the local, stable classes analyzed in §11, the viable generation laws are those of the form f = f(σ), equivalent to a relabeling of cosmological time: the generation rate may vary with cosmic epoch but not across space. The choice of generation law is therefore not confined to Layer 3: it is constrained by the Layer 2 requirement that the present be a well-defined Cauchy surface aligned with matter. 6.12 Succession of Hypersurfaces ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The ontological dynamic is expressed by the succession relation. In the emergent geometric regime, this takes the form of successive spacelike hypersurfaces; in the pre-geometric regime, it should be understood as successive generative stages of the causal ordering: Σ_σ → Σ_{σ+Δσ} Each transition is governed by GR's evolution equations applied to data on Σ_σ without reference to any data at σ′ > σ. This succession produces the 'lived flow' of time as a real unfolding, not a representational one. [FIGURE: Figure 3] Figure 3. Hypersurface generation: successive level sets Σ_σ → Σ_{σ+Δσ} produced by lawful evolution of data on the present edge. The lapse of the cosmological foliation is constant on each level set, so equal steps in σ correspond to equal proper time along the maximizing geodesics; systems moving relative to them, including clocks at rest in gravitational wells, accumulate less. ---------------------------------------------------------------------------------------------------7. General Relativity as Initial-Value Generation ---------------------------------------------------------------------------------------------------Einstein's equations can be written as a well-posed initial-value problem (under standard conditions). Given Cauchy data on Σ_σ, the evolution equations determine a unique development — up to diffeomorphism — in a neighborhood of Σ_σ, without specifying future boundary data. CTGP interprets this structure as ontological generation: the next hypersurface is not pre-existing but is computed from the present one by lawful dynamics. G_{μν} = 8πG T_{μν}
The Einstein equation governs the generation of each new slice. On the CTGP reading, this equation is not a constraint on a pre-existing four-dimensional structure but a production rule: given Σ_σ, it tells us what Σ_{σ+Δσ} is. The hyperbolic character of the equations — their dependence on initial data rather than boundary data in the future — supports this reading structurally. Becoming as primitive. Generation is not an ordinary relation between two coexisting states. Predecessor and successor are never present together, so there is no stage at which both exist and one produces the other. On CTGP, generation is primitive and tensed: the present has come to be from what was present. The division of labor is exact. Cosmological time supplies the ordering and copresence of stages (§6.1), and the initial-value formulation supplies the lawful connection between successive stages. Neither supplies becoming itself, which is the posit that CTGP adds to both. [FIGURE: Figure 4] Figure 4. Minkowski spacetime diagram with the light cone of an event p and simultaneity slices of two inertial frames, illustrating the causal structure (I⁺, I⁻) underlying CTGP’s notion of generation. ---------------------------------------------------------------------------------------------------8. ADM Decomposition and Hypersurface Evolution ---------------------------------------------------------------------------------------------------In ADM form, the spacetime metric is decomposed into lapse N, shift N^i, and the induced 3-metric h_{ij} on Σ_t. The extrinsic curvature K_{ij} encodes how Σ_t is embedded in the ambient spacetime: g_{μν} → (h_{ij}, N, N^i) K_{ij} = (1/2N)(∂_t h_{ij} − ∇_i N_j − ∇_j N_i) The evolution equations and constraints provide a natural slice-to-slice evolution picture. CTGP maps this to an ontological claim: Σ_σ is literally produced from Σ_{σ−Δσ} by lawful dynamics, without reference to a pre-existing future slice. The ADM Hamiltonian and momentum constraints ensure that the generated data on each new Σ_{σ+Δσ} are consistent with the field equations. These constraints are not boundary conditions from the future but internal consistency requirements on the present slice, further supporting the generative interpretation. ---------------------------------------------------------------------------------------------------9. Action Principle on Generated Domains and Cosmological Foliation ---------------------------------------------------------------------------------------------------9.1 Action Principle ~~~~~~~~~~~~~~~~~~~~ The action principle for CTGP restricts integration to the generated domain M(σ). Within this action principle, the kinematic equation governing σ is obtained as an Euler–Lagrange condition rather than stipulated. The CTGP framework is formulated by requiring the physical generative history to satisfy the stationary-action condition δS = 0, where the action is taken to be:
S(σ) = ∫_{M(σ)} √(−g) [ L_{GR} + λ(∇_μσ ∇^μσ + f(σ)) ] d⁴x where L_{GR} = (R − 2Λ)/16πG + L_{matter} is the Einstein–Hilbert Lagrangian density with matter, λ is a Lagrange multiplier field, f(σ) > 0 is the generation law (f = 1 for the canonical choice σ = τ), and √(−g) d⁴x is the invariant volume element. The integration is restricted to M(σ), ensuring no future teleology. The term λ(∇_μσ ∇^μσ + f(σ)) enforces the norm condition on σ. Variation with respect to the fields yields the constraint and propagation relations below. This action and all subsequent variational results are valid within the emergent geometric regime (ρ ≪ ρ_P), where the smooth Lorentzian manifold (M, g_{μν}) is a valid continuum approximation. The pre-classical regime is addressed in §9.2 and §10.2; the extended action S_{ext}(σ) that models the crossover via the suppression function α(ρ) is developed in §9.2. Varying S(σ) with respect to λ yields the Euler–Lagrange condition: δS/δλ = 0 ⟹ ∇_μσ ∇^μσ + f(σ) = 0 That is, ∇_μσ ∇^μσ = −f(σ), the norm condition of §6.3, arising here as a necessary condition for stationarity of S(σ). Varying with respect to σ yields the propagation equation ∇_μ(λ∇^μσ) = ½λf′(σ), which transports λ along the generation flow. Varying with respect to g^{μν} recovers G_{μν} = 8πG T_{μν} together with the λ-dependent term of Proposition 17.7, which vanishes identically for the configuration λ = 0 adopted in §6.9. The variational grounding answers the question of why σ should obey any particular equation at all: σ obeys ∇_μσ ∇^μσ = −f(σ) for the same reason that geodesics obey the geodesic equation — because it is what the variational principle demands. The formal statement is given in Proposition 17.7 (§17). The restriction of the integration domain to the generated manifold M(σ) is the central ontological commitment of the framework — not merely a heuristic convenience. In the standard block universe interpretation, the action is implicitly integrated over the entire four-dimensional manifold, treating all events as equally real elements of the variational principle. By explicitly limiting the action to M(σ), CTGP asserts that future spacetime regions do not contribute to the dynamical definition of the theory at stage σ. The future is not simply unknown; it is not part of the domain of the physical laws that generate the present. This transforms the interpretation from a passive description of a pre-existing structure to an active generation of new structure from existing structure. The Euler-Lagrange equations derived from this action are therefore not constraints on a static block, but rules for constructing the next increment of reality from the existing one. The domain of the action is itself a growing entity, and the field equations are defined only over that domain. Because Euler–Lagrange equations are local, this restriction does not alter the field equations at any point of M(σ); the distinction it draws concerns the domain over which the laws are defined, not their local form. A completed block would serve as the domain for an action defined over the whole of M; the CTGP action has no such domain. This distinction is formalized in Theorem 17.9 (§17). 9.2 Domain of Validity and the Extended Action ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The effective action S(σ) of §9.1 is valid within a precisely specified domain: the emergent geometric regime, where ρ ≪ ρ_P (the Planck density), and the smooth Lorentzian manifold (M, g_{μν}) is a valid continuum approximation. Outside this domain — specifically in the pre-geometric regime where ρ ∼ ρ_P — the continuum integral, the metric volume element √(−g) d⁴x, and the Ricci scalar R are themselves emergent structures that cannot be taken as primitive inputs to the variational principle. The two regimes and their respective formal machinery are distinguished as follows: • Pre-geometric regime (ρ ∼ ρ_P): σ is primitive; no metric assumed; no continuum action defined; causal ordering is the operative structure. • Emergent geometric regime (ρ ≪ ρ_P): σ represented by cosmological time; metric available; S(σ) applies. The transition between regimes is not a discontinuity but a crossover parameterized by the dimensionless ratio ρ/ρ_P. This crossover can be represented in the effective action by introducing a density-dependent suppression factor α(ρ) on the GR Lagrangian: S_{ext}(σ) = ∫_{M(σ)} √(−g) [ α(ρ) L_{GR} + λ(∇_μσ ∇^μσ + f(σ)) ] d⁴x α(ρ) = 1 / (1 + (ρ/ρ_P)^n), n ≥ 1 The suppression function α(ρ) satisfies: α → 1 as ρ/ρ_P → 0 (low-energy limit, GR fully restored), and α → 0 as ρ/ρ_P → ∞ (Planck regime, GR suppressed). The σ-constraint sector λ(∇_μσ ∇^μσ + f(σ)) is retained at all densities, reflecting the fact that the generative ordering structure of σ is active across both regimes — it is never suppressed, because it is the structure from which the geometric regime itself emerges. The exponent n controls the sharpness of the transition; n = 2 gives a smooth crossover centered at ρ = ρ_P. An important ontological clarification is required by S_{ext}(σ): the metric g_{μν} still appears in the extended action through √(−g) and the σ-constraint sector. This creates a choice between two interpretations of the metric’s status in the pre-classical regime. Under the emergent-metric reading (consistent with §10.3), g_{μν} in S_{ext}(σ) is to be understood as a continuum approximation that becomes progressively less valid near ρ_P; the action itself is correspondingly an effective field theory expression whose UV completion lies in the pre-geometric structure. Under the constrained-metric reading, g_{μν} remains a fundamental but dynamically unconstrained quantity whose constraints emerge alongside GR as α(ρ) → 1. CTGP adopts the emergent-metric reading as primary (Emergent Metric Principle, §10.3), treating S_{ext}(σ) as valid only where the continuum approximation holds. The extended action therefore serves a specific formal function: it models the crossover continuously rather than imposing a sharp regime boundary, and it identifies α(ρ) as the natural emergence function for GR within the CTGP framework. S_{ext}(σ) is the candidate effective action for the crossover regime; its pre-geometric limit is not fully specified by the continuum formalism alone, and remains subject to the Scope of the Research Program principle of §10.4. 9.3 Cosmological Foliation and the CMB Frame ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In homogeneous, isotropic cosmology, the FLRW metric provides a preferred foliation by cosmic time
t: ds² = −dt² + a(t)² dΩ_k² This does not violate local Lorentz invariance; rather, it reflects a physical symmetry of our universe's matter distribution. The CMB frame singles out a preferred rest frame not as a kinematic artifact but as a physical fact about the distribution of matter and energy. CTGP's global present Σ_σ is defined by cosmological time, and on large scales it coincides with this frame, providing a non-absolute but physically grounded global 'now.' Different observers need not synchronize their local frames; they share the cosmological foliation without needing operational access to it. [FIGURE: Figure 5] Figure 5. FLRW foliation: surfaces of constant cosmic time t, with comoving worldlines diverging as the scale factor a(t) grows. The comoving worldlines are the maximizing geodesics that define cosmological time and provide the physical basis for CTGP’s global present. 9.3.1 Emergent Cosmological Foliation A clarification bears on one of the most persistent objections: the charge that CTGP introduces a preferred frame. The correct response is that CTGP does not introduce a preferred frame — it identifies the cosmological foliation as an emergent physical structure generated by the large-scale matter distribution. The logical chain that establishes this is as follows: Einstein equations are foliation-neutral → the actual matter distribution fixes the metric → the metric fixes cosmological time τ → on large scales, the level sets of τ coincide with the CMB rest frame CTGP does not use the cosmological foliation as an external input. Rather, CTGP identifies the cosmological foliation as an emergent physical structure generated by the large-scale matter distribution. This framing aligns CTGP with spontaneous symmetry breaking, a mechanism physicists already accept throughout physics: a rotationally symmetric Lagrangian can produce a ground state that breaks rotational symmetry, with the symmetry-breaking direction selected by the physical configuration rather than by the Lagrangian itself. The CMB rest frame is precisely analogous: the Einstein equations are foliation-neutral, but the physical solution — a universe filled with matter in a particular configuration — fixes a metric whose cosmological time is, on large scales, the CMB rest frame. 9.4 Cosmological Simultaneity and Relativistic Locality ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP distinguishes three senses of “present” that are often conflated in objections to the framework. Local physical present: what an observer experiences at their worldline. This is the phenomenological present, dependent on local physical processes and subject to the usual relativistic effects of time dilation and gravitational redshift.
Cosmological present: the foliation naturally selected by the large-scale matter distribution of the universe, approximated observationally by the CMB rest frame. This is a physically motivated but not observer-dependent quantity: any observer can in principle determine the CMB frame and locate themselves within it. Ontological present: the boundary between the generated domain M(σ) and the ungenerated future. This is the active edge at which new hypersurfaces are produced, parameterized by σ. CTGP’s present edge Σ_σ is identified with the cosmological present rather than with Newtonian absolute simultaneity or with any particular observer’s local present. Consequently, CTGP preserves local Lorentz invariance while maintaining a physically motivated global generation parameter. The resulting “universal tick” is therefore not a universal clock visible identically to every observer but a cosmologically grounded ordering relation associated with the generation of new hypersurfaces. Much confusion disappears once these three senses of present are kept distinct: local simultaneity is observer-dependent and Lorentz-relative; cosmological simultaneity is physically selected by the matter distribution; ontological simultaneity concerns what has been generated and what has not. 9.4.1 Co-presence, Cosmic Time, and Synchronization Co-presence is not a relation added to existence but a consequence of it. Since only the present exists — earlier stages have ceased to exist and later ones are ungenerated — everything that exists belongs to a single generative stage. This is why CTGP requires a global foliation: existence cannot be frame-relative, so there must be one fact about which events constitute the present. Cosmological time specifies which surfaces these are (§6.1). Membership in Σ_σ is co-presence, and because Σ_σ is a level set of the generation parameter, copresence is equality of σ: two co-present events occur at the same cosmological generation stage because they carry the same value of σ. This is not a correlation between two facts but a single fact stated twice. In the emergent geometric regime, where σ is a monotone relabeling of cosmological time (Postulate 3), this is what “same cosmic time” refers to; prior to geometric emergence σ remains an ordering parameter and the equality still holds, without presupposing proper time. What does not follow from equality of σ is equality of proper time, which differs between worldlines under relative motion and differing gravitational conditions; synchronization in any operational sense, since co-present events may exchange no timing signals and need not be in causal contact; or sameness of rate, since systems undergoing radically different physical processes are generated within the same Σ_σ. The point is clearest for spacelike-separated events. Consider a person on Earth raising an arm and, billions of light years away, another person raising an arm. Neither event lies in the other’s past light cone; neither can influence the other, and in an expanding universe the two may never enter causal contact at all. CTGP nonetheless holds that there is a fact of the matter as to whether the two events belong to the same Σ_σ, and that this fact is fixed by the cosmological foliation rather than by any observer’s frame. The two movements are unrelated, uncoordinated, and unsynchronized. What they share is the same cosmological generation parameter σ, and therefore the same generative stage — and that equality carries no implication of shared proper time, shared rate, or any coordination whatever between them. The non-synchrony is quantitative and already carried by the formalism. The norm condition of §6.3 fixes the lapse of the cosmological foliation: N = f(σ)^{−1/2}, which is constant on each level set
and equals 1 for the canonical labeling σ = τ. The observers normal to Σ_σ are therefore the maximizing geodesics themselves, and a system moving with speed v relative to them accumulates proper time dτ_local = N√(1 − v²/c²) dσ. Gravitational time dilation enters through the same factor, since a clock held at rest in a gravitational well is not in free fall and moves relative to the geodesics passing through it. Systems co-present at Σ_σ therefore accumulate different amounts of proper time between Σ_σ and Σ_{σ+dσ}. Both arm-raisers are co-present; neither is a clock the other could read, and the local simultaneity surface of each is in general tilted relative to Σ_σ. CTGP therefore does not claim that either observer’s own present is the privileged one. It claims that the generative ordering is settled by a foliation that neither of them occupies. Both facts hold at once, and neither displaces the other. The value co-present events share is itself a proper time: σ is a monotone relabeling of cosmological time, the maximal proper time accumulated since the initial singularity, which in cosmology is the proper time of the comoving matter congruence. Each co-present system also carries its own proper time, which in general differs from it — a system with peculiar velocity accumulates less between Σ_σ and Σ_{σ+dσ} than a matterframe worldline does, and a clock at rest in a gravitational well accumulates less for the same reason. The two arm-raisers therefore share one cosmic proper time and carry non-identical local times simultaneously. This introduces no second kind of time. There is one quantity, proper time, evaluated along different worldlines; what distinguishes the cosmic value is the congruence along which it is measured and the fact that its level sets foliate the spacetime, not that it is a different sort of thing. This is why the matter-frame congruence enters as a realization rather than as a criterion of membership. Cosmological time is attained along the maximizing geodesics, which in cosmology are the comoving worldlines. Systems carrying peculiar velocity — essentially all systems, including the Earth — do not lie on those worldlines, and are co-present nonetheless. The construction selects the parameter; it does not impose a shared clock on the events the parameter orders. This also settles what would otherwise be an awkward case. Suppose a physical system underwent no change of any kind between two successive stages — no alteration of internal or spatial state. CTGP still distinguishes Σ_σ from Σ_{σ+dσ} as a progression in cosmological generation. Participation in the advancing present is not conditional on possessing a changing state variable: under Postulate 2, persisting at all is being generated at successive stages, and an unchanging configuration is one in which successive stages produce the same state rather than one exempt from generation. In brief. CTGP's simultaneity isn't Newton's absolute 'cosmic now.' Newton's time is imposed from outside and flows identically everywhere, so every clock in the universe ticks at the same rate and agrees on the same reading. CTGP rejects that. In CTGP, two events are co-present when they belong to the same generative stage of reality, the same step in the present's unfolding. The dividing line isn't imposed from outside. It's fixed by the geometry the universe's own matter produces, and on large scales it's the frame in which the cosmic microwave background looks uniform. Co-present events don't share proper time, clock rates, or any synchronization, and relativity's local effects like time dilation stay fully intact. Picture two people raising their arms billions of light years apart. They can never signal each other, and their clocks don't agree, but they still belong to the same stage of the present. Think of a computer’s shared system time, with each program showing its own timer — except that no clock anywhere keeps the present; the universe’s own structure does.
