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-of-
physics 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.
Proof sketch. Restrict to the open region I⁺(S), itself globally hyperbolic. Every past-inextendible
causal curve in it approaches S, and J⁻(p) ∩ J⁺(S) is compact for each p, so the supremum defining
τ_S(p) is finite and τ_S → 0 along curves approaching S: the cosmological time of I⁺(S) is regular.
Theorem 17.4 then applies to that region, and the same argument with time reversed applies to I⁻(S).
□
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References
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Andersson, L., Galloway, G. J., & Howard, R. (1998). The cosmological time function. Classical and
Quantum Gravity, 15, 309–322.
Arkani-Hamed, N., Cheng, H.-C., Luty, M. A., & Mukohyama, S. (2004). Ghost condensation and a
consistent infrared modification of gravity. Journal of High Energy Physics, 2004(05), 074.
Arnowitt, R., Deser, S., & Misner, C. W. (1962). The dynamics of general relativity. In L. Witten
(Ed.), Gravitation: An Introduction to Current Research (pp. 227–265). Wiley. (arXiv:gr-
qc/0405109)
Babichev, E., & Ramazanov, S. (2017). Caustic free completion of pressureless perfect fluid and
k-essence. Journal of High Energy Physics, 2017(08), 040.
Bassi, A., Lochan, K., Satin, S., Singh, T. P., & Ulbricht, H. (2013). Models of wave-function
collapse, underlying theories, and experimental tests. Reviews of Modern Physics, 85, 471–527.
Bell, J. S. (1987). Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.
Bernal, A. N., & Sánchez, M. (2003). On smooth Cauchy hypersurfaces and Geroch’s splitting theorem.
Communications in Mathematical Physics, 243(3), 461–470.
Bernal, A. N., & Sánchez, M. (2005). Smoothness of time functions and the metric splitting of
globally hyperbolic spacetimes. Communications in Mathematical Physics, 257(1), 43–50.
Bigelow, J. (1996). Presentism and properties. Philosophical Perspectives, 10, 35–52.
Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of 'hidden' variables.
Physical Review, 85, 166–179; 180–193.
Bombelli, L., Lee, J., Meyer, D., & Sorkin, R. D. (1987). Space-time as a causal set. Physical
Review Letters, 59(5), 521–524.
Broad, C. D. (1923). Scientific Thought. Routledge.
Brown, H. R. (2005). Physical Relativity: Space-Time Structure from a Dynamical Perspective. Oxford
University Press.
Callender, C. (2017). What Makes Time Special? Oxford University Press.
Carroll, S. M. (2004). Spacetime and Geometry: An Introduction to General Relativity. Addison-
Wesley.
Carroll, S. M. (2010). From Eternity to Here: The Quest for the Ultimate Theory of Time. Dutton.
Chalmers, D. J. (1996). The Conscious Mind. Oxford University Press.
Chamseddine, A. H., & Mukhanov, V. (2013). Mimetic dark matter. Journal of High Energy Physics,
2013(11), 135.
Chiribella, G., D’Ariano, G. M., Perinotti, P., & Valiron, B. (2013). Quantum computations without
definite causal structure. Physical Review A, 88, 022318.
Colladay, D., & Kostelecký, V. A. (1998). Lorentz-violating extension of the standard model.
Physical Review D, 58, 116002.
Craig, W. L. (2001). Time and the Metaphysics of Relativity. Kluwer Academic Publishers.
Dehaene, S. (2014). Consciousness and the Brain. Viking.
Deng, N. (2013). The block universe: A metaphysical analysis. In R. Ciuni et al. (Eds.), New Papers
on the Present. Philosophia Verlag.
Dieks, D. (Ed.). (2006). The Ontology of Spacetime. Elsevier.
Dowker, F. (2005). Causal sets and the deep structure of spacetime. In 100 Years of Relativity.
World Scientific.
Earman, J. (2006). Two challenges to the requirement of global hyperbolicity. Synthese, 148,
443–468.
Einstein, A. (1915). Die Feldgleichungen der Gravitation. Sitzungsberichte der Preussischen Akademie
der Wissenschaften.
Ellis, G. F. R. (2014). The evolving block universe and the meshing together of times. Annals of the
New York Academy of Sciences, 1326, 26–41.
Ellis, G. F. R., & Rothman, T. (2010). Time and spacetime: The crystallizing block universe.
International Journal of Theoretical Physics, 49, 988–1003.
Ellis, G. F. R., Maartens, R., & MacCallum, M. A. H. (2012). Relativistic Cosmology. Cambridge
University Press.
Erickcek, A. L., Carroll, S. M., & Kamionkowski, M. (2008). Superhorizon perturbations and the
cosmic microwave background. Physical Review D, 78, 083012.
Everett, H. (1957). 'Relative state' formulation of quantum mechanics. Reviews of Modern Physics,
29, 454–462.
Friston, K. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience,
11, 127–138.
Garriga, J., & Mukhanov, V. F. (1999). Perturbations in k-inflation. Physics Letters B, 458,
219–225.
Geroch, R. (1970). Domain of dependence. Journal of Mathematical Physics, 11(2), 437–449.
Ghirardi, G. C., Rimini, A., & Weber, T. (1986). Unified dynamics for microscopic and macroscopic
systems. Physical Review D, 34(2), 470–491.
Gödel, K. (1949). An example of a new type of cosmological solutions of Einstein’s field equations.
Reviews of Modern Physics, 21, 447–450.
Gottesman, D., & Preskill, J. (2004). Comment on “The black hole final state.” Journal of High
Energy Physics, 2004(03), 026.
Hartle, J. B., & Hawking, S. W. (1983). Wave function of the universe. Physical Review D, 28(12),
2960–2975.
Hawking, S. W., & Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge
University Press.
Hawking, S. W., King, A. R., & McCarthy, P. J. (1976). A new topology for curved space-time which
incorporates the causal, differential, and conformal structures. Journal of Mathematical
Physics, 17(2), 174–181.
Hoftuft, J., Eriksen, H. K., Banday, A. J., Górski, K. M., Hansen, F. K., & Lilje, P. B. (2009).
Increasing evidence for hemispherical power asymmetry in the five-year WMAP data. The
Astrophysical Journal, 699, 985–989.
Hořava, P. (2009). Quantum gravity at a Lifshitz point. Physical Review D, 79, 084008.
Horowitz, G. T., & Maldacena, J. (2004). The black hole final state. Journal of High Energy Physics,
2004(02), 008.
Ismael, J. (2016). How Physics Makes Us Free. Oxford University Press.
Jacobson, T., & Mattingly, D. (2001). Gravity with a dynamical preferred frame. Physical Review D,
64, 024028.
Joos, E., Zeh, H. D., Kiefer, C., Giulini, D., Kupsch, J., & Stamatescu, I.-O. (2003). Decoherence
and the Appearance of a Classical World in Quantum Theory (2nd ed.). Springer.
Kiefer, C. (2012). Quantum Gravity (3rd ed.). Oxford University Press.
Kostelecký, V. A., & Russell, N. (2011). Data tables for Lorentz and CPT violation. Reviews of
Modern Physics, 83, 11–31.
Malament, D. B. (1977). The class of continuous timelike curves determines the topology of
spacetime. Journal of Mathematical Physics, 18(7), 1399–1404.