16. Conclusion
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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 single-
outcome 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.
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17. Formal Theorems and Propositions
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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 non-
teleological 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.