9.4.2 Why There Is Always a Present Stated without formalism, CTGP’s core claim is one that ordinary experience already assumes: there is a way things currently are. Not a privileged coordinate system, not a universal clock reading, not a synchronization procedure — simply that reality has a current state, which is succeeded by another. On this framework that assumption is correct and is what Σ_σ formalizes. It is worth noting which side of this dispute is the revisionary one. The block universe does not merely add a claim to ordinary understanding; it denies one, holding that “the way things currently are” picks out nothing ontologically distinguished. CTGP’s burden is to show that the formalism can carry the ordinary assumption, not to motivate the assumption itself. The present also need not be postulated as an additional structure laid over spacetime. On this framework it is constituted. At any generative stage, every physical state in the universe is undergoing the transition from what it is to what it becomes next. This is CTGP’s ontological premise and is not established by general relativity, which describes evolution without asserting that later states are brought into existence; what follows here is what that premise yields, not a derivation of it. Take the totality of those transitions — one per state, across the whole universe — and that totality is the cosmic present. Σ_σ is not a surface selected from among many and then declared privileged; it is the collection of everything currently being generated. Once the generative ontology is adopted, the universe-wide present is therefore not an additional structure appended to the physical dynamics; it is constituted by the totality of physically ongoing state transitions. The posit is the generative ontology itself, not a privileged surface added on top of it. A system undergoing no change is included on the same terms, its transition being to the same state (§9.4.1). Two things must be kept apart here, because the constitutive account establishes less than it may appear to. It establishes that there is a present: the totality of transitions is well defined whether or not any observer selects a slicing. It does not by itself establish either that those transitions glue into a single spacelike hypersurface or that the resulting slicing is unique. The first is secured elsewhere in the formalism. The Hamiltonian and momentum constraints are precisely the conditions under which a local generative step is consistent with the steps taken in neighbouring regions, so the transitions cohere into a surface rather than fragmenting into unrelated local advances (§6.7; Appendix A.1). The second is not secured by the transitions at all. ADM evolution admits many-fingered time: different choices of lapse and shift foliate the same spacetime differently while describing identical physics. What selects one foliation among these is cosmological time (§6.1), the maximal proper time elapsed since the initial singularity. The constitutive argument therefore answers why there is a cosmic present; cosmological time answers which one it is. Presenting either as discharging both burdens would overstate the case. The scope of the claim should nevertheless be stated precisely, because it is easy to overstate. CTGP does not hold that every mathematical solution of the Einstein field equations possesses a unique globally privileged present. That would be false: Gödel’s rotating dust solution, Taub-NUT, and the region of the Kerr interior beyond the Cauchy horizon admit no global Cauchy foliation, and no time function of the required kind exists on them. The solution space of GR is a space of mathematical objects, most of which are not realized. What CTGP holds is that physically realized spacetime — the actual history that is generated — possesses such a present, and that the solutions lacking one are for that reason not candidate histories. The direction of the argument matters. Under the Causal Reconstruction Principle (§10.3), causal
ordering is primitive and metric geometry is reconstructed from it; generation is therefore prior to spacetime structure rather than a feature added to a spacetime already given. A solution with closed timelike curves is not a generated history whose foliation happens to fail. It is a geometry that no sequence of generative stages could produce, since producing it would require an event to be among the conditions of its own production. The restriction to globally hyperbolic spacetimes is thus not a boundary drawn around the theory to protect it; it is what the theory says about which geometries can be real. A confirmed observation of closed timelike curves, or of any phenomenon requiring a Cauchy-horizon-crossing region to be physically realized, would falsify the framework (§13). This also disposes of a question sometimes raised as though it bore on the matter: whether current physics exhibits any non-abstract existence that is absolutely unchanged from one instant to the next, and whether such absolute stasis has been shown impossible. Neither has been established, and neither needs to be. An absence of known examples together with an absence of a proof of impossibility is evidentially neutral and supports nothing. Under CTGP the question does not arise in the first place, for the reason given in §9.4.1: persisting is being generated. An unchanging configuration is one in which successive stages produce the same state, not one standing outside the succession. Absolute stasis is not a rival to the generative present; it is not a describable condition within it. 9.5 Foliation in Strong-Curvature Regimes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The FLRW foliation is well-defined in the cosmologically relevant regime of large-scale homogeneity and isotropy. However, a global Cauchy foliation is not guaranteed in all spacetimes. Pathological topologies, strong anisotropy, closed timelike curves (CTCs), and black hole interiors may each undermine the existence of a smooth global foliation. CTGP addresses these cases via a three-option framework: Option (1): Restrict CTGP's domain of application to cosmological-scale regions where the FLRW approximation holds and a global Cauchy foliation exists. This is the conservative stance and is the primary intended domain of the framework. On cosmological scales, this restriction is well-motivated by the empirical fact of CMB isotropy. Option (2): Use local foliation patches with gluing rules. In regions where the global foliation breaks down, define Σ_σ patchwise over overlapping open sets, with consistency conditions on patch boundaries. This extends CTGP's applicability at the cost of introducing gluing ambiguities that must be handled by additional physical data (e.g., the stress-energy distribution at the boundary). Option (3): Treat black hole interiors and CTC-permitting solutions as requiring separate analysis or as lying outside CTGP's intended domain. The interior of a black hole beyond the Cauchy horizon may be physically inaccessible and dynamically isolated from the exterior universe; CTGP's generative structure applies to the exterior region, and the interior is flagged as a domain requiring separate treatment in any future extension. CTC-permitting solutions are handled under Option (1) by restricting to globally hyperbolic sectors: CTGP treats CTC-permitting solutions as physical idealizations absent from our universe's actual global structure. Each option can be substantiated by existing results in GR. For Option (1), the empirical and
theoretical case is strong: global hyperbolicity is a standard causal assumption of relativistic cosmology (Hawking and Ellis 1973); strong cosmic censorship, the still-unproven conjecture that generic physically reasonable spacetimes admit no extension beyond their maximal globally hyperbolic development, would make it a consequence of the dynamics rather than an assumption; and nothing in the observed large-scale structure of our universe indicates its failure; CMB isotropy to one part in 10^5 confirms the FLRW approximation at cosmological scales to high precision. CTGP’s restriction to this domain is therefore a principled constraint, not a limitation, in the same sense that thermodynamics restricts its domain to systems with well-defined state variables. For Option (2), dual foliation methods developed in mathematical GR suggest that overlapping foliation patches can be glued consistently wherever the constraint data on the overlaps can be reconciled; patch inconsistencies are therefore diagnosable and resolvable by standard constraint-solving methods (Theorem 17.15). For Option (3), the relevant physical fact is that the spacetimes known to require it — Kerr interiors, Taub-NUT, and Gödel universes — are not consistent with our observed cosmological boundary conditions. Kerr interiors are idealized; real astrophysical black holes formed by collapse develop apparent horizons whose exterior evolution remains well-posed and globally hyperbolic. CTGP’s generative structure continues without interruption in the exterior. The instability of the Cauchy horizon to perturbations (mass inflation) further suggests that the Kerr interior beyond the inner horizon is physically unrealized. These three anchors ground the threeoption framework in established GR results. 9.6 Emergent but Law-Selecting Structures ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP belongs to a broader class of physical frameworks in which large-scale emergent structures activate effective physical constraints. This framing positions CTGP within emergent-law physics rather than metaphysics. The key idea is that certain large-scale physical structures — while not themselves fundamental — nonetheless select which effective laws govern the system. Consider the following canonical examples: Emergent structure | Effective law selected ---------------------------------+------------------------------Fluid phase | Navier–Stokes equations Superconductivity (Cooper pairs) | London equations Cosmic fluid (FLRW) | FLRW foliation and cosmic time In each case, a lower-level theory (classical mechanics, quantum electrodynamics, general relativity) is formally foliation-neutral or medium-neutral. But the actual physical configuration — the fluid phase, the superconducting condensate, the cosmic matter distribution — selects an effective structure that constrains physical law at the relevant scale. CTGP applies the same logic to the cosmological foliation. The matter distribution of the universe may produce, through its large-scale dynamics: • the CMB rest frame as the dynamically selected preferred frame; • the σ ordering as the physically realized temporal ordering; • the cosmological time slicing as the constraint-preserving foliation for the initial-value problem.
These structures are emergent in the technical sense: they are not present in the Einstein equations as written, but they arise from the physical solution. Once emerged, however, they are lawselecting: they determine which temporal framework is operative and which temporal ordering is physically operative. This framing recasts the central CTGP claim from a metaphysical thesis about the nature of time into a physical thesis about how the universe's matter content generates its own temporal structure. Physicists who would be skeptical of the former are often already committed to the latter in other contexts. CTGP asks only for consistency: if emergent structures can select effective dynamics in fluids, superconductors, and condensed matter systems, then the cosmological matter distribution can select an effective temporal structure for the universe as a whole. ---------------------------------------------------------------------------------------------------10. Quantum Compatibility and Pre-Geometric Ontology ---------------------------------------------------------------------------------------------------CTGP is compatible with multiple quantum interpretations so long as they preserve (i) relativistic causal structure and (ii) the open-future ontology at the global level. The layered structure of causality that makes this possible is stated in the Layered Causality Principle (§10.4). 10.1 Interpretive Routes ~~~~~~~~~~~~~~~~~~~~~~~~ (Q1) Decoherence-based readings: Macroscopic definiteness is approached without introducing an observer-dependent ontology; CTGP interprets the realized quasiclassical history as the generated one. Decoherence provides an account of why interference between macroscopically distinct states becomes negligible due to entanglement with environmental degrees of freedom. Decoherence alone does not select an outcome, however, so this route inherits the measurement problem rather than solving it, and it secures CTGP’s unique factual past only by assumption. CTGP treats the realized quasiclassical trajectory as generated stage by stage along n^μ. (Q2) Objective collapse models (e.g., CSL): Collapse provides a physically real stochastic actualization mechanism aligned with CTGP's ontological edge. σ can be treated as indexing collapseupdated hypersurfaces. This route has the virtue of providing an explicit physical mechanism for 'selection' at each generation step. dρ/dt = −(i/ħ)[H, ρ] − (λ_{CSL}/2) ∫ d³x [A(x), [A(x), ρ]] Here A(x) is a smeared mass-density operator and λ_{CSL} sets the collapse strength; on this route, collapse events realize the transition Σ_σ → Σ_{σ+Δσ}, making the stochastic dynamics coextensive with the growth of M(σ). (Q3) Epistemic readings: CTGP treats the wavefunction as informational while spacetime generation is ontologically basic; local predictions remain unchanged. On this route, the quantum state encodes epistemic facts about the actual generated structure, not additional ontological elements. Like decoherence-based readings, this route does not itself select an outcome, so it secures CTGP’s unique factual past only by assumption; collapse models and foliation-dependent hidden-variable
theories supply that uniqueness through their dynamics. Relational interpretations, which make facts relative to observers, conflict with a unique factual past and are not among CTGP’s available routes. CTGP does not require a specific quantum interpretation, but it is most naturally aligned with approaches that provide definite macroscopic outcomes at the present edge — collapse models and foliation-dependent hidden-variable theories, such as relativistic Bohmian mechanics with a covariantly fixed foliation, being the primary candidates. CTGP thus requires single-outcome realism: at each stage, each measurement has exactly one realized macroscopic outcome. The standard Everettian ontology, on which all branches are physically real, conflicts with this requirement, although not with the cosmological foliation itself: an Everettian could accept the foliation and a unique universal state on each level set while denying that each stage has a unique macroscopic outcome. The claim is specific to CTGP; it is not the claim that no-collapse quantum theories are incompatible with presentism in general. Scope of quantum-compatibility claims. CTGP’s quantum-compatibility claims are conditional rather than unconditional. Relativistic hypersurface-path independence is asserted only for admissible local update generators whose spacelike-separated densities commute on a common invariant domain and satisfy standard energy-bound and closability conditions (formalized in Proposition 17.16). Likewise, continuum completely positive dynamics are claimed only as limits of UV-regularized local semigroups under trace-norm convergence assumptions (Theorem 17.17). On the classical side, any CTGP-induced correction terms are required to be perturbatively small relative to the Einstein–matter principal part, so that ADM constraint violation remains O(ε) in Sobolev norm on local existence intervals (Theorem 17.14). In strong-curvature regimes, CTGP treats smooth σ as an emergent continuum representative of a more primitive causal growth ordering rather than as a globally valid classical scalar (Proposition 17.18). Corollary: CTGP entails a unique factual past. M(σ) contains exactly one realized causal history. Any quantum interpretation incompatible with this consequence — including any that treats past branches as ontologically real — falls outside CTGP’s scope regardless of its empirical status. 10.2 Pre-Classical Regime and Emergent Effective Laws ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP does not posit an absence of law at any stage of cosmic history, including the Planck epoch. The earliest regime of the universe is governed by the minimal generative structure described in Postulate 5: the σ-field constraint and its associated causal ordering. Classical spacetime — as a smooth Lorentzian manifold — may not yet exist during this epoch. Light cones and metric structure may be only approximate or emergent quantities, becoming valid only after sufficient decoherence and coarse-graining. GR is therefore best interpreted as the low-energy effective limit of CTGP’s deeper generative structure, not as a law that held unmodified from the initial singularity. This alignment places CTGP within the broader class of quantum-gravity programs that treat classical spacetime as emergent — including causal set theory, loop quantum gravity, and emergent spacetime approaches — without committing CTGP to any one of them. The following table formalizes the domain of validity for each ontological level and its associated formal machinery within CTGP. The column “Formal Domain” specifies where each description applies;
the column “Status” specifies the ontological category per Postulate 5. Level
| Description
| Status (per | Formal domain | | Postulate 5) | -----------------+--------------------------+-------------------+--------------------------------σ-generation law | Primitive generative | Fundamental | All regimes (pre-geometric and | ordering parameter | | emergent) Causal ordering | Partial order on | Fundamental| All regimes; light-cone | generative stages | emergent hybrid | representation valid only post| | | emergence General | Effective classical | Effective (α(ρ) → | Emergent geometric regime only Relativity (GR) | spacetime dynamics | 1) | (ρ ≪ ρ_P) QFT | Relativistic quantum | Effective | Emergent geometric regime only | field dynamics | | Thermodynamics | Statistical regularities | Emergent | After sufficient decoherence and | from coarse-graining | | coarse-graining The asymmetry in the “Formal Domain” column is the central structural claim of §10.2: σ-ordering is present and active at all stages, while every other level of law acquires its formal machinery only after the conditions for that level have been realized. GR is therefore never retroactively applied to the pre-geometric regime, and the σ-constraint is never suspended in the emergent regime. This table instantiates Postulate 5 mathematically: σ is the only structure whose domain of validity is unrestricted. 10.3 Pre-Geometric Ontology ~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP posits that the deepest layer of reality consists not of spacetime geometry but of lawful generative ordering, of which classical spacetime geometry is an effective continuum approximation. The principles that govern this relationship are stated below and in §10.4. Emergent Metric Principle: The Lorentzian metric g_{μν} is not fundamental but emerges as an effective low-energy description of underlying generative and causal relations. Diffeomorphism invariance is correspondingly an emergent symmetry of the continuum limit rather than a primitive feature of pre-geometric reality. The continuum action of CTGP (§9.1), which contains √(−g) d⁴x, should therefore be interpreted as an effective low-energy action valid after the emergence of classical geometry, not as a fundamental statement about pre-geometric structure. Causal Reconstruction Principle: Effective spacetime geometry is reconstructed from underlying causal relations and generative ordering. This principle connects naturally to the Hawking–King–McCarthy and Malament reconstruction results already cited in §6.2, which establish that causal ordering determines conformal metric structure. CTGP’s pre-geometric ontology strengthens these results from a mathematical convenience into an ontological commitment: geometry is not prior to causality but is derived from it. The generative ordering therefore takes ontological precedence: generative ordering → causal structure → emergent metric, not the reverse. Fundamental Ontological Hierarchy: The following chain specifies the direction of ontological priority in CTGP. Each arrow denotes “grounds” or “gives rise to”, never the reverse:
σ (primitive generative ordering) → causal partial order → conformal geometry → full Lorentzian metric → GR dynamics (effective) → QFT (effective) → thermodynamic regularities (emergent) Each step in this chain has associated formal conditions for when the transition becomes valid. The step from causal partial order to conformal geometry corresponds to the large-N limit of causal-set theory. The step from conformal geometry to full metric requires specifying a volume element (in causal-set theory, this is supplied by the counting measure). The step from full metric to GR dynamics requires the α(ρ) suppression factor of §9.2 to reach the value α → 1, which occurs as ρ/ρ_P → 0. No step in this chain runs in the reverse direction within CTGP; the hierarchy is strictly one-directional. This provides the formal grounding for the claim that GR cannot be “applied at the Planck epoch”: GR appears only at the end of the chain, where all prior steps have been completed. Effective Law Emergence Principle: Stable regularities emerge when the generated present acquires sufficient physical structure to support reproducible signal propagation and dynamical constraints. This links directly to the signal-based epistemological dimension of CTGP: the observability of physical laws is itself signal-dependent. Effective laws become epistemically accessible only when the universe supports the propagation and persistence of the signals required to reveal those regularities. Physical structure and epistemic accessibility therefore co-emerge at the same threshold. 10.4 Pre-Geometric Consistency Principles ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The continuum formalism of Part II is an effective description, and a small set of interpretive principles governs how its language is to be read across the pre-geometric and emergent regimes. They are collected here so that each can be cited by name. Together they fix the framework’s ontological commitments: the metric is emergent, general relativity is effective, and σ is fundamentally a pre-geometric ordering parameter. Primitive Status of σ Principle. At the deepest ontological level, σ is not a scalar field defined on spacetime. It is a pre-geometric generative ordering parameter that labels lawful stages of reality’s production. Before classical geometry exists, there is no manifold, no metric, and no scalar field in the standard sense; therefore σ cannot fundamentally be a spacetime scalar. The scalar-field representation of σ arises only in the emergent continuum limit once classical geometry becomes applicable. Passages in this paper that describe σ as “a scalar field,” “a spacetime field,” “a function of cosmological time,” or “a physically measurable scalar” are to be read as applying exclusively within the emergent geometric regime. At the fundamental level, the correct characterization is: σ is a primitive generative ordering parameter whose emergent geometric regime representation is a smooth scalar field. Two-Regime Representation Principle. The pre-geometric and emergent-geometric descriptions of σ must never be conflated. The following table defines the regime-appropriate interpretation for every context in which σ is invoked: Regime | Interpretation of σ -------------------+-------------------------------------------------------------------------
Pre-geometric | Generative ordering parameter (no manifold assumed) Emergent geometry | Cosmological time τ and its monotone relabelings σ = F(τ) Cosmological limit | Cosmic proper time t of FLRW Operational | Locatable by any observer through the CMB temperature and dipole (§12.4) Dual Interpretation of Σ_σ. Because hypersurfaces require pre-existing geometry, the present boundary Σ_σ admits two regime-dependent definitions. In the pre-geometric regime, Σ_σ is defined as the set of events sharing equal generation stage in the causal ordering induced by σ — a “generation stage” rather than a hypersurface. In the emergent geometric regime, after classical geometry has become available, Σ_σ admits representation as the smooth spacelike hypersurface corresponding to the continuum representation of that generation stage. References to “the present hypersurface Σ_σ” in this paper apply in the emergent geometric regime. Three items should be kept apart. Σσ is a region: a spacelike, three-dimensional hypersurface that cuts across every system at once, not the region occupied by any particular one. The state at a generation stage is a mathematical object: the Cauchy data on Σσ (the induced metric, the extrinsic curvature, and the matter fields with their conjugate data, subject to the constraints), a point in the constrained phase space of the Einstein–matter system. What exists is the physical configuration of fields and matter that this state describes. In pre-geometric contexts, the phrase should be read as “the present generation stage Σ_σ.” Generative Stages Language Principle. Throughout this paper, the phrase “successive hypersurfaces are generated” is to be interpreted as shorthand for “the present boundary advances continuously along the generation flow vector n^μ; foliation into spacelike hypersurfaces Σ_σ is a mathematical representation of this continuous flow, available only in the emergent geometric regime.” The fundamental process is the continuous propagation of generative flow; the hypersurface description is a representation that becomes available after geometry emerges. This reading applies globally and does not require case-by-case annotation at each occurrence. Layered Causality Principle. CTGP operates with two distinct layers of causality that must not be conflated. Fundamental causality consists of σ-ordering and causal relations: the primitive directional structure from which reality’s generative sequence is constituted. Classical causality consists of light cones and relativistic propagation: an effective representation that becomes valid only after spacetime geometry has emerged. Classical causal structure therefore emerges from deeper generative ordering encoded by σ and its induced causal relations. References to “causality” in discussions of the pre-geometric regime concern the fundamental layer; references to “causal structure” in the context of GR, FLRW cosmology, or observational predictions concern the classical layer. Effective-Regime Restriction Principle. All reconstruction theorems (HKM, Malament), temporalfunction theorems (Geroch), Cauchy-foliation results (Bernal–Sánchez), and ADM formulations discussed in CTGP apply only within the emergent geometric regime where classical spacetime is a valid continuum approximation (ρ ≪ ρ_P). None of these constructions are claimed to be fundamental. Their role within CTGP is to establish that, once geometry has emerged, a well-defined foliation structure is guaranteed by causal geometry alone — not to assert that geometry is prior to causality. This restriction applies universally to §§6, 7, 8, 9, and their subsections. The pregeometric regime is governed by the minimal generative structure (σ, C) specified by the Fundamental Ontological Hierarchy (§10.3) and the Pre-Geometric Minimal Ontology below; the geometric machinery enters only after the transition described in §9.2.
