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 co-
present. 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 stage-
relative 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 quantum-
gravity 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-path-
independent. 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. □