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. Higher-
level 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.
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PART II. Formal Framework
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6. Mathematical Framework
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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 closed-
timelike-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.