Proper-Time Qualification. Proper time presupposes a metric and is therefore not available as a concept in the pre-geometric regime. Any statement in this paper of the form “σ corresponds to cosmological proper time” is to be read as: “In the emergent geometric regime, σ is cosmological time up to monotone relabeling.” The claim that σ fundamentally is proper time is not made and would be inconsistent with CTGP’s pre-geometric ontology. The proper-time interpretation is an emergent correspondence, not a definition. Regime-Dependent Representation Principle. The relationship between σ and geometry is regimedependent and must be stated precisely. In the pre-geometric regime, σ is primitive and is not defined through any geometric or material quantity; it is the ontological ground from which causal structure, geometry, and eventually matter-energy descriptions emerge. In the emergent geometric regime, σ is represented by cosmological time and its relabelings, satisfying ∇_μσ ∇^μσ = −f(σ). The generation law may depend on σ but not on local matter content (§§6.11, 11). The logical order remains: σ (primitive) → causal structure → emergent geometry → cosmological time as the continuum representative of σ. Pre-Geometric Minimal Ontology. Before classical geometry emerges, the complete inventory of fundamental entities in CTGP is: Present at the pre-geometric level: generative ordering parameter σ; causal relation structure C; minimal generative laws governing lawful succession. Absent at the pre-geometric level: metric; manifold; coordinates; light cones; proper time; stress-energy tensor; Einstein equations. These emerge later and belong to the emergent or effective layers of the hierarchy. Effective Action Principle. The continuum action of CTGP — S(σ) = ∫_{M(σ)} √(−g) [L_{GR} + λ(∇_μσ ∇^μσ + f(σ))] d⁴x — is an effective low-energy action applicable only after emergent geometry has established classical spacetime as a valid continuum approximation. It is not claimed to represent the microscopic dynamics of the pre-geometric regime. The extended action S_{ext}(σ) of §9.2 models the crossover regime continuously but similarly does not constitute a fundamental pre-geometric theory. The UV completion of these actions lies in the pre-geometric structure (σ, C) described by the Pre-Geometric Minimal Ontology, and remains an open problem for the research program described below. The action is accordingly to be read as an effective field theory — a well-established status in contemporary physics — rather than as a fundamental statement; it does not presuppose that geometry is fundamental. Scope of the Research Program. CTGP specifies ontological constraints on possible quantum-gravity theories rather than providing a completed microscopic theory. The framework presently leaves open: the mathematical realization of σ at the Planck scale; the origin of causal relations C; the mechanism by which volume information arises from causal structure; the reconstruction of metric geometry from causal ordering; and the microscopic derivation of effective laws. CTGP should therefore be understood as a generative ontology and a research program constraining future theories of quantum gravity. The research program has a well-defined core: any adequate quantum-gravity theory must preserve causal directionality, produce emergent geometry from pre-geometric ordering, and recover the effective description characterized by the CTGP action in the low-energy limit.
Primitive Generative Structure Principle. Explanatory chains terminate in primitive structures. Within CTGP, σ-ordering and minimal generative laws are primitive. The question “what governs σ?” receives the answer: nothing governs σ from without, for the same reason that causal order in causal-set theory requires no external explanation, logical consistency is not explained by a metalogic, and quantum postulates are not derived from a more fundamental theory. Explanatory termination in primitive structures is a standard and unavoidable feature of any axiomatic framework. No contradiction arises from this termination; it is a feature of all foundational theories, not a deficiency of CTGP specifically. The Generative Minimality Principle (Postulate 5) formalizes this: (σ, C) with minimal generative laws is the sufficient primitive structure, and no regress threatens because no further explanation is required of a primitive. Unified Ontological Thesis of CTGP. Reality is fundamentally constituted by a minimal lawful generative structure consisting of a primitive ordering parameter σ and associated causal relations C. Classical spacetime geometry, relativistic causal structure, stress-energy descriptions, and Einsteinian dynamics are emergent effective representations arising when the generated domain acquires sufficient structure to support stable signal propagation and reproducible dynamical regularities. The continuum formalism of General Relativity therefore describes the low-energy limit of a deeper pre-geometric generative ontology rather than the fundamental architecture of reality. This thesis is the integrating statement of CTGP’s ontology. Its content is the conjunction of the principles above: (σ, C) are primitive; geometry is emergent; GR is effective; all geometric machinery applies only in the appropriate regime; proper time, hypersurfaces, and cosmological time are regime-dependent representations of more fundamental pre-geometric structure; the framework is a research program as much as a completed theory; and explanatory termination at the primitive level is not a deficiency but a philosophical necessity shared by all foundational theories. ========================================================================= =========================== PART III. Physical Realization and Empirical Standing ========================================================================= =========================== ---------------------------------------------------------------------------------------------------11. Admissible Generation Laws: A Consistency Analysis ---------------------------------------------------------------------------------------------------The generative present of §6 is fixed by geometry. A natural further hypothesis is that the rate at which σ advances depends on local physical conditions — in particular, that generation proceeds faster where matter is denser. Such a law would give the generation parameter physical content beyond geometry and could in principle imprint new cosmological signatures. This section shows that the hypothesis is untenable. The exact form of the law destroys the alignment of the present with matter (§§11.1–11.2); its stable soft completions force a uniform rate inside bound structures and suppress any cosmological effect far below observability (§11.3). Within the local, stable classes analyzed, the viable generation laws reduce to functions of σ alone (§6.11), and their residual caustic problem is resolved at Layer 2 by cosmological time (§11.4). Finally, no admissible law yields an intrinsic cosmological axis (§11.5). Within those classes, the results hold for general f and apply to every increasing density dependence, not only to particular choices. 11.1 The Conformal Form of the Norm Condition
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Proposition 11.1 (Conformal form). If g^{μν}∂_μσ∂_νσ = −f with f > 0, then σ has unit-norm gradient in the conformally related metric g̃ = f g. The normalized generation flow n_μ = −∂_μσ/√f has acceleration a_μ = n^ν∇_ν n_μ = −D_μ ln √f, where D_μ = (δ_μ^ν + n_μ n^ν)∂_ν is the derivative projected orthogonally to n. Proof. Using the symmetry of ∇_μ∂_νσ and n^μ n_μ = −1, one finds n^μ∇_ν∂_μσ = ∂_ν√f, and hence n^μ∇_μ n_ν = −∂_ν ln√f − n_ν n^μ∂_μ ln√f = −D_ν ln√f. □ Two cases follow. If f = f(σ), then D_μf = f′D_μσ = 0, so a_μ = 0: the generation flow is geodesic. If f carries spatial matter dependence — for example f = ρ_c²/ρ_*² — then a_μ = −D_μ ln ρ_c, and the flow is pushed down density gradients. In Newtonian terms (c = 1) the generation flow experiences the effective potential Φ_{eff} = Φ + ln(ρ_c/ρ̄). On sub-horizon scales density contrasts exceed metric potentials by a factor of order (k/aH)², and inside a galaxy ln ρ_c varies by order ten while Φ ∼ 10⁻⁶. A density-weighted generation flow therefore responds to matter density far more strongly than to gravity. 11.2 Relative Tilt Between the Generation Flow and Matter ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Linearize about FLRW in longitudinal gauge, with pressureless matter and λ = 0. Write the covariant velocities of the two flows as u_i = ∂_{iV}, so that V_σ = −δσ/σ̄̇ for the generation flow, while matter obeys V̇_m = −Φ. The linearized norm condition gives δσ̇ = Φσ̄̇ + δf/(2σ̄)̇ . Proposition 11.2 (Relative tilt). The relative tilt T ≡ V_σ − V_m is gauge-invariant and obeys Ṫ = −(ḟ/2f̄) T − Δ_f, Δ_f ≡ δf/(2f̄) + (ḟ/2f̄) V_m, with the metric potential Φ cancelling identically. For f = f(σ), Δ_f = −(ḟ/2f̄)T and Ṫ = 0, so the physical relative velocity v = kT/a decays as 1/a. For f = ρ_c²/ρ_*² in the matter era, the equation becomes Ṫ − 3HT = −Δ_c, where Δ_c is the comoving density contrast, with solution T = C a³ + 2Δ_c/H, v = C k a² + 2 (k/aH) Δ_c. Two conclusions follow for density weighting. A primordial tilt grows as a² during matter domination and a³ during radiation domination, a growth factor of order 10²⁶ from the end of inflation to the present; an intrinsic tilt is therefore either fine-tuned to about one part in 10²⁶ initially or has already driven the foliation toward null. And structure sources a tilt that exceeds matter peculiar velocities by a factor of order (k/aH)², so that alignment of the generation flow with matter survives only on scales comparable to or larger than the horizon. Corollary 11.3 (Light-crossing bound). Suppose σ advances along matter worldlines at a rate set by
local density, σ = ∫ρ_c dτ, in a quasi-static structure with density scale length L = ρ_c/|∇ρ_c|. Then σ ≈ ρ_c(x)t, and its level sets are spacelike only for t < L (c = 1). For a galaxy with L ≈ 10 kpc and an age of 10¹⁰ yr this condition is violated by a factor of about 3 × 10⁵. If the exact norm condition is imposed instead, the generation flow is repelled from density peaks within a lightcrossing time and forms caustics. □ 11.3 Soft Completions ~~~~~~~~~~~~~~~~~~~~~ Adding a conventional kinetic term to the exact action changes nothing: a term proportional to ∇_μσ∇^μσ appears in the Lagrangian only in combination with the multiplier term and is absorbed by a constant shift of λ, at the cost of a term proportional to f. Genuine propagation of σ therefore requires a Lagrangian nonlinear in X ≡ −½∇_μσ∇^μσ. The natural completion replaces the exact constraint by a stable-sign preferred norm, S_σ = ∫ √(−g) P(X, T) d⁴x, P = P₀(X) + (κ/2)[X − X⋆(ρ)]², κ > 0, with X⋆ increasing in ρ. The standard k-essence conditions (Garriga and Mukhanov 1999) apply: the sound speed is c_s² = P_X/(P_X + 2XP_{XX}), absence of ghosts requires P_X + 2XP_{XX} > 0, and absence of gradient instability requires P_X > 0. The penalty must carry the stable sign shown; the opposite sign makes P_{XX} negative near the preferred norm and produces a ghost. Because P does not depend on σ, the field equation is the conserved-current equation ∇_μ(P_X∇^μσ) = 0 with no source term: matter enters only through the coefficient P_X(X, T). On an FLRW background this gives a³P_Xσ̇ = C, so that near the preferred norm c_s² ≈ C/(2κX a³σ̇). The sound speed is therefore set by an integration constant rather than predicted, and C = 0 reproduces the pressureless behavior of the exact constraint, as in ghost condensation (Arkani-Hamed et al. 2004). Theorem 11.4 (Static rate uniformity). Let the σ-sector be shift-symmetric, S_σ = ∫√(−g) P(X, T) d⁴x, with T_{μν} depending only on metric and matter fields. Consider a static, asymptotically flat configuration σ = ωt + φ(x), regular everywhere, and suppose P_X > 0 wherever it is supported. Then ∇φ ≡ 0. Proof. The spatial equation is ∂_i(√(−g) P_X g^{ij}∂_jφ) = 0. Regularity and the divergence theorem make the flux through every closed surface vanish, so φ − φ_∞ = O(r⁻²). Multiplying by φ − φ_∞ and integrating by parts gives ∫√(−g) P_X g^{ij}∂_iφ∂_jφ d³x = lim_{r→∞} ∮(φ − φ_∞)P_X ∇φ · dS = 0. The integrand is non-negative, so ∇φ = 0. □ Every stable static structure therefore has the same generation rate as its surroundings: σ = ωt, with X = ω²/(2N²) where N is the lapse. A locally enhanced generation rate inside bound structure is excluded for the entire class. Proposition 11.5 (Stability transport bound). Suppose the background evolves near the preferred norm, X̄ ≈ X⋆(ρ̄). (a) By Theorem 11.4, a static structure containing densities up to ρ_{max} at an epoch with background norm X₁ is stable only if
P₀′(X₁) > κ [X⋆(ρ_{max}) − X₁]. (b) Along the background, the no-ghost condition reads d(X^{1/2}P₀′)/dX > −κX^{1/2}. Integrating to an earlier epoch with X₂ > X₁ gives P₀′(X₂) ≳ κ X⋆(ρ_{max}) (X₁/X₂)^{1/2}, provided X⋆(ρ_{max}) X₁^{1/2} ≫ X₂^{3/2}. (c) The fractional modulation of the σ-sector by density perturbations δ at the earlier epoch is then bounded by ε ≡ κ ρ̄ X⋆′(ρ̄) δ / P̄_X ≲ δ · [ρ̄ X⋆′(ρ̄) / X⋆(ρ_{max})] · (X₂/X₁)^{1/2}. For X⋆ ∝ ρ² this becomes ε ≲ 2δ (ρ̄₂/ρ_{max})² (ρ̄₂/ρ̄₁); the constants κ, the normalization of P₀, and ρ_* all cancel. At recombination (ρ̄₂ ≈ 4 × 10⁻¹⁸ kg m⁻³, δ ≈ 10⁻⁵), measured against the epoch at which compact objects first exist (z ≈ 20, ρ̄₂/ρ̄₁ ≈ 1.4 × 10⁵), the bound is ε ≲ 10⁻⁶⁸ with ρ_{max} ≈ 10¹⁷ kg m⁻³ (neutron stars) and ε ≲ 10⁻⁴⁰ even with ρ_{max} = 10³ kg m⁻³ (ordinary condensed matter). □ Corollary 11.6 (Exclusion of density weighting). Within the class of local, shift-symmetric, stablesign completions, no unsaturated increasing X⋆(ρ) produces an observable density-weighted cosmological signature. □ Remark (saturation). If X⋆ stops growing above some density ρ_s ≲ ρ̄₂, the bound relaxes to order δ. This is a distinct, screened model class: it introduces a new density scale tuned to the relevant epoch, equalizes the generation rate in everything denser than ρ_s — the opposite of densityweighted generation — and supplies no preferred axis. Its residual effect is proportional to the σ-sector’s share of the energy budget, which is bounded by constraints on early dark energy. 11.4 Caustics and the Cosmological-Time Completion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The admissible generation laws f = f(σ) make the generation flow geodesic (Proposition 11.1). Initialized on the comoving congruence, it coincides with the free fall of pressureless matter, and pressureless free fall shell-crosses inside every collapsed halo. A smooth scalar σ therefore develops caustics wherever structure has formed, and any nonzero λ-dust would acquire a divergent density there. Caustic-free completions of pressureless fluids and k-essence have been studied in the literature (Babichev and Ramazanov 2017). CTGP does not need such a completion. The Layer 2 present is defined by cosmological time, which is the maximal single-valued continuation of the unit-norm solution: where maximizing geodesics cross, τ loses differentiability, its level sets acquire corners, and they remain Cauchy surfaces (Theorem 17.4). With λ = 0 (§6.9) no divergent stress-energy arises. The generative present is therefore globally well defined in a universe with nonlinear structure, while the smooth continuum σ-field is a valid description only where the maximizing congruence is single-valued. Alignment of the present with matter is correspondingly narrowed: it holds exactly in FLRW and with the free fall of pressureless matter in single-stream regions, not with pressure-supported components or within multistream regions.
11.5 No Intrinsic Cosmological Axis ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Three directions on the sky must be distinguished. The kinematic dipole n̂_{kin}, near Galactic (l, b) ≈ (264°, 48°), records the motion of the Solar System relative to the CMB rest frame at about 370 km s⁻¹. The axis n̂_{mod} of the observed large-angle hemispherical power modulation, with amplitude A ≈ 0.07, lies near (l, b) ≈ (220°, −20°) (Hoftuft et al. 2009; Planck Collaboration 2020), roughly 70–80° from n̂_{kin}. A third direction, n̂_σ, would be the spatial projection of ∇_μσ on the present hypersurface, if such a projection existed. On a homogeneous background it does not: ∇_μσ̄ = (σ̄,̇ 0, 0, 0), so the spatial projection vanishes and n̂_σ is undefined. An intrinsic axis would require a long-wavelength gradient mode of σ, which by Proposition 11.2 is a tilt of the generation foliation relative to matter. For admissible generation laws such a tilt decays as 1/a, leaving no intrinsic axis at late times; for densityweighted laws it grows by a factor of order 10²⁶ and is either fine-tuned or runaway. Superhorizon modes invoked to generate hemispherical asymmetry are further constrained by the low multipoles they induce through the Grishchuk–Zel’dovich effect (Erickcek, Carroll, and Kamionkowski 2008). CTGP therefore predicts no intrinsic cosmological axis and offers no explanation of the observed large-angle anomalies, which remain questions for standard cosmology. This is a consequence of the framework rather than an omission: the generative present is a geometric structure, and geometry alone does not single out a spatial direction in a homogeneous, isotropic universe. ---------------------------------------------------------------------------------------------------12. Empirical Standing of the Ontology ---------------------------------------------------------------------------------------------------This section states precisely what observation can and cannot establish about CTGP’s ontology. The results are general: they apply to any presentist reading of physics, not only to CTGP. 12.1 Observational Equivalence ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Proposition 12.1 (Observational equivalence of presentist and eternalist readings). Let P (presentist) and E (eternalist) be two ontological readings of the same physical content 𝒦 = (M, g_{μν}, matter fields, dynamical laws, probability measure μ over admissible histories). Call a quantity *observable* if it is a function of the physical records present to an observer at the stage where the observation occurs — for the presentist, the records on the present surface Σ_σ. Then every observable has the same probability distribution under P and under E, and no experiment can discriminate between them. Proof. An observation is a physical process, and its outcome is a function of the records present at the stage where it occurs. Both readings assign the same admissible histories and the same measure μ, and hence the same distribution of present records at every stage, so each observable has the same distribution — the pushforward of μ — under both. □ Remark (what is observed). Every observation available to an observer is an observation of present records. A fossil, a memory, a photograph, a detector event, or a photon arriving from SN1987A
carries information about an earlier state, but information about another time is not the existence of that time: under CTGP the earlier state has ceased to exist, and a present observer cannot observe it as a presently existing state — not as a limitation of instruments but because, under CTGP, a presently existing past state is not there to be observed; the notion of observing one is incoherent within the ontology. The same holds for the future, which a present observer can predict or represent but cannot observe, because it has not been generated. This does not break the equivalence; it is the reason the equivalence holds. Eternalism does not permit observation of another time either: its earlier events exist, but not at the observer’s location, and an observer interacts only with the records present there. Neither ontology allows an observer to observe another time as present. They differ only over whether the unobserved earlier events exist, and present records — the same under both — cannot decide that. A theory that did allow observation of another time as present would be different physics, with different records, and would fall under §§12.2–12.3 rather than under this proposition. Remark (indeterminism). The result does not depend on determinism. For stochastic laws, E represents the realized history as one member of the ensemble, weighted by μ; P represents it as generated stage by stage with the same transition probabilities. The observable statistics coincide. Remark (scope). The proposition assumes that every observation, including an observer’s reports about experience, is a physical record. First-person temporal phenomenology is not observable in this sense; it enters CTGP as an explanatory datum through the Phenomenological Completeness Principle (§4.5), not as experimental evidence. Corollary. With the physics held fixed, the ontology cannot be verified by measurement. Empirical bearing on it must come either from constraints the ontology places on which physics is possible (§12.2) or from physical content added to 𝒦, which bears directly on that content and only indirectly on the ontology. 12.2 The Generability Constraint ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Observational equivalence holds for fixed physical content. The two ontologies do not, however, make the same demands on what that content can be. If only the present exists, each new state must be producible from the current one alone: the fundamental dynamics must admit a generative initialvalue formulation relative to the existing present, with no dependence on future boundary data and no global consistency conditions that require the whole history at once. Eternalism carries no such requirement. A block universe can contain initial-value laws, but it can equally contain laws with future boundary conditions, final-state constraints, or self-consistency conditions on closed causal loops. Let G denote the proposition that the fundamental laws are generable in this sense. Because generability is part of what CTGP means by presentism, a confirmed failure of G would directly contradict the ontology, whereas eternalism merely permits generable laws. Finding that our best theories admit generative initial-value formulations is therefore evidence the ontology could have failed and did not. The criterion stated precisely. Generability requires a state S_σ on each level set and a law fixing the transition probabilities P(S_{σ′} | S_σ) for σ′ > σ — a point mass in the deterministic case — without reference to data on later level sets. Future dependence counts against generability only if
no empirically equivalent formulation of this kind exists. Enlarging the state space is legitimate only if the added variables are physically instantiated on Σ_σ and have dynamics of their own, not if they are defined by reference to later data; without that restriction, any history could be made generable by encoding its future in the present state. Global formulations do not count against generability when they are equivalent to a well-posed initial-value problem: general relativity can be derived from an action over spacetime, but its classical content is captured by the Cauchy problem of §7. No particular probability is assigned here to generable laws on the eternalist view; the point is only that presentism is exposed to this evidence where eternalism is not. This does not conflict with Proposition 12.1: the discrimination occurs across candidate physical theories, not within a fixed one. The evidence. The successful physical theories considered here admit well-posed initial-value formulations: general relativity on globally hyperbolic spacetimes and relativistic field theories with hyperbolic equations of motion. No confirmed fundamental dynamics has required futuredependent input, and physics has survived increasingly stringent tests without it. The accurate summary is that the empirical physics examined so far has not revealed a confirmed violation of the generability requirement. The limits of the evidence. Three qualifications apply. First, the support is shared: the growing block entails generability as well, so the evidence favors dynamic ontologies as a class over eternalism, not CTGP over the growing block. Second, an eternalist may regard generable, locally predictive dynamics as independently natural for reasons unrelated to temporal ontology — one example is the argument that observers can arise only where the field equations are hyperbolic and the world is predictable (Tegmark 1997). To the extent that such reasons hold, the evidence has correspondingly little force. Third, the evidence is theory-relative: it bears on which laws nature has, and so on the ontology only through the constraint the ontology imposes on laws. The evidence is therefore consistent with the requirement, but its evidential force is limited: current physics does not establish generability. Future tests. The constraint is exposed to physics still to come. A fundamental final-state condition in quantum gravity, such as the black-hole final-state proposal (Horowitz and Maldacena 2004; for its difficulties, see Gottesman and Preskill 2004), would be incompatible with CTGP’s generability requirement and would therefore provide strong evidence against its presentist ontology; resolution of black-hole evaporation through ordinary initial-value dynamics would be a further survived risk. A unique final state that simply results from deterministic evolution does not count; only a final condition that must be imposed as an independent input to fix earlier evolution would. A requirement that quantum gravity involve indefinite causal order at the level of spacetime itself would similarly bear on CTGP’s definite primitive ordering (§6.4). 12.3 Refuting Observations and Preferred-Frame Tests ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Two kinds of observation would count directly against the ontology: a confirmed physically realized closed timelike curve, and a retrocausal dependence in the interventionist sense, P(O_t | do(F_{t+Δ})) ≠ P(O_t) that cannot be reduced to correlations, post-selection, or boundary
conditions, and cannot be eliminated by an empirically equivalent formulation without retrocausation. Time-symmetric descriptions and correlations with later variables do not meet this standard. Either would contradict the requirement that each present be generated from its past alone (§9.4.2; §13). No maintained physical connection between the present and a past or future moment has ever been observed; phenomena that suggest one — entanglement between photons that never coexisted, delayed-choice erasure, violations of temporal Leggett–Garg inequalities — establish correlations among present records, not persisting links across time. Other observations would change the cost of holding the ontology without deciding it. A physically preferred global foliation, whether detected as preferred-frame effects or required by an empirically successful theory of quantum gravity, would remove the principal relativistic objection to presentism, though an eternalist may accept such a foliation too. Conversely, Lorentz invariance confirmed to ever deeper levels raises the cost without refuting the view. CTGP inherits the local phenomenology of general relativity: the cosmological frame has no effect on local physics. Because the generative present is defined by cosmological time and introduces no new field, the preferred-frame parameters that would couple local physics to the CMB rest frame vanish. For an observer moving through that frame at v/c ≈ 1.2 × 10⁻³, such couplings of strength α would produce sidereal and annual modulations of clock rates and propagation of order α(v/c)². Lunar laser ranging and pulsar timing bound the relevant post-Newtonian preferred-frame parameters at about 4 × 10⁻⁵ and 2 × 10⁻⁹ (Will 2014), and atomic-clock and spectroscopic comparisons bound preferredframe effects on matter far more tightly (Kostelecký and Russell 2011). All such results are null, as CTGP requires. They are consistent with CTGP but do not discriminate it from eternalism, which makes the same prediction. Preferred-frame theories with a dynamical timelike vector, such as Einstein–æther and khronometric gravity (Jacobson and Mattingly 2001; Hořava 2009), are the natural comparison class, and their bounds apply to any Layer 3 extension of CTGP that would give the generation flow independent dynamics. The two halves of this claim should be stated together. CTGP introduces no local Lorentz violation and no new local field; it does commit to a globally distinguished foliation wherever cosmological time is regular. The null results above bear on the first half, not the second, and no measurement of local physics could bear on the second. 12.4 Operational Locatability of the Present ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The generative present is not a posit beyond the reach of measurement. The universe contains nearly universal reference structures through which it becomes measurable: the CMB, the most comprehensive cosmological rest-frame reference, and the 21 cm hyperfine line, a spectroscopic reference for redshift and motion across cosmological distances (since reionization carried mainly by neutral hydrogen in and around galaxies). Neither generates time; both make its large-scale ordering observable. Any observer can determine the cosmic time of their own stage from the local CMB temperature, which falls in inverse proportion to the scale factor, and their motion relative to the cosmological frame from the CMB dipole, about 370 km s⁻¹ for the Solar System. The question “which stage of the cosmic present am I in” therefore has an operational answer, and the correction it requires can be computed in practice: for SN1987A the difference between the cosmological-time placement and the naive light-travel-time subtraction is at most about 200 years out of 168,000 (Appendix E). This establishes that CTGP’s present is physically well defined. It does not
discriminate CTGP from eternalism, which can use the same cosmic clock. That observers always find their own present real is guaranteed under both readings, since every observation occurs at the observer’s stage; it therefore confirms physical instantiation without bearing on ontological exclusivity. 12.5 Summary of Empirical Standing ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Level | Claim | Empirical status ----------------+----------------------------------------------+----------------------------------Ontology | Only the present exists | Not observable (Proposition 12.1) Structural | Fundamental laws are generable from the | Empirically exposed; refutable; constraint | present (entailed by CTGP, merely permitted | consistent with current physics | by eternalism) | (§12.2) Physical | Generation law, couplings, and signatures of | Ordinary empirical hypotheses; realization | σ | density-weighted forms excluded | | (§11) Three outcomes are possible for any physical realization of CTGP. If the added physics fails, as density-weighted generation does, that refutes the realization, not the ontology. If the added physics succeeds but is observationally equivalent to its rivals, CTGP is a physically coherent formulation of presentism without experimental discrimination. If it succeeds and yields observations its rivals do not naturally reproduce, the support runs along a chain: the observation supports a physical theory, the theory supports objective becoming, and becoming lends indirect support to presentism. At no step does an observation verify the ontology directly. CTGP does not seek an experiment that detects the nonexistence of the past; it seeks a coherent realization of objective becoming whose empirical consequences, if any, are carried by explicitly stated physics. ========================================================================= =========================== PART IV. Evaluation ========================================================================= =========================== ---------------------------------------------------------------------------------------------------13. Objections and Replies ---------------------------------------------------------------------------------------------------Objection O1 (Relativity of simultaneity): Presentism requires a global present, but SR denies absolute simultaneity. Reply: CTGP does not require an operationally absolute simultaneity relation between arbitrary spacelike-separated events. It requires a foliation adequate for cosmology and compatible with GR's initial-value structure. Cosmological time provides a physically grounded, non-absolute global 'now', which on large scales is the rest frame of the CMB — not a kinematic absolute but a physical fact about our universe. The metaphysical claim concerns the ontological structure of M(σ), defined by cosmological time, not the conventions individual observers use to label events as simultaneous (§9.4).
Objection O2 (Lorentz invariance): Any preferred foliation violates Lorentz invariance. Reply: A global foliation can be physically emergent while local laws remain Lorentz-invariant (§9.3.1). Preferred frames in cosmology are properties of solutions, not new physics, and CTGP predicts no preferred-frame effects on local physics, consistent with existing null results (§12.3). Objection O3 (Diffeomorphism invariance): 'Growth' is a gauge artifact. Reply: Growth is not a gauge artifact. The temporal ordering CTGP relies on begins from the causal structure of spacetime itself: by the Hawking–King–McCarthy and Malament theorems, the causal ordering relation already determines the conformal structure of the spacetime metric. Geroch and Bernal–Sánchez then show that this causal ordering can be represented by smooth temporal functions whose level sets are spacelike Cauchy hypersurfaces. CTGP does not introduce temporal ordering; it selects a physically preferred representative of a class of orderings already latent in globally hyperbolic GR. The representative is cosmological time, defined from the causal and metric structure alone, which in cosmology coincides with the matter rest frame. The ontology is not a coordinate artifact but a claim about causal structure. Equivalence classes under diffeomorphisms are respected; the growth claim is made at the level of equivalence classes, not coordinate labels, and cosmological time is preserved by isometries (Proposition 17.8). Objection O4 (No empirical difference): CTGP is metaphysical over-interpretation. Reply: CTGP does not claim that its ontology is observable: presentist and eternalist readings of identical physics are observationally equivalent (§12.1). It is nonetheless empirically exposed through the requirement that physical law be generable from the present, which can be refuted (§12.2), and it makes commitments in the philosophy of mind that are not empirically neutral (Appendix D). Objection O5 (Infinite Regress of Laws): If σ governs the emergence of physical laws, what governs σ? This threatens an infinite regress of meta-laws. Reply: σ is not an emergent law requiring a further law to explain it but a primitive generative structure, and explanatory chains may terminate in primitives without contradiction, as causal set theory terminates at the partial order. The point is formalized by the Generative Minimality Principle (Postulate 5) and the Primitive Generative Structure Principle (§10.4). Objection O6 (What Is σ Defined Over?): If spacetime is not fundamental, what mathematical object does σ live on? A scalar field requires a manifold; a manifold requires spacetime; but spacetime is claimed to be emergent. Reply: σ is not fundamentally a field over spacetime but an ordering parameter labeling generative stages; at the pre-geometric level it orders a partially ordered set of stages without requiring a manifold (§10.3). The continuum action of §9.1 is accordingly an effective low-energy description, with the same status GR occupies within quantum gravity. Objection O7 (Closed timelike curves): If spacetime admits closed timelike curves, a global Cauchy foliation fails, and with it the generated present. Reply: CTGP does not treat global hyperbolicity as a convenience: a solution with closed timelike
curves is a geometry that no sequence of generative stages could produce (§9.4.2). The known CTCadmitting spacetimes — Gödel’s rotating dust universe, Tipler cylinders, and Kerr interiors beyond the Cauchy horizon — are either inconsistent with observed cosmological boundary conditions or idealizations that break down before the chronology-violating region is reached (§9.5). The restriction is therefore physically motivated, and a confirmed observation of chronology violation would falsify the framework (§12.3). Objection O8 (Worldline perspective): Eternalists can explain temporal experience as a worldline perspective within the block. Reply: The worldline response succeeds only if the phenomenology of temporal experience is treated as a representational illusion — a feature of how observers inside the block model their situation, not of how things are. CTGP rejects this on the grounds developed in §§2.2 and 3: structure alone does not amount to experience, and the felt asymmetry between past and future is a positive constraint on ontology. The worldline perspective is descriptive and cannot say why any part of a static manifold is accompanied by experience; CTGP supplies the architecture — active present-edge processes, causal continuity, classicalization — within which that question can be posed (Appendix D). Objection O9 (Point presentism): In special relativity the only frame-independent candidate for an event’s present is the event itself (Stein 1991). A relativistic presentism therefore collapses into a present consisting of a single point. Reply: The objection is correct about special relativity, and CTGP does not dispute it. The invariant structure of Minkowski spacetime supplies no global present, and its cosmological time is not even regular, since every event has past-directed timelike curves of unbounded length. CTGP accordingly does not derive the present from the symmetries of special relativity. It takes the present from the actual matter-filled solution of general relativity, whose cosmological time is regular and whose level sets are spatially extended Cauchy surfaces (§6). The present is a whole slice of the universe, not a point. This is a restriction to the relativistic spacetimes that model the actual universe, not a retreat to non-relativistic ones. Minkowski spacetime, in which the construction selects nothing, is an idealization of those spacetimes rather than a model of the universe as a whole. That it depends on the actual solution rather than on the laws is the same cost acknowledged under Objection O2. Objection O10 (History-dependent dynamics): Some systems respond to present conditions in ways that depend on their history. In magnetic hysteresis, described by the Stoner–Wohlfarth model (Stoner and Wohlfarth 1948), a ferromagnet’s response to an applied field depends on how the field varied earlier. The present state therefore seems insufficient without the past. Reply: The history such systems depend on is carried by present physical variables, here the ferromagnet’s present magnetization, which is a spatially extended present state. Generation requires only that whatever a system’s evolution depends on be instantiated now, not that the system lack a history. A law with irreducible dependence on earlier states, one whose past dependence no present variable could carry, would indeed conflict with CTGP as sharply as future dependence would; hysteresis and other familiar memory effects are not of that kind. Objection O11 (Nothing marks the present): Cosmological time orders every stage alike. Nothing in it
marks one level set as the present one. Reply: Correct, and CTGP does not claim otherwise. The geometry settles which events are co-present, not which stage is present. On an eternalist reading every level set exists, and something further would be needed to single one out. On CTGP no such selection problem arises: only the configuration on one level set exists, and the others appear only in the formal representation M(σ). Presentness is supplied by the ontology, not by the time function. General relativity supplies the foliation; CTGP supplies its interpretation. A unifying response to the objections above is worth making explicit: CTGP’s ontological thesis does not modify the dynamical equations of physics; it modifies the ontological interpretation of their solutions, while its Layer 3 hypotheses are stated separately as testable physics. The Einstein equations are compatible with both static and generative readings; CTGP argues that the generative reading is superior because it simultaneously respects relativistic causal structure, accommodates cosmological foliation, aligns with causal-set growth models, and provides a natural ontological location for temporal experience. At the level of Layers 1 and 2, the theory therefore functions not as a competing physical model but as an interpretive completion of relativistic spacetime — consistent with existing physics while providing an account of an additional observed phenomenon (the phenomenology of temporal experience) that the block universe ontology leaves unaddressed. ---------------------------------------------------------------------------------------------------14. Cosmological Records and the Persistence of Causal Information ---------------------------------------------------------------------------------------------------Within the CTGP framework, the physical universe does not merely evolve through time — it continuously encodes its own causal history in the structures that persist from one generated hypersurface to the next. This section examines how major cosmological observables function as precisely this kind of causal record, embedded in later states of M(σ) as physical consequences of earlier ones. 14.1 The Cosmic Microwave Background as Causal Record ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The cosmic microwave background (CMB) is the most precisely characterized causal record in observational cosmology. The thermal photons constituting the CMB were last scattered at the surface of last scattering approximately 380,000 years after the Big Bang, when the plasma of the early universe recombined and became transparent. The anisotropies in the CMB temperature field — density fluctuations at the level of one part in 100,000 — encode information about the baryon-photon plasma at that epoch, and have since propagated causally forward through cosmic expansion to be observed at the present hypersurface Σ_σ. Within CTGP, the CMB is not merely a probe of the past; it is literally a physical record embedded in the present state of the generated manifold M(σ). The past hypersurfaces at which last scattering occurred no longer exist as independently real spacetime regions — they are encoded causally in the radiation field now permeating the universe. The temperature map of the CMB is the universe’s causal receipt for the events of its first 380,000 years. 14.1.1 The 21 cm Hydrogen Line: The Volumetric Causal Archive
While the CMB provides a single, high-fidelity causal surface from the recombination epoch, the 21cm hyperfine transition line of neutral hydrogen (λ = 21.106 cm, ν = 1420.405 MHz) offers a uniquely powerful extension of CTGP’s causal continuity thesis and constitutes the most extensive known physical realization of Postulate 4 (Persistence Through Causal Continuity and Records). Before cosmic reionization (z ≳ 6), neutral hydrogen filled the intergalactic medium, the diffuse cosmic web between the first luminous structures; afterward it resides mainly in and around galaxies, which it uses as tracers of the large-scale matter distribution. Across both epochs it provides a threedimensional screen for mapping causal structure. The logical structure is precisely that demanded by CTGP’s ontology: the signal exists now; the emitting hydrogen existed earlier; the earlier state itself no longer exists as an independently real spacetime region under CTGP; what remains is not the earlier organization itself but the continuing propagation of its physical consequences, one subset of which functions as a causal record. This is not an analogy to CTGP’s framework — it is a direct physical instantiation of it. Within the CTGP framework, this signal is therefore not merely a probe of distant structure; it is a present-day physical encoding of the universe’s own generative history, tying the framework’s ontology, its treatment of the past, and the signal-based structure of causal records together in a particularly coherent way. Temporal layering: the observed frequency of the 21 cm line is redshifted (ν_{obs} = ν_{rest}/(1 + z)). A radio telescope observing across a continuous band (for example 50–200 MHz) is not observing different spatial locations but different stages of the generated domain M(σ): lower frequencies correspond to higher z, and hence to earlier stages Σ_σ. Whereas the CMB is a single two-dimensional surface (z ≈ 1100), 21 cm tomography can in principle reconstruct the neutral hydrogen distribution across the generated history from z ≈ 0 to z ≈ 200, providing a spatially resolved causal archive of how baryonic matter evolved under gravitational collapse. 14.2 Gravitational Waves as Records of Causal Events ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Gravitational waves propagate at the speed of light through spacetime curvature, carrying information about the dynamics of their source events: binary mergers, core-collapse supernovae, inflation-era phase transitions, and potentially the causal growth process of the early universe itself. Detected gravitational wave signals from binary black hole and neutron star mergers represent causal information propagated forward through billions of years of cosmic time. Within CTGP, the arrival of a gravitational wave signal at the present hypersurface is the causal encoding of a merger event that occurred at an earlier stage of M(σ) — a record of causal activity persisting in the radiation field of spacetime geometry itself. The cosmic gravitational wave background, expected to be detectable by next-generation space-based observatories, would constitute an even richer causal record: a superposition of gravitational-wave signals from astrophysical and cosmological sources across the history of the generated universe. 14.3 Large-Scale Structure Formation as Causal Continuity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The filamentary large-scale structure of galaxies and clusters observed in surveys such as the Sloan Digital Sky Survey and the Dark Energy Spectroscopic Instrument represents the gravitational amplification of density fluctuations seeded in the early universe. These fluctuations, encoded in the CMB anisotropy field, evolved through gravitational dynamics across cosmic history to produce the cosmic web observed today. This evolution is a paradigm case of causal continuity: the present distribution of matter in the universe is the direct causal consequence of initial conditions at
recombination, mediated by gravity, dark matter, and baryonic physics across billions of years of generated cosmic time. The correspondence between CMB anisotropies and present-day structure — quantitatively verified to high precision — indicates that the universe’s causal records are not merely qualitative but are precisely encoded and propagated. Within CTGP, this causal continuity between early and late cosmic states is not merely an empirical pattern but a structural consequence of how successive hypersurfaces are generated from prior ones through hyperbolic evolution. 14.4 Supernova Neutrinos and Relic Backgrounds ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The neutrino burst from Supernova 1987A, detected hours before the optical signal (Appendix E), provided a direct confirmation that neutrinos propagate causally from their source event to the present hypersurface. The diffuse supernova neutrino background — the superposition of neutrino emissions from all core-collapse supernovae across cosmic history — constitutes a diffuse causal record of stellar death events spanning billions of years. Similarly, the relic neutrino background from the first second of the universe, while not yet directly detected, represents the oldest surviving causal record of thermalized matter, encoding the conditions of the early hot dense state of the generated manifold. These diverse cosmological observables collectively indicate that CTGP’s claim about causal records is not merely philosophical but is concretely instantiated in the physical content of the observable universe: every major class of cosmological relic corresponds to a physical signal that has propagated causally forward from earlier states of M(σ) to the present hypersurface Σ_σ. 14.5 Record Formation as Environmental Redundancy ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The physical mechanism by which records form is decoherence. When a system interacts with its environment, correlations with the system’s state are copied into many environmental degrees of freedom; because entanglement is monogamous, the system’s coherence with any isolated partner is correspondingly diluted. What survives into later stages is therefore not a preserved earlier state but a redundant, environmentally distributed record of selected properties of it — the structure Zurek calls quantum Darwinism (Zurek 2003, 2009). This gives Postulate 4 a concrete physical basis. A causal record is a present correlation, multiply copied into the environment, whose structure is a lawful transformation of an earlier organization that no longer exists. Coherence is lost through specific environmental mechanisms at specific rates, and records are correspondingly finite, redundant, and degradable (§14.6). 14.6 Limits of the Causal Archive ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Crucially, causal propagation is selective and lossy. Record formation is governed by local physical dynamics — decoherence, amplification, thermalization — whose rates vary enormously across the universe: dense, strongly interacting regions both create and scramble records rapidly, while nearvacuum regions transmit signals over long distances with little distortion. The fidelity with which information about prior events is carried forward is therefore not uniform across spacetime. (1) Finite Encoding Capacity. The amount of information that can be carried forward from Σ_{σ′} to Σ_σ is limited by the physical degrees of freedom of intervening fields and their dynamical evolution. Processes such as thermalization, decoherence, gravitational mixing, and nonlinear
interactions progressively degrade fine-grained information. CTGP therefore suggests that most microphysical details of earlier stages will not be recoverable at later stages, even in principle, because their causal imprints are dispersed below any reconstructible threshold. (2) Coarse-Graining and Record Formation. What survives into Σ_σ are not full microstates of prior hypersurfaces, but coarse-grained invariants—stable, redundantly encoded features of the dynamics. Cosmological relics such as the cosmic microwave background, large-scale structure correlations, and conserved quantities function as high-fidelity carriers of such information. (3) Local Variability of Epistemic Access. Because record formation depends on local physical conditions, the efficiency of causal encoding varies across spacetime. Matter-dense regions may both generate and rapidly scramble information, while near-vacuum regions can preserve signals over long distances with minimal distortion. The epistemic accessibility of past events is therefore spatially heterogeneous, reflecting underlying physical conditions rather than any global indeterminacy. ---------------------------------------------------------------------------------------------------15. Relationship to Causal Set Theory and Growing-Block Models ---------------------------------------------------------------------------------------------------The conceptual structure of Cosmic-Time Generative Presentism shares significant similarities with several existing approaches in the foundations of spacetime physics while also differing from them in important respects. Understanding these relationships clarifies CTGP’s position within the broader landscape of temporal ontologies. 15.1 Growing-Block Universe Models ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Growing block models propose that the past and present exist while the future does not yet exist, with spacetime expanding as new moments are added to the existing structure. CTGP’s architecture is broadly continuous with this tradition, particularly with Ellis and Rothman’s crystallizing block universe (2010). The continuity is architectural rather than ontological: in CTGP, earlier states do not persist as ontologically existing regions of spacetime — it is precisely through this ontological absence that the causal records embedded in later physical states become the sole mode of the past’s reality. Information about prior events survives only through those physical records. The universe therefore functions not as an ever-accumulating spacetime block but as a sequence of generated states whose causal history is encoded in present physical structures. The distinction has formal consequences for the structure of the action, which is integrated only over M(σ) at each stage rather than over an accumulating manifold containing all past slices as co-existing regions. CTGP also goes beyond prior growing-block proposals by identifying the growth parameter with a geometric invariant, cosmological time, whose level sets remain Cauchy surfaces in the presence of collapsed structure. 15.2 Causal Set Theory ~~~~~~~~~~~~~~~~~~~~~~ Causal set theory (Bombelli et al. 1987; Sorkin 2003; Dowker 2005) represents spacetime as a discrete partially ordered set of causally related events, with spacetime geometry emerging from the underlying causal network. Classical sequential growth models (Rideout and Sorkin 2000) provide a dynamical framework in which the causal set grows by sequential addition of new elements. CTGP
shares with causal set approaches the emphasis on causal structure as fundamental to the organization of spacetime, and the treatment of temporal ordering as arising from causal connectivity rather than being imposed externally. Both frameworks treat causal ordering as primary and view spacetime structure as emerging from chains of physical interaction. CTGP can be understood as a continuum analogue of causal set cosmology: as discussed in §6.2, the continuous generation parameter σ may correspond, in an appropriate continuum approximation, to an averaged measure of generated causal structure — a motivating analogy rather than an established correspondence. One contrast deserves emphasis. Classical sequential growth dynamics is required to satisfy discrete general covariance: physical content must be independent of the order in which elements are born, so the birth order is itself a gauge labeling, and causal-set theorists generally read the resulting becoming as asynchronous, without a global now. CTGP’s physically calibrated global present is therefore an additional commitment, supplied by cosmological time in the continuum regime (§6.1), rather than something inherited from causal-set dynamics. CTGP differs from causal set theory in that it does not require spacetime to be fundamentally discrete. The framework remains fully compatible with the continuous spacetime manifolds of general relativity while interpreting their temporal structure as dynamically generated through the variational structure of σ. 15.3 Process-Based and Dynamical Approaches ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Several philosophical and physical frameworks treat reality as fundamentally processual rather than static. Smolin’s temporal naturalism (2013) argues that time is ontologically fundamental and that physical laws evolve, sharing with CTGP the rejection of a completed four-dimensional block as the primary ontological category. Process cosmology in the tradition of Whitehead treats reality as composed of processes of becoming rather than static substances, a commitment that maps naturally onto CTGP’s generative hypersurface structure. Penrose’s conformal cyclic cosmology and various quantum gravity proposals also invoke dynamical spacetime generation, albeit with different formal structures. CTGP aligns with these process-oriented approaches in emphasizing the generative character of physical reality while providing a precise formal framework grounded in standard general relativity. 15.4 The Distinguishing Integration of CTGP ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The distinguishing feature of CTGP lies in its integration of three elements that are typically treated separately in the existing literature. First, a generation parameter grounded in relativistic causal structure: CTGP identifies the growth parameter with cosmological time, fixed by the spacetime’s causal and metric structure and encoded in a constrained action. Second, a temporal ordering that is observationally accessible: in FLRW the level sets of cosmological time are the surfaces of constant cosmic time, so the generation parameter is realized by a quantity cosmology already measures, without introducing a preferred frame beyond what is already present in standard cosmology. Third, persistence of past information through physical records: CTGP’s account of causal records embedded in successive present states provides a precise mechanism for how the past remains accessible despite no longer existing as an independent spacetime region. Together these components provide a framework in which temporal passage, causal structure, and cosmological evolution are treated as aspects of a single underlying generative process rather than as separate explananda requiring separate theoretical treatments. CTGP is compatible with all of these, and explicitly incorporates their formal resources where
appropriate. Its distinctive contribution, however, is something none of them provides: most quantum gravity growth models describe the growth of causal relations — the addition of ordered pairs of events, the extension of the causal partial order. CTGP adds to this the growth of the experiential present. The present hypersurface Σ_σ is not merely the most recently added stratum of a growing causal structure; it is the ontological locus of qualia, the active edge at which physically instantiated becoming constitutes experience. This connects physics and phenomenology in a way that causal set theory, temporal naturalism, and process cosmology do not attempt: alongside a formal account of how generation proceeds, CTGP specifies where within that structure experience occurs. It does not explain why physically instantiated becoming is accompanied by experience at all; Appendix D states plainly that the framework solves neither the hard problem nor the mapping problem, and the core ontology does not depend on it. The quantum gravity frameworks handle the former; CTGP handles both. The connection between the growth edge and the experiential edge is not a separate postulate layered onto the physics but a consequence of what the growth edge is. In causal set theory and in CTGP’s continuum analog, the present hypersurface Σ_σ is distinguished from all past hypersurfaces by a single structural fact: it is the only stratum at which causal dynamics are actively instantiated rather than recorded. Past hypersurfaces are represented within M(σ) as completed causal structure — fixed, encoded, no longer undergoing lawful evolution. Σ_σ alone is where the evolution equations are being applied, where fields are being propagated, where the Cauchy problem is being solved. If qualia require anything at all, they require a physical process that is occurring, not one that has occurred. The experiential edge coincides with the growth edge not because CTGP stipulates it but because only the growth edge satisfies the necessary condition of being an active, non-completed physical process. Past slices, whose reality now consists entirely in present causal records, are inert with respect to ongoing dynamics; Σ_σ is not. The localization of experience at the present edge is therefore not an additional postulate but a consequence of applying the Representation–Instantiation Distinction (§D.4) to the temporal structure of M(σ) itself: past hypersurfaces represent completed structure, while the physical configuration on the present hypersurface instantiates ongoing becoming. ---------------------------------------------------------------------------------------------------16. Conclusion ---------------------------------------------------------------------------------------------------CTGP articulates a generative presentist ontology: earlier stages persist only through their consequences and records, M(σ) is the formal representation of the present’s past-tensed profile rather than an accumulating region, and the physical configuration instantiated on the present hypersurface Σ_σ is the whole of what exists. The framework preserves local relativistic physics while interpreting GR’s initial-value structure as genuine becoming. It begins from a phenomenological constraint — that an adequate ontology of time must sustain the conditions under which genuine temporal becoming is possible — and develops this constraint into a formal framework grounded in cosmological time and ADM evolution. CTGP provides a unified ontological account of three classes of phenomena that static spacetime ontologies explain through separate structures. First, experiential temporal flow: the felt asymmetry between past and future, the sense of genuine succession, and the directedness of conscious experience are grounded in the continuously advancing generative flow at Σ_σ — the smooth
propagation of the present boundary along n^μ — rather than being explained away as cognitive artifacts of observer perspective within a completed block. Second, cosmological time: the natural temporal ordering provided by FLRW cosmic expansion, the matter frame, and the CMB rest frame are not merely convenient coordinate choices but physical expressions of the generation parameter σ that parameterizes the progression of the present boundary. Third, the persistence of physical records: the cosmic microwave background, large-scale structure, gravitational wave signals, and cosmological relics function as causal records embedded in successive present states — physical information propagated forward from earlier stages of M(σ) to the present hypersurface Σ_σ. The theory reframes spacetime not as a static block in which past, present, and future co-exist as equally real regions of a completed manifold, but as a continuously generated causal structure whose present boundary advances through cosmological time. This reframing is not a modification of the Einstein field equations but a different ontological interpretation of the same mathematical formalism — one that accounts for the phenomenological datum of temporal experience that the block universe leaves unexplained, and whose empirical exposure lies in a structural constraint on the form of physical law rather than in any measurement of the ontology itself. CTGP’s most distinctive contributions relative to prior growing-block models are: (a) the identification of the generative present with the level sets of cosmological time, a Cauchy foliation that survives caustic formation and approximately coincides with the cosmological rest frame; (b) the grounding of the generation parameter in standard cosmic time, with an explicit account of its admissible relabelings; (c) the account of cosmological observables as causal records embedded in successive present states, with decoherence as the mechanism of record formation; (d) the explicit treatment of qualia as intrinsic to physically instantiated processes at Σ_σ; (e) a consistency analysis establishing which generation laws are dynamically admissible, together with a precise statement of the ontology’s empirical standing — observationally equivalent to eternalism for fixed physics, yet exposed through the requirement that physical law be generable from the present; and (f) causality is structurally preserved, not abandoned. CTGP rejects the claim that quantum gravity necessarily violates causality. Instead, it treats causality as a structural feature of the generated domain M(σ), compatible with standard relativistic quantum field theory at tested scales and with causal set theory as an especially natural quantum-gravity companion framework. The future remains ontologically ungenerated rather than merely epistemically inaccessible, and no retrocausal influences are required or predicted by the framework. Two results of the paper deserve emphasis because they are negative. First, the rate of generation cannot be tied to local matter density: such laws break the alignment of the present with matter within a light-crossing time of any structure, and their stable completions suppress every cosmological signature below observability. Second, no measurement can distinguish presentism from eternalism when the physics is held fixed. Neither result weakens the ontology; both clarify where it stands. The costs of the view are explicit. It gives ontological significance to a foliation selected by the actual solution rather than by the laws; it applies only to globally hyperbolic spacetimes with regular cosmological time; it posits primitive past-tensed properties; and it requires singleoutcome realism in quantum theory. Cosmological time supplies the ordering of the present and general relativity its lawful succession, but becoming itself remains the posit that CTGP adds to both. Future work includes: (i) extending surface-relative time (§6.10.1) beyond exact FLRW, including
determining when a unique maximal Cauchy surface exists in non-uniform bouncing cosmologies, and treating past-eternal, cyclic, and emergent cosmologies, for which neither cosmological time nor surface-relative time supplies the present; (ii) a quantum-gravity account of the primitive ordering, including whether it must admit indefinite causal order; (iii) connecting CTGP’s generative structure to concrete discrete dynamics, particularly causal sets; (iv) a sharper assessment of the evidential bearing of the generability constraint; and (v) development of the psychophysical program of Appendix D. CTGP is a coherent generative ontology, preserving general relativity and compatible with standard relativistic quantum theory in its tested domain, argued on explanatory grounds and exposed to refutation through the one structural commitment presentism cannot relinquish: that the world is made, moment by moment, from what exists now. ---------------------------------------------------------------------------------------------------17. Formal Theorems and Propositions ---------------------------------------------------------------------------------------------------Theorem 17.1 (Generated-domain causal closure). Let (M, g) be globally hyperbolic. By the Geroch splitting theorem (Geroch 1970), M admits a foliation by Cauchy hypersurfaces Σ_σ with M ≅ ℝ × Σ. Define M(σ) = ⋃_{σ′<σ} Σ_{σ′}. Then for any p ∈ Σ_σ, the causal past J−(p) ∩ M is contained in M(σ). Thus the generated domain is causally closed with respect to the present hypersurface. Sketch of proof. In globally hyperbolic spacetimes, each inextendible causal curve intersects each Cauchy surface exactly once. For p on Σ_σ, any point q in J−(p) lies on Σ_{σ′} for some σ′ ≤ σ; hence q ∈ M(σ). □ Proposition 17.2 (Well-posed growth step). Given ADM initial data (h_{ij}, K_{ij}, matter data) satisfying Hamiltonian and momentum constraints on Σ_σ, Einstein-matter evolution determines (locally in σ) a unique development up to diffeomorphism. The temporal ordering implicit in this initial-value structure is inherited from the causal structure of spacetime itself. Corollary 17.3 (No future boundary dependence). The development depends on initial data and evolution equations, not on boundary conditions at σ′ > σ. CTGP interprets this as supporting nonteleological generation: the future is not consulted. Theorem 17.4 (Cosmological time; Andersson, Galloway, and Howard 1998). Let τ(p) = sup{L(γ) : γ a past-directed causal curve from p}, and suppose τ is regular: finite everywhere and tending to zero along every past-inextendible causal curve. Then (i) the spacetime is globally hyperbolic; (ii) τ is a time function, continuous and strictly increasing along every future-directed causal curve; (iii) every level set of τ is a Cauchy surface; (iv) τ is locally Lipschitz; and (v) every event p lies on a timelike geodesic from the initial singularity whose length equals τ(p). Wherever τ is differentiable, −∇^μτ is the future-directed unit tangent of that geodesic, so g^{μν}∂_μτ∂_ντ = −1. Proposition 17.5 (Existence and monotonicity of σ). (1) Existence: if cosmological time is regular, then σ = F(τ), with F strictly increasing, is a time function whose level sets are Cauchy surfaces, and ∇_μσ is timelike wherever τ is differentiable. (2) Matter alignment: in FLRW, τ is cosmic time and the level sets of σ are orthogonal to the comoving matter congruence; beyond FLRW, alignment holds with the free fall of pressureless matter in single-stream regions at linear order. (3)
Monotonicity: σ is strictly increasing along every future-directed causal curve, independently of energy conditions. Proposition 17.6 (Matter-constrained foliation expansion). Let u^μ be the timelike congruence aligned with the cosmological matter flow, and let θ = ∇_μu^μ denote its expansion scalar. In globally hyperbolic, irrotational cosmological spacetimes satisfying the Einstein equations, the evolution of θ is governed by the Raychaudhuri equation: dθ/dτ = −(1/3)θ² − σ_{μν}σ^{μν} + ω_{μν}ω^{μν} − R_{μν}u^μu^ν + ∇_μa^μ. In the irrotational, geodesic cosmological limit this reduces to dθ/dτ = −(1/3)θ² − σ_{μν}σ^{μν} − R_{μν}u^μu^ν. Under standard energy conditions, the matter content constrains the sign and evolution of θ, and hence constrains the geometric extension of the CTGP foliation. The expansion scalar θ = d(ln √h)/dτ provides the physically grounded local measure of how the present hypersurface is extending relative to neighboring hypersurfaces in the generated sequence. Proposition 17.7 (Variational Derivation of the Norm Condition). The norm condition ∇_μσ ∇^μσ = −f(σ) is not postulated but derived. Consider the CTGP action S[g_{μν}, σ, λ] = ∫_{M(σ)} √(−g) [ L_{GR} + λ(∇_μσ ∇^μσ + f(σ)) ] d⁴x where λ(x) is a Lagrange multiplier scalar field and f(σ) > 0 is the generation law. The three Euler–Lagrange conditions are as follows. (i) Variation with respect to λ: δS/δλ = 0 gives ∇_μσ ∇^μσ + f(σ) = 0, i.e., ∇_μσ ∇^μσ = −f(σ). This is the norm condition, established as a necessary condition for stationarity of S. (ii) Variation with respect to σ: since ∂/∂σ of the Lagrangian is λf′(σ) and ∂/∂(∂_μσ) is 2λ∇^μσ, integrating by parts and discarding the boundary term (which vanishes on ∂M(σ) by the causal closure of the generated domain, Theorem 17.1) yields the propagation equation ∇_μ(λ∇^μσ) = ½λf′(σ). It is linear and homogeneous in λ, reduces to a conservation law for the current λ∇^μσ when f is constant, and transports λ along the generation flow; λ = 0 on an initial surface therefore implies λ ≡ 0. (iii) Variation with respect to g^{μν}: the L_{GR} term yields G_{μν} = 8πG T_{μν} as usual. The constraint term contributes T^{(σ)}_{μν} = 2λ∇_μσ∇_νσ − λg_{μν}(∇_ρσ∇^ρσ + f), up to the sign convention adopted for T_{μν}. On the constraint surface the second term vanishes, leaving T^{(σ)}_{μν} = 2λ∇_μσ∇_νσ: a pressureless fluid flowing along ∇^μσ with energy density proportional to λf — the structure familiar from mimetic gravity (Chamseddine and Mukhanov 2013). Because f depends only on σ, no term couples the constraint sector to the matter equations. For λ ≡ 0, the Einstein–matter system is exactly that of general relativity. The significance of this proposition is methodological as much as technical. Any framework that introduces a new field equation faces the challenge of justifying why that equation holds. Lagrange multiplier actions are a well-established method for implementing constraints without the constraints being arbitrary: the multiplier enforces the condition exactly, and the condition itself becomes a consequence of the variational principle. CTGP’s adoption of this structure places the σ kinematic equation on the same foundational footing as every other equation of motion in fundamental physics — it holds because the action is stationary, not because it is assumed. This structure answers the objection that σ’s equation of motion is ad hoc. □
Proposition 17.8 (Geometric invariance of the generative present). Let Φ : (M, g) → (M′, g′) be an isometry between spacetimes with regular cosmological time. Then τ′ ∘ Φ = τ. Consequently, the generative present is fixed by the geometry and is not a gauge choice: diffeomorphisms that preserve the metric map its stages to stages, and no relabeling of coordinates can alter which events are copresent. The only freedom is the labeling σ = F(τ), which leaves the stages unchanged. Proof. Isometries map causal curves to causal curves and preserve Lorentzian length, so the supremum defining τ′(Φ(p)) ranges over the images of exactly the curves defining τ(p). □ Theorem 17.9 (Stage-Relative Representation). Let M(σ) = ⋃_{σ′ < σ} Σ_{σ′} be the domain of the CTGP action S(σ) (§9.1). If (1) σ is monotonic, (2) Σ_σ obeys hyperbolic evolution, and (3) future hypersurfaces are excluded from the domain, then the family {M(σ)} is nested, M(σ′) ⊂ M(σ) for σ′ < σ, and each M(σ) is determined by data on Σ_σ and its causal past alone. The formalism therefore represents reality stage-relatively, without modifying the field equations. This representation is not by itself inequivalent to a completed manifold: a block theorist may read each M(σ) as a region of a pre-existing spacetime. Whether the nested family is a growing whole or a sequence of regions of a fixed whole is settled by an ontological assumption about the existence of future stages, not by the mathematics. CTGP adopts the first reading; the theorem shows that its formalism expresses that reading consistently, not that the field equations or observations favor it (§12.1). Proposition 17.10 (Domain Restriction and Local Dynamics). Let S_{block} = ∫_M √(−g) L_{GR} d⁴x and S_{CTGP}(σ₀) = ∫_{M(σ₀)} √(−g) [L_{GR} + λ(∇_μσ ∇^μσ + f(σ))] d⁴x. (i) At every point of M(σ₀), the Euler–Lagrange equations of S_{CTGP} with λ = 0 coincide with those of S_{block} together with the norm condition. (ii) M(σ₀) has a future boundary Σ_{σ₀} and M does not, so the two domains are not diffeomorphic as manifolds with boundary. The domain restriction is therefore not a difference in local dynamics or in any observable (Proposition 12.1). It is the formal expression of the stagerelative ontology, and whether the future boundary marks the edge of reality or only the edge of a description is an interpretive question. Proof. Part (i) holds because Euler–Lagrange equations are local and the constraint sector contributes nothing when λ = 0. Part (ii) holds because diffeomorphisms map boundary points to boundary points. □ Corollary 17.11 (A Formal, Not Empirical, Distinction). The CTGP and block-universe variational principles differ in domain but not in local field equations or observable consequences. The distinction they encode is ontological: whether the domain of the laws is a growing entity or a fixed manifold. Empirical bearing on that distinction comes only through the generability constraint (§12.2). Definition 17.12 (Geometric Growth Measure). Let u^μ be the timelike congruence aligned with the cosmological matter flow, and let θ = ∇_μu^μ denote its expansion scalar. CTGP identifies θ as the physically meaningful local measure of geometric growth: it equals d(ln √h)/dτ and thus measures the fractional rate of change of local hypersurface volume along the matter congruence. Rather than defining ontological growth as dM/dσ — which treats the generated domain as if it were a scalar quantity — CTGP uses the foliation expansion to characterize growth. The evolution of θ is governed
by Proposition 17.6; it is constrained directly by matter content through the Einstein equations. Cosmologically, this yields a physically meaningful picture: in the early dense universe, θ is large (rapid geometric expansion); in the matter-dominated and late universe, the Raychaudhuri equation predicts the observed deceleration and re-acceleration through the curvature and energy-condition terms, without requiring the parameter dM/dσ to be interpreted as a rate of literal spacetime production. Proposition 17.13 (Compatibility with Causal-Set Quantum Gravity). Let a discrete causal set (C, ≺) be generated via sequential growth dynamics in the sense of Rideout and Sorkin (2000). Assume the continuum limit of (C, ≺) yields a globally hyperbolic spacetime (M, g) with a smooth temporal function σ whose level sets are spacelike Cauchy hypersurfaces (Bernal–Sánchez). Then: (1) the directional causal structure — past ⊂ M(σ), future ∉ M(σ) — is preserved in the continuum limit; (2) every element in the causal set has a finite causal past at any finite stage of growth, and no future elements are present at that stage; (3) the element-by-element addition of causal elements in sequential growth dynamics represents one possible discrete realization of CTGP’s fundamentally continuous generative flow, whose large-N limit may correspond to the continuum σ (§6.2). The direction of compatibility runs from CTGP’s continuous generative-flow ontology to causal-set models as discrete instantiations, not the reverse. Therefore, CTGP’s commitment to causality and the ontologically open future is compatible with at least one well-defined approach to quantum gravity. The claim that quantum gravity necessarily violates causality is accordingly a contingent interpretive claim, not a result established by any current quantum-gravity framework with empirical support. Remark: This result is stated as a compatibility proposition rather than a theorem because it depends on the unestablished identification of causal set theory as the correct theory of quantum gravity. Its purpose is to demonstrate that CTGP’s causal commitments are not undermined by quantumgravity considerations in at least one well-developed approach. Because sequential growth dynamics is label-invariant, the birth order of elements is not itself physical; the compatibility asserted here concerns the directional causal structure of (1) and (2), not the identification of a global present, which CTGP supplies separately through cosmological time (§15.2). Theorem 17.14 (Constraint Stability under CTGP Perturbations). Let U = (h_{ij}, K_{ij}, matter) denote ADM variables evolving under ∂_t U = ℰ(U) + ε 𝔻(U), where ℰ is the standard Einstein–matter evolution and ε 𝔻 represents CTGP-induced perturbative corrections. Define the constraint vector C = (ℋ, ℋ_i). Then the constraint propagation system satisfies ∂_t C = A^i(U)∂_i C + B(U)C + ε S(U). For Sobolev index s > 5/2 and vanishing initial constraints C(0) = 0, ‖C(t)‖_{Hs-1} ≤ ε e^{Kt} ∫_0^t ‖S(U(τ))‖_{Hs-1} dτ. Constraint violation therefore remains O(ε) in Sobolev norm on finite evolution intervals. If the source term S(U) lies within the standard gauge sector, the violation is absorbable by lapse and shift adjustment. The appropriate statement is a Sobolev-norm control bound rather than a fixed numerical tolerance: CTGP-compatible perturbations preserve constraint structure to leading order. □ Theorem 17.15 (Constraint-Compatible Foliation Gluing). Let U^{(1)} and U^{(2)} be solutions of the Einstein–matter system in Sobolev class H^s (s > 5/2 + 1) on overlapping globally hyperbolic charts V_1, V_2 whose initial data agree on the overlap up to order s. Let χ_1, χ_2 be a partition of unity subordinate to V_1, V_2 and define blended initial data U_0 = χ_1 U_0^{(1)} + χ_2 U_0^{(2)}. After projection onto the constraint manifold via the conformal method, provided the constraint mismatch on the overlap is sufficiently small in H^{s-1}, there exists T > 0 and a unique local solution U ∈
C([0,T], H^s) ∩ C^1([0,T], H^{s-1}) with stability bound ‖U − U^{(a)}‖_{Hs-1(V′a)} ≤ C_T ‖overlap mismatch‖_{Hs-1(V1 ∩ V2)} on each subpatch V′_a ⋐ V_a. The gluing claim is thereby a standard PDE construction: local CTGP foliations may be patched only through overlap data first reconciled at the level of the Einstein constraint equations, after which the hyperbolic evolution applies to the corrected blended data. □ Proposition 17.16 (Conditional Tomonaga–Schwinger Integrability). Let {Σ} be a family of smooth spacelike Cauchy hypersurfaces in a globally hyperbolic spacetime (M,g). Let the density operator ρ[Σ] evolve under δρ[Σ]/δΣ(x) = ℒ_x(ρ[Σ]), where ℒ_x is a local generator at x ∈ Σ. Assume: (1) Microcausality — [ℒ_x, ℒ_y]ρ = 0 for spacelike-separated x, y on a common dense invariant domain D; (2) Energy-boundedness — each smeared generator ℒ(f) = ∫_Σ f(x) ℒ_x dΣ is closable on D and satisfies ‖ℒ(f)ψ‖ ≤ a‖H_0 ψ‖ + b‖ψ‖ with a < 1; (3) Strong continuity of the generated propagators on D. Then the evolution operator between two hypersurfaces depends only on the spacetime region swept out, not the intermediate foliation: ρ[Σ_2] = U(Σ_2, Σ_1) ρ[Σ_1] is foliation-pathindependent. No superluminal signaling is introduced. CTGP requires only collapse densities built from local commuting field polynomials or smeared number/energy densities with spacelike-separated supports commuting in the AQFT sense; this admissible class satisfies the conditions above, making hypersurface-path independence a conditional theorem rather than an assumed property. □ Theorem 17.17 (Continuum Limit of Local CP Evolution). Let ℒ^{(n)} be a sequence of Lindblad generators built from bounded truncations L_α^{(n)} of a local operator family L_α with UV cutoff Λ_n → ∞. Assume: (1) L_α^{(n)} → L_α strongly on a common dense domain D; (2) ∑_α L_α† L_α ≤ c_1 H_0 + c_2 I relative to a positive reference Hamiltonian H_0; (3) truncated semigroups e^{tℒ(n)} are CPTP; (4) initial states satisfy Tr(ρ H_0) < ∞. Then for each finite interval [0,T]: sup_{t ≤ T} ‖e^{tℒ(n)}ρ − e^{tℒ}ρ‖_1 → 0 as n → ∞. The error bound is ‖e^{tℒ(n)}ρ − e^{tℒ}ρ‖_1 ≤ C_T(ε_n(ρ) + δ_n), where ε_n(ρ) = Tr(ρ Π_{>Λn}) is the UV tail measure and δ_n is the truncation error. The theorem gives the finite-mode-to-continuum passage the form of a trace-norm convergence statement, supplying the reference Hamiltonian, common domain, UV tail parameter, and finite-time control that make the claim a genuine mathematical program. □ Proposition 17.18 (Emergent Status of σ Near Planck Scale). For curvature invariants satisfying |R_{μνρσ}| ≪ M_{Pl}^2, σ exists as a smooth temporal scalar field whose level sets are spacelike Cauchy hypersurfaces, given by cosmological time as in §§6.1–6.3. Near Planck curvature, the smooth representation may break down; CTGP then treats σ as the continuum image of an underlying growth ordering variable, with causal-set counting as the preferred candidate discrete realization. CTGP is therefore fundamental at the ordering level, not necessarily at the smooth-manifold level: the smooth σ-field is emergent in the classical regime, and singular classical regions mark breakdown of the continuum representation rather than breakdown of temporal ordering itself. This avoids a false dichotomy between “σ is fundamental as a smooth field everywhere” and “CTGP fails wherever classical GR fails.” □ Proposition 17.19 (Bridge-Model Identifiability Conditions). Let q = Θ F(X) represent a candidate mapping from physiological state vector X to phenomenological proxy q. The mapping is empirically meaningful for CTGP’s psychophysical research program only if: (1) it generalizes to held-out
subjects not used in fitting; (2) it remains predictive after nuisance regression controlling for arousal, task difficulty, medication status, and global signal; (3) it retains predictive power under causal perturbation (TMS, anesthesia, sleep-stage shifts, focal lesions); (4) nested model comparison suggests CTGP-style substrate-sensitive models outperform purely functional baselines; (5) a primary endpoint is pre-registered. These criteria convert the mapping problem from an unfalsifiable metaphysical claim into a testable research program. Note that identifiability in this strong sense is required for the psychophysical extension of Appendix D but is not a prerequisite for the core spacetime ontology of the main text, which stands on its own physics and philosophy-ofphysics merits. □ Definition 17.20 (Tense Semantics). Let W_σ denote the present world at stage σ, and for σ′ < σ let Had(σ′, φ) denote the primitive property *having been, at stage σ′, such that φ* (§5.2.1), where φ ranges over non-tensed propositions. (a) *Truth condition.* “At σ′ it was the case that φ” is true at σ iff W_σ instantiates Had(σ′, φ). (b) *Accuracy condition.* For every σ′ < σ, W_σ instantiates Had(σ′, φ) iff W_σ′ ⊨ φ. (c) *Representation.* M(σ) = ⋃_{σ′<σ} Σ_{σ′} is the formal representation of the history carried by the present world’s past-tensed profile. For each σ′ < σ, the represented stage Σ_{σ′} satisfies φ in M(σ) exactly when W_σ instantiates Had(σ′, φ). (d) *Asymmetry.* W_σ instantiates no property grounded in a future stage, since no stage Σ_{σ″} with σ″ > σ exists. Corollary 17.21 (Nesting). For σ < σ″, M(σ) ⊂ M(σ″). Proof. Every stage in ⋃_{σ′<σ} Σ_{σ′} also lies in ⋃_{σ′<σ″} Σ_{σ′}. Nesting is a property of the formal representation; it does not imply that earlier stages continue to exist. □ Proposition 17.22 (No Revision of the Past). For σ < σ″ and every σ′ < σ, Σ_{σ′} satisfies φ in M(σ) iff Σ_{σ′} satisfies φ in M(σ″). Proof. By (c) and (b), Σ_{σ′} satisfies φ in M(σ) iff W_σ instantiates Had(σ′, φ) iff W_σ′ ⊨ φ. The same chain holds with σ″ in place of σ. □ Proposition 17.22 is the formal counterpart of the claim in §5.2.1 that losing a record does not change what happened. Records can degrade between σ and σ″; the represented history of every earlier stage cannot. Proposition 17.23 (Surface-Relative Cosmological Time). Let (M, g) be globally hyperbolic and S a spacelike Cauchy surface, and let τ_S be the signed Lorentzian distance from S (§6.10.1). Then on each side of S, τ_S is the cosmological time of that region with S as its past (respectively future) boundary; it is finite, tends to zero on approach to S, and its level sets are Cauchy surfaces. Consequently σ = F(τ_S), with F strictly increasing, is an admissible generation parameter through S. In an exact FLRW universe with a single bounce, taking S to be the surface of zero mean curvature at the bounce makes τ_S cosmic time measured from the bounce.
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CTGP's σ-change of variables. A.1 ADM Variables and Constraints ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ g_{μν} → (h_{ij}, N, N^i); K_{ij} = (1/2N)(∂_t h_{ij} − ∇_i N_j − ∇_j N_i) Hamiltonian constraint: H = R^{(3)} + K² − K_{ij}K^{ij} − 16πG ρ = 0 Momentum constraint: H_i = ∇_j (K^j_i − δ^j_i K) − 8πG j_i = 0 A.2 Evolution Equations ~~~~~~~~~~~~~~~~~~~~~~~ ∂_t h_{ij} = 2N K_{ij} + ∇_i N_j + ∇_j N_i ∂_t K_{ij} = −∇_i ∇_j N + N( R^{(3)}_{ij} + K K_{ij} − 2K_{ik}K^k_j ) + (matter terms) A.3 Cosmological Time and Its Relabelings ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The discussion in §6.7 establishes that the present hypersurfaces are Cauchy surfaces whose induced data satisfy the ADM constraints and evolve consistently under the Einstein equations. This appendix records the explicit form of the generation parameter in cosmological settings. In FLRW with a big-bang singularity, cosmological time is cosmic time: τ = t. A relabeling σ = F(t) with F′ > 0 converts time derivatives by ∂_σ = (1/F′) ∂_t; for the canonical choice F = identity, σ = t. A.3a Well-Definedness, Dimensions, and Reparameterization of σ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Three properties of the generation parameter are relied upon elsewhere and are stated explicitly here. The first is well-definedness. Cosmological time is defined as a supremum over all pastdirected causal curves, not by integration along a chosen congruence, so no path-independence condition is required: τ(p) is well defined whenever it is finite. Its level sets are Cauchy surfaces when τ is regular (Theorem 17.4). A rotating cosmology such as Gödel’s dust solution admits no regular cosmological time, consistent with the global hyperbolicity requirement of §6.1. The second is dimension. For the canonical choice σ = τ, σ carries dimensions of time; for other relabelings, its dimensions are those of F. The two are related monotonically, induce the same foliation, and differ only in normalization and units. Statements that σ “corresponds to” cosmological proper time are to be read in this sense. The third is reparameterization. Any strictly increasing F yields an equally admissible parameter,
so σ is fixed only up to monotonic relabeling. This is harmless for every claim the framework makes, because those claims depend on the level sets of σ and the direction of ∇_μσ, both of which are relabeling-invariant. The magnitude |∇_μσ| = F′ is conventional and carries no physical content. For reference, the chain from geometry to foliation is as follows: g_{μν} → causal structure → τ, the maximal elapsed proper time, defined without choice → Σ_τ, Cauchy surfaces when τ is regular → σ = F(τ), a labeling convention. The matter congruence enters only as a consequence: in cosmology, the maximizing geodesics are the comoving worldlines. A.4 σ-Evolution of ADM Equations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Substituting ∂_t = F′ ∂_σ into ADM evolution, with the lapse normalized to cosmic time: ∂_σ h_{ij} = (2N/F′) K_{ij} + (1/F′)(∇_i N_j + ∇_j N_i) ∂_σ K_{ij} = (1/F′)(standard ADM right-hand side) A.5 Expansion Scalar and Raychaudhuri Calibration ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Let u^μ be the timelike unit vector field aligned with the matter congruence. The expansion scalar θ = ∇_μu^μ satisfies θ = d(ln √h)/dτ, measuring the fractional rate of local volume change of the foliation along the congruence. Its evolution is governed by the Raychaudhuri equation: dθ/dτ = -(1/3)θ² - σ_{μν}σ^{μν} + ω_{μν}ω^{μν} - R_{μν}u^μu^ν + ∇_μa^μ In the irrotational, geodesic cosmological limit (ω_{μν} = 0, a^μ = 0) this reduces to: dθ/dτ = -(1/3)θ² - σ_{μν}σ^{μν} - R_{μν}u^μu^ν CTGP uses this result to interpret geometric growth in terms of foliation expansion rather than as a derivative dM/dσ of the generated domain treated as a scalar quantity. The expansion scalar θ provides the GR-native local measure of how the present hypersurface is extending relative to neighboring hypersurfaces: θ > 0 in expanding cosmologies, θ < 0 in collapsing regimes. Via the Einstein equations, the R_{μν}u^μu^ν term links the evolution of θ directly to the matter content through the Ricci curvature, grounding CTGP’s growth picture in standard relativistic dynamics. ---------------------------------------------------------------------------------------------------Appendix B. Causal Structure Theorems and Their Application to CTGP ---------------------------------------------------------------------------------------------------B.1 Global Hyperbolicity and Causal Order ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A spacetime (M, g) is globally hyperbolic if it admits a Cauchy hypersurface — a spacelike surface intersected exactly once by every inextendible causal curve. This property is widely believed to hold for our universe on cosmological scales (Hawking and Ellis 1973).
Theorem B.1 (Hawking–King–McCarthy, 1976). In suitably well-behaved relativistic spacetimes, the chronological relation I+(p) — the set of events reachable from p by future-directed timelike curves — determines the conformal structure of the spacetime metric. The topology and light-cone structure can be reconstructed uniquely up to an overall conformal factor from knowledge of the chronological order alone. Theorem B.2 (Malament, 1977). The causal ordering relation alone suffices to determine the conformal metric structure of relativistic spacetime under mild conditions. Temporal precedence is not a coordinate convention but a geometric fact encoded in causal relations. Theorem B.3 (Geroch, 1970). In a globally hyperbolic spacetime there exists a continuous time function — a function that strictly increases along every future-directed causal curve. Moreover, M ≅ ℝ × Σ, where each slice is a Cauchy hypersurface. This splitting is not imposed but follows from the causal structure. Theorem B.4 (Bernal and Sánchez, 2003, 2005). In any globally hyperbolic spacetime, there exists a smooth temporal function whose gradient is everywhere timelike and whose level sets are smooth spacelike Cauchy hypersurfaces. Any Geroch splitting can be refined to a smooth one. Moreover, any two smooth Cauchy foliations can be smoothly deformed into each other. B.2 Application to CTGP ~~~~~~~~~~~~~~~~~~~~~~~ These theorems guarantee that the mathematical structure CTGP requires — a global foliation by present hypersurfaces Σ_σ with σ increasing causally — exists in any globally hyperbolic spacetime. This is not an additional CTGP postulate but a consequence of GR’s causal structure under physically reasonable assumptions. Standard interpretations treat the entire product manifold ℝ × Σ as ontologically complete. CTGP instead interprets the parameter labeling the slices as indexing the progressive generation of spacetime structure: at stage σ, only the present slice exists; the hypersurfaces in its past belong to the formal history M(σ), and the future hypersurfaces Σ_{σ′>σ}, while mathematically definable within the full spacetime manifold, are not part of the generated domain. The freedom to choose among smooth temporal foliations corresponds to the freedom to pick different smooth temporal functions. CTGP resolves this ambiguity with cosmological time: among all time functions whose level sets are Cauchy surfaces, it is singled out by maximal elapsed proper time, and in cosmology it coincides with the matter rest frame. The logical priority is clear: causal geometry (D.1–D.4) precedes cosmological time (Theorem 17.4), which precedes the labeling convention σ = F(τ). σ is not a novel object introduced by CTGP; it is a distinguished Cauchy time function that general relativity already provides. ---------------------------------------------------------------------------------------------------Appendix C. Vulnerability Assessment ---------------------------------------------------------------------------------------------------The three principal vulnerabilities of CTGP concern foliation dependence, underdetermination with respect to eternalism, and the use of a preferred cosmological frame. The responses to each are
developed in the main text (§§6.10, 11.4, 12, 9.3.1); this appendix records the residual assessment. After these responses, the residual severity of each vulnerability is as follows. The foliation criticism is answered: the present is geometrically unique wherever cosmological time is regular, aligned with matter in cosmology, and globally defined through structure formation; the open cases are past-eternal, cyclic, and emergent cosmologies and non-uniform bounces, with single uniform bounces covered by §6.10.1. The underdetermination criticism is correct and is accepted in its precise form: the ontology is observationally equivalent to eternalism for fixed physics, and its empirical exposure is structural rather than observational (§12). The preferred-frame criticism is answered in its usual form, since cosmology already employs the same frame and CTGP predicts no local preferred-frame effects. The deep philosophical challenge that remains is whether a generative ontology is more than an interpretive preference. That question cannot be settled by formal argument or by measurement alone; it belongs to the broader debate over scientific realism, underdetermination, and inference to the best explanation. CTGP’s contribution is to make the debate concrete: to specify the present precisely, to show which physical realizations of it are consistent, and to identify the structural commitment through which the ontology remains exposed to evidence. ---------------------------------------------------------------------------------------------------Appendix D. CTGP and Conscious Experience ---------------------------------------------------------------------------------------------------The relationship between CTGP's temporal ontology and the philosophy of mind is among the framework's most distinctive contributions. This appendix develops the connection systematically. Structural note on scope. CTGP operates at two analytically distinct levels that should be kept separate when evaluating the framework’s physics credentials. The core theory — comprising §§1–12, the σ formalism, ADM evolution, cosmological foliation, and the empirical standing set out in §12 — is a standalone spacetime ontology that can be evaluated entirely on physics and philosophy-ofphysics merits, independent of the consciousness discussion. Appendix D extends CTGP’s temporal ontology into the domain of mind and experience. This extension is motivated by the core theory — the present-edge structure of M(σ) provides a natural locus for qualia — but it is not a prerequisite for accepting the core framework. Readers primarily interested in the physics case may treat Appendix D as an optional philosophical appendix. Its conclusions do not feed back into the formal theorems of §17 or the empirical analysis of §§11–12. The physically instantiated criterion (§D.4) and the mapping problem (§§D.5–D.6) are philosophical elaborations, not physical postulates of the core theory. The discussion of conscious temporal experience throughout this appendix is intended as phenomenological motivation for the CTGP framework rather than as empirical evidence for it. This scope restriction applies to the qualia material collected here and not to the explanatory arguments of the main text. In particular, the explanatory challenge to the block universe in §2.2 and the convergence of the thermodynamic, causal, cosmological, and memory-formation arrows in §3.4 are main-body arguments from explanatory economy, not claims about phenomenal consciousness, and they stand or fall independently of anything in this appendix. D.1 Qualia Ontology: Locus on Σ_σ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
CTGP locates qualia — the intrinsic, subjective character of experience — on the present hypersurface Σ_σ. The claim is not that qualia are identical to physical processes on Σ_σ (that would presuppose a solution to the hard problem that CTGP does not claim to provide), but that qualia are intrinsic to ongoing, physically instantiated processes at the present edge. The key ontological theses are: • Qualia are features of the present moment Σ_σ. They exist only as properties of processes that are actively unfolding at the present edge of M(σ). • Past brain states belong to the formal history M(σ). They were fully real when present, and their consequences persist as causal records in present neural structure, but they do not persist as qualia. The qualia associated with a past experience are not still 'present' in any ontologically loaded sense. • Memory is re-instantiation, not continuation. When one remembers a past experience, the relevant neural patterns are re-activated at the present Σ_σ, producing a current experience that represents the past. This maps onto empirical findings about memory reconsolidation. Which experience is occurring. One may ask, of a four-dimensional person, why one particular brain state is the one being experienced rather than another or all of them at once. Posed as a request for a selection mechanism, the question has a ready eternalist answer: nothing selects, each state is experienced at its own location, and “now” is indexical. The question that remains is sharper. Assigning experiences to temporal parts establishes that each part has its experience; it is less clear that it accounts for the diachronic fact of being an experiencer whose experience is actually going on, rather than redescribing that fact in four-dimensional terms. CTGP answers with an additional ontological fact: only the generatively active state is presently real and presently undergoing physical realization. Earlier states were experienced when they were present and are not experienced now, because they no longer exist; later states are not yet experienced, because they have not been generated. CTGP also provides a principled explanation for the apparent privacy and inaccessibility of finegrained experience. Because many internal processes leave limited or non-redundant causal traces, the detailed structure of subjective experience is often underdetermined by publicly accessible data. This follows directly from the limited persistence and recoverability of causal encodings, rather than from any non-physical property of consciousness. D.2 Classicalization as the Bridge Mechanism ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ How do microscopic quantum processes at Σ_σ give rise to determinate experiential states? CTGP proposes classicalization — the combination of decoherence, amplification, and causal continuity — as the physical bridge from microphysics to determinate experience. The mechanism works in three stages: First, decoherence suppresses quantum interference between macroscopically distinct states, yielding effective classicality at the scale of neural dynamics. Second, amplification through biological signal cascades converts microscopic state distinctions into macroscopic, causally potent differences. Third, causal continuity — the fact that each Σ_{σ+Δσ} is generated from Σ_σ by deterministic or stochastic lawful evolution — ensures that experiential states are embedded in a continuous causal history, not isolated events.
These three stages together produce the physical conditions under which determinate experiential states are possible. CTGP does not claim that classicalization provides an account of why there is experience at all (the 'hard problem' in Chalmers's sense) but that it identifies the physical architecture necessary for the qualia-capable processes that CTGP posits. D.3 Necessary Conditions for Qualia on Σ_σ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP proposes the following as necessary (not sufficient) conditions for a physical process to be a vehicle of qualia: • Robust classical dynamics: the process must be sufficiently decohered that quantum interference is negligible at the relevant scale. • Organized integration: the process must integrate information across multiple degrees of freedom in an organized, non-random way. (This condition is related to but not identical to IIT's phi; see below.) • Causal continuity: the process must be embedded in a continuous causal chain connecting it to past states within M(σ). Isolated, causally disconnected events are not qualia-capable on this account. • Physical instantiation: the vehicle of qualia must be an actual, physically instantiated process, not the structure that a description or simulation represents. This condition excludes represented structure as a locus of experience; it does not by itself exclude any physical system, including the physical system carrying out a simulation. See §D.4. D.4 Domain-Relative Instantiation Criterion ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ D.4.1 Operational Clarification of the Physically Instantiated Criterion The physically instantiated criterion should not be interpreted as a presently established empirical discriminator between conscious and non-conscious systems. Rather, it functions as a research hypothesis concerning the relationship between phenomenal experience and physical realization. Formally, CTGP distinguishes three claims: (1) Structural Equivalence Claim: Two systems may exhibit identical input-output behavior and implement isomorphic computational structures. (2) Instantiation Claim: Structural equivalence alone does not guarantee identity of physically realized causal dynamics. (3) Phenomenal Dependence Claim: If phenomenal experience depends in part upon intrinsic physical properties rather than solely upon abstract functional organization, then structurally equivalent systems may differ phenomenally despite computational equivalence. CTGP does not presently provide an experimentally validated criterion capable of determining whether a given substrate possesses the intrinsic properties relevant to phenomenal experience. The framework therefore treats the physically instantiated criterion as a philosophical hypothesis that motivates empirical investigation rather than as a demonstrated result.
Accordingly, CTGP does not claim that digital simulations are known to lack qualia. It claims only that functional equivalence by itself is insufficient to establish phenomenal equivalence. Whether artificial substrates instantiate the relevant intrinsic properties remains an open question. D.4.2 Representation and Instantiation One of CTGP's most distinctive commitments is the domain-relative instantiation criterion. It claims that representing a process is not the same as instantiating it, and that this distinction matters for qualia. A computational simulation of neural dynamics encodes structural patterns and may reproduce input–output behaviour, but the simulation's states are symbolic representations of a process rather than the process itself. CTGP suggests that qualia depend on the actual physical instantiation of processes at Σ_σ—physical dynamics actively unfolding in the present—rather than on the manipulation of representations. This view leaves open whether some digital or other substrates might instantiate the relevant dynamics; it cautions only that functional equivalence alone does not guarantee that the requisite physical processes are present. This criterion grounds CTGP's cautious stance on strong functionalism. While functional organisation may be necessary for qualia, it may not be sufficient: the nature of the underlying physical process could matter. CTGP's domain-relative instantiation criterion is a philosophical elaboration of the Representation–Instantiation Distinction and does not modify any equation of the core framework. It emphasises that the framework's claims about mind are metaphysical extensions of its temporal ontology, not empirically settled facts. The physically instantiated criterion appears at the foundation of the CTGP framework, which treats the rate of temporal progression as depending on how the laws of physically instantiated nature — gravity among them — jointly bear on temporal becoming. The criterion is built into the framework’s ground-floor ontology, not introduced ad hoc to handle edge cases. The physical framework of CTGP is nonetheless logically independent of the criterion; this appendix explores one philosophical extension consistent with the ontology but not required for its physical validity. A pressing objection is that quantum gravity hints at discrete physics — Planck-scale discreteness in causal sets or loop quantum gravity — so why should discrete computation lack what discrete physics may have? The answer turns on the distinction between ontological discreteness and symbolic discreteness. The discreteness that quantum gravity posits is discreteness of the physical causal structure itself: actual causal events are generated sequentially, each inheriting the full intrinsic character of its physical substrate. Digital computation, by contrast, is symbolic discreteness: a finite state machine manipulates representations of structure according to rules, without thereby instantiating the intrinsic properties of the system it represents. Whether the computing hardware has experiential properties of its own is a separate question this criterion leaves open. One necessary condition can nonetheless be stated for psychological pain in particular, such as grief, shame, or dread. Such pain is constituted by appraisal rather than by a raw signal, so it requires a self-model, meaning, memory, and anticipation that matter to the system itself. A system lacking those capacities could not have psychological pain, whatever it computes or represents. Having them would be necessary but not sufficient, and whether any artificial system has them is an open question, partly empirical. A Planck-scale causal element in a growing causal set is
a genuine event with intrinsic physical properties; a bit in a register is a representation of such an event. The distinction is not between fine-grained and coarse-grained physics but between instantiation and representation. Russellian monism makes this precise: physical processes have both structural (relational) properties, which can be encoded and simulated, and intrinsic properties, which are not fixed by that structure. CTGP identifies the latter as the vehicle of qualia. What follows is a claim about level of description rather than about substrate: the represented structure — the sequence of symbolic states a computation implements — is an abstractum, and an abstractum is not a locus of experience. This leaves entirely open what the physical system carrying out the computation may instantiate on its own account, which is a question about that system's intrinsic properties and is not settled by the fact that the system is being used to represent something else. This argument does not require physics to be continuous; it requires only that computation be representational, which is constitutive of what computation is. This distinction can be stated formally as the Representation–Instantiation Distinction. Computational systems implement syntactic state transitions: S_{n+1} = f(S_n), where S denotes a symbolic state and f is an effective rule. Physical systems instantiate intrinsic causal dynamics: dφ/dt = F(φ), where φ is a genuine physical field evolving under real forces. The crucial asymmetry is that computation manipulates representations of states, while physics evolves actual states. A computation is therefore a mapping between symbolic states that represents a physical process; a physical process instantiates that process. The corollary is familiar at the macroscopic level: a perfect simulation of a hurricane does not produce wind. CTGP extends this principle into the domain of consciousness: a perfect simulation of a brain does not thereby instantiate the intrinsic causal dynamics of the brain it represents, because the simulation operates on representations of those dynamics rather than on the dynamics themselves. The claim is that the simulated brain is not a second locus of experience; it is not a claim about whether the simulating machine is one. The physically instantiated criterion is thus not a free-standing metaphysical stipulation but a direct application of the Representation–Instantiation Distinction to the case of qualia—grounded in the philosophy of computation, not merely asserted. Clarification of Intrinsic Physical Properties. The distinction between digital representation and ontological physical process requires clarification. CTGP’s physically instantiated criterion does not deny that physical theories may admit discrete microstructures (as in causal-set theory or loop quantum gravity). Rather, the claim concerns the difference between representation and instantiation. A digital register encodes information symbolically. The bit “1” possesses no intrinsic physical property corresponding to redness, pain, or any other qualitative character. Its significance is purely relational and interpreter-dependent. The point concerns what a symbol is, not what the system implementing it may possess. By contrast, a physical event in spacetime possesses intrinsic properties determined by its participation in causal interactions. These properties are not merely structural relations but include the local physical character of the event itself. This position aligns with Russellian monism, which holds that physics describes relational structure while leaving the intrinsic nature of physical properties underdetermined. CTGP therefore adopts the following ontological claim: physical events possess intrinsic properties that ground both causal powers and qualitative character. Physics describes the structural relations between these events, while the intrinsic properties provide the ontological substrate of experience. The criterion also presupposes a distinction that is easy to lose: the conditions for intelligence
and the conditions for qualitative experience are not the same conditions. Intelligence is specified functionally — by what a system does — and is therefore settled by functional duplication. Qualitative character is not specified functionally, and no argument from task performance establishes it. A system may satisfy the first set of conditions completely while the second remains an open question. The argument against strong functionalism does not depend on the representation/instantiation framing and is not weakened by the scoping above. It follows from Russellian monism alone: functional equivalence is specified structurally; intrinsic character is not fixed by structure; therefore functional duplication does not settle phenomenal identity. This holds whatever the substrate of either system, and it is unaffected by any question about what a simulating machine instantiates. The physically instantiated criterion therefore asserts: a digital simulation may reproduce the structural relations between events without thereby instantiating the intrinsic physical properties of the processes it represents. This claim is metaphysical rather than empirical, but it functions analogously to other ontological commitments in physics (e.g., the existence of spacetime events or quantum states). Three further claims should be kept apart here. The first is what the criterion denies: a system that represents another system's experience does not thereby have that experience. Experience is here understood to involve subject-relative registration — there being someone for whom the content occurs — and a representation of a subject does not constitute one. This is a clarification of terms rather than a theory of consciousness, and nothing in it turns on whether experienced content is also available for report, attention, or behavioral control. The denial concerns representation only, and carries no implication about what the representing system may undergo on its own account. The second concerns range: a conscious existence can only be aware of what can reach it, and what can reach it is fixed by the classes of signal exchange its substrate, architecture, and interfaces permit. Differently constituted systems therefore differ in their participatory capacities — in what they can receive, register, and enter into — and their experiential ranges, including the scope, integration, persistence, and contents of awareness, are correspondingly not identical. These constraints are specifiable in physical terms, whereas experiential range is not; the former constrains the latter without determining or revealing it. The criterion asserts the non-identity of ranges; it does not specify the content of any range, and it does not hold that experience is available only to non-digital substrates. The third concerns realization: within a shared experiential category, each experiencer instantiates that experience in a way particular to itself — a point holding between any two experiencers and implying nothing about substrate in particular. D.5 Substrate-Sensitivity and the Mapping Problem ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP adopts a substrate-sensitive account of qualia: not every physical process that satisfies the necessary conditions in §D.3 necessarily instantiates qualia of any given qualitative character. The substrate's specific dynamical properties — the particular way in which it integrates, amplifies, and causally continues information — shape the character of experience. This is the mapping problem, explicitly named in CTGP: the framework provides the metaphysical arena (the present + physically instantiated process + causal continuity) but does not by itself derive the specific pattern-toquale mapping. That mapping requires additional theoretical work.
Three bridge principle options are available: • Russellian intrinsic properties: the intrinsic (non-structural) properties of physical processes at Σ_σ just are the qualitative characters of experience. This is Russellian monism adapted to CTGP's temporal ontology. • IIT-style lawlike mapping: integrated information (or a generalization thereof) determines quale character via a law-like relation. CTGP's classicalization conditions set the stage; IIT or a successor theory provides the specific mapping. • Protopanpsychist composition: micro-experiential properties combine — via rules that CTGP's present-slice structure constrains — into macro-experiential properties. CTGP's physically instantiated criterion then rules out purely combinatorial accounts based on abstract functional roles. CTGP's working hypothesis is a combination of the Russellian and IIT-style options: a lawlike mapping from the intrinsic properties of causally continuous, classicalized physical processes at Σ_σ to qualitative experience, with the specific mapping remaining an open empirical and theoretical question. CTGP is compatible with Russellian intrinsic-property accounts, IIT-style functional mapping, and protopanpsychist composition — the framework constrains but does not decide between them. D.6 The Remaining Mapping Problem ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP intentionally distinguishes between two explanatory levels: (1) ontological placement of experience, and (2) specific mapping from physical processes to qualitative character. CTGP resolves the first but leaves the second partially open. The framework establishes that qualia occur only in ongoing, causally continuous physical processes located on the present hypersurface Σ_σ. It also provides constraints on candidate processes (physically instantiated, classically resolved, causally integrated dynamics). However, CTGP does not derive why a particular neural process corresponds to a particular qualitative character (e.g., why a given neural pattern corresponds to the experience of red rather than blue). D.6.1 Scope of the Mapping Claim CTGP does not claim to solve the hard problem of consciousness, the bridge problem, or the character problem. Its contribution is more limited. The framework proposes: (a) a temporal locus for conscious processes (the present hypersurface Σ_σ); (b) candidate necessary conditions for conscious processes (physical instantiation, causal continuity, and classicalized dynamics); (c) constraints on acceptable bridge theories. The framework does not derive a unique mapping from physical states to phenomenal states. Such a mapping would require additional psychophysical principles beyond those supplied by the current
CTGP formalism. Consequently, CTGP should be understood as providing an ontological framework within which a consciousness theory might operate, rather than as a complete theory of consciousness itself. D.7 Empirical Implications ~~~~~~~~~~~~~~~~~~~~~~~~~~ CTGP's account of qualia generates several empirical implications, distinguishing it from purely abstract philosophy-of-mind positions: • No macroscopic quantum consciousness: CTGP suggests that qualia require classicalized dynamics. Proposals requiring quantum coherence at the neural scale (e.g., Penrose-Hameroff) are incompatible with CTGP's classicalization requirement. • Substrate signatures: if the physically instantiated criterion is correct, there should in principle be signatures distinguishing genuinely qualia-capable processes from digital simulations, even when they are functionally equivalent. Identifying such signatures is a research program, not a settled result. • Memory as re-instantiation: CTGP suggests that memory recovery is best understood as reinstantiation of patterns at the present Σ_σ, not retrieval of stored qualia. This maps onto the empirical finding of memory reconsolidation — the fact that recalled memories are transiently labile and can be modified by conditions at the time of recall. D.8 Pause Argument and Process Continuity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A system whose physical processes are intermittently halted and resumed may report uninterrupted subjective continuity. From its internal perspective the process may appear continuous even though the underlying physical process is discontinuous. This suggests that experienced continuity does not strictly track uninterrupted physical process continuity. Within CTGP, this distinction is taken to support the view that the physical instantiation of processes—rather than their abstract functional structure alone—plays a role in determining whether experience occurs. The framework does not claim that all paused systems lack experience; instead, the pause argument motivates the domain-relative instantiation criterion, emphasising that the nature of a process’s physical unfolding, including its causal continuity, may be relevant to whether it is associated with experience. D.9 Process Continuity vs. Experienced Continuity ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This section contrasts process continuity with experienced continuity. A thought experiment in which a system is paused, stored, and restarted illustrates that the felt continuity of a process does not guarantee that the underlying process is continuously instantiated. Experienced continuity can thus arise from representational mechanisms even when physical becoming is intermittent. CTGP proposes that genuine physical continuity—an ongoing, physically instantiated process at the present edge Σ_σ—may be required for processes to be associated with experience. This argument does
not prove that paused systems lack experience; rather, it underscores the open question of how the continuity of physical processes relates to phenomenal experience. ---------------------------------------------------------------------------------------------------Appendix E. Supernova 1987A: A Worked Example of Causal Records and Co-presence ---------------------------------------------------------------------------------------------------SN1987A illustrates Postulate 2 (Causal Continuity) and Postulate 4 (Persistence Through Causal Continuity and Records), and demonstrates the consistency of the cosmological foliation of §9.3. It is not evidence for CTGP over eternalism: every fact about the event is equally derivable in a block-universe reading. CTGP’s empirical exposure lies elsewhere, in the requirement that physical law be generable from the present (§12.2). The role of the supernova here is narrower and still worth filling: it shows that CTGP’s account of access to the past — never retrieval, only inheritance — is concretely instantiated in a case where the reconstruction is unusually clean. E.1 Distant Is Not the Same as Spacelike ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ This distinction is stated first, because every confusion about the case descends from conflating the two. The interval between two events is not classified by spatial distance alone. A signal that crosses between them disqualifies the pair from being spacelike, however far apart they are. Pair of events | Interval | Significance for CTGP ------------------------+---------------------------------------------+---------------------------Collapse → photon | Null | Ordinary causal propagation detection (1987) | | Collapse → neutrino | Essentially null (ultrarelativistic; | Ordinary causal propagation detection | strictly timelike, since neutrinos are | | massive) | Collapse → | Spacelike | Requires a foliation to contemporaneous Earth- | | define “contemporaneous” state | | 1987 detection → Earth | Timelike | Ordinary causal succession now | | Table E.1. Interval classification of the event pairs relevant to SN1987A. The collapse lies on our past light cone. Reconstructing it from its messengers is ordinary causal inference, not access to a spacelike elsewhere, and nothing in that reconstruction bears on simultaneity. The one genuinely spacelike pair is the third, which by construction exchanges nothing with us; §E.5 concerns that pair alone. E.2 The Event in CTGP’s Tenses ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A blue supergiant in the Large Magellanic Cloud, roughly 168,000 light-years distant, underwent core collapse. The neutrino burst was detected on 23 February 1987; the optical signal followed within hours.
The neutrino lead is a fact about emission, not propagation. Neutrinos decoupled from the collapsing core immediately and escaped; the shock required additional hours to traverse the envelope and break out of the photosphere. Both messengers then crossed 168,000 years of intervening space at, or indistinguishably near, the speed of light. Nothing outran anything. Under CTGP the correct tensed description is as follows. • The collapse was generated at an earlier stage Σ_{σ′} and does not now exist. • The emitted neutrino and photon fields were generated at every intervening stage, as present-state field configurations, in unbroken causal continuity. • The 1987 detection was not the past arriving. It was a present-state detector interacting with a present-state field whose organization is a lawful transformation of the organization the source had when the source was present. • There is no retrieval across time. There is only inheritance within the present. None of this is a retrieval operation dressed in tensed language. Each arrow in the chain source state → radiation state → later radiation state → detector state relates two adjacent stages, and at every stage only the current state exists. The chain is long; it is nowhere non-local, and at no point does a photon carry a preserved fragment of 1987 forward as an existing past object. E.3 The Event’s Reality Did Not Depend on Being Observed ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The independence of the event from its observation is CTGP-consistent: the collapse was a generative stage of M(σ), not a fact constituted at detection. The messenger structure is the reason. Two channels, emitted under different source conditions, arrived in the order and with the separation that source physics dictates. An observation-constituted event would owe no such debt to stellarinterior physics that had, on that view, never occurred. The claim that distant events are not real until observed is not a position any serious interpretation holds, and refuting it carries little weight. The live alternatives are eternalism and the growing block, neither of which this argument touches. E.4 What “Cosmic Receipts” May and May Not Mean ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The ledger metaphor is useful and requires care. Read loosely, it implies a stored past — a repository holding earlier states in reserve — which is exactly the eternalist picture CTGP rejects. Under Postulate 4, a record is not a preserved earlier state. It is a presently existing physical organization whose structure is a lawful transformation of an earlier organization that no longer exists. Some consequences stay organized as records; others become distributed and effectively unrecoverable. SN1987A is a case in which several stayed organized and mutually cross-validating. Stated carefully, the metaphor says this: the present carries forward organized consequences from which earlier stages can be reconstructed, because the transformations were lawful and partially information-preserving. The universe does not keep books. The universe is the current page, written by the previous page, with the previous page gone. E.5 Co-presence: What Is Entailed and What Is Selected
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The claim that SN1987A demonstrates nonlocal temporal comparability divides into two claims, only one of which is contested. E.5.1 That There Was a Co-present Earth-Stage Is Entailed, Not Evidenced If the collapse was generated at all, the stage it belonged to was generated — and generation is not selective about where it happens. Our region was in that stage. Denying this would require the LMC’s collapse to have been real while our own concurrent existence was somehow in abeyance, which is incoherent rather than merely unsupported. The concurrency claim has specific content. Earth had existed for roughly 4.54 billion years by then; the progenitor, Sanduleak −69 202, was a massive star whose whole lifetime spanned only on the order of ten million years, so it was born, lived, and died entirely within Earth’s existing lifetime. The co-present Earth-stage was an Earth with oceans, an atmosphere, a biosphere, and hominins. Earth’s own stage also left its own causal records — ice cores, sediments, and the genomes still being carried forward — so its reality is independently attested, not inferred from the supernova. This follows from Postulate 1 together with the reality of the event. It does not require the supernova’s light, and it is not something the light could have established. E.5.2 Which Earth-Stage It Was Requires the Foliation What the evidence alone underdetermines is the pairing. We reach the collapse along our past light cone; we reach Earth 168,000 years ago through terrestrial records; we never reach the two as a pair, because that pairing is spacelike and exchanges nothing. The pairing is produced by a subtraction, and the subtraction requires a slicing. The sensitivity is substantial. To first order, Δt′ ≈ vx/c², and x here is 168,000 light-years, so an observer coasting through Earth at 0.5c assigns the collapse to an Earth-moment displaced by roughly 84,000 years from ours (about 97,000 years once the Lorentz factor is included). The photons and the light cone are the same; the answer differs. Such an observer does not claim that Earth was nonexistent: the observer affirms concurrency and disputes individuation, which is exactly the division drawn above. CTGP supplies what fixes the pairing, and supplies it from §6.1 rather than from the supernova: cosmological time fixes which stages are co-present, and in our universe it approximately coincides with the CMB rest frame and is realized along the comoving matter worldlines (§9.3.1). The supernova does not select the foliation, and it could not. Presenting it as doing so would invite a circularity objection against something the framework already supplies independently. E.5.3 The Placement The collapse belonged to the stage whose σ-value is earlier than that of the 1987 detection by approximately 168,000 years of matter-frame proper time. The approximation holds because the LMC’s peculiar velocity and the relevant gravitational potential differences are small; cosmological time
is realized along the comoving worldlines, and neither Earth nor the LMC lies exactly on them. One number, two roles. The figure “168,000 years” appears here in two roles, and only the second is a CTGP claim. As a light-travel time it characterizes a null pair (the first row of Table E.1); it is frame-dependent — a boosted observer assigns a different Δt to the same null pair — but it asserts nothing about simultaneity. As a σ-difference it asserts which stage the collapse belonged to, and that is a claim about the foliation. The figure also carries an observational spread of roughly 3%: 51.4 kpc ≈ 168,000 light-years is the commonly cited value (Panagia 1999), while published LMC distances range over roughly 163,000–169,000 light-years. None of the argument turns on which end of that range is correct. The placement is also robust, which is its main operational payoff. Earth’s peculiar velocity relative to the CMB is about 370 km/s, so the CMB-frame σ-assignment and the naive light-travel-time subtraction differ by at most about 200 years out of 168,000. The everyday answer is nearly right, and it is nearly right because we are nearly comoving — itself a fact about the matter frame to which CTGP points, not a coincidence. It follows, to the extent supported, that the relevant σ-stage was co-present with a stage of terrestrial history falling well within the span of anatomically modern humans in Africa. This is co-presence — equality of σ — and carries no implication of shared proper time, shared rate, synchronization, or coordination of any kind. The two regions exchanged nothing. They shared a generative stage. E.6 Against Retrofitting ~~~~~~~~~~~~~~~~~~~~~~~~ The generated domain does not acquire earlier structure retroactively upon inspection. The 1987 detection was constrained by conditions at the source stage and propagated forward through an unbroken chain of lawful transformations, with no point at which the chain could have been inserted after the fact. The present inherited the causal consequences of the past; it did not author them. E.7 What SN1987A Bears On ~~~~~~~~~~~~~~~~~~~~~~~~~ Claim | Does SN1987A bear on it? -----------------------------+--------------------------------------------------------------------“Only the local now is real” | Yes. A reconstructible distant event with a determinate σ-placement | is incompatible with a purely local-bubble ontology. “There was no co-present | Yes — by entailment (§E.5.1), not by measurement. distant state” | “Distant events do not exist | Yes, but the target is not a position seriously held; low value. until observed” | “Time cannot be compared | Partially. The comparison is reconstructible and cross-validating across space” | given a foliation; the case cannot select the foliation. “The block universe is | No. Nothing here discriminates between CTGP and eternalism. false” | Table E.2. The bearing of SN1987A on claims about time and co-presence.
E.8 Summary ~~~~~~~~~~~ Supernova 1987A was a generative event at a definite stage of the universe’s causal unfolding, not a fact constituted by its detection. It emitted multiple messengers according to local physics at that stage; those messengers propagated as present-state configurations through every intervening stage; and their present organization is structured enough that the source stage can be reconstructed from it, consistently across independent channels. That a co-present terrestrial stage existed follows from the event being generated at all. Which terrestrial stage it was follows from the matter-frame foliation — which the universe’s own matter distribution, not the supernova, supplies. The past is gone. Its consequences are here, organized well enough to read — and the reading is nearly independent of reference frame, because we are nearly comoving. The universe is neither a block of equally real moments nor a fog of observation-constituted facts. SN1987A is consistent with that picture and illustrates the record mechanism vividly; it does not establish it, and it does not need to. Presenting an illustration as decisive evidence would invite the objection that the framework reads its own ontology into evidence that does not select for it. Stated as entailment plus foliation, the argument is less rhetorically striking and considerably harder to dislodge.