9.4 What Is Postulated: The Identification of t With Cosmic Time
The continuous parameter t in Definition 9.1 is the time variable of a non-relativistic dynamics.
Identifying it with CTGP’s generation parameter σ, which in the emergent geometric regime is the
cosmological time function (CTGP – explained, §6), is a postulate of the application, not a result.
Nothing in the model derives cosmic time. What the model shows is conditional: if the generation
parameter is a global time with respect to which the present state evolves by a law of this form,
then the continuous-time model has every property established in Part II and §9.3. Deriving the
identification would require at least a relativistic version of the dynamics and an account of how
its time parameter is fixed by the cosmological matter distribution; these are the subjects of §13
and §14, and they remain open (§17.3, O2 and O3). The regularity of cosmological time that the
identification requires is stated precisely in §13.1: the level sets are future Cauchy surfaces in
general, and Cauchy surfaces under the condition of Proposition 13.1.
9.5 The Discrete Stages as a Coarse-Grained Description
The discrete-stage model of Part II is a coarse-grained description of the continuous one. With
stage duration τ and λ = 1 − e^{−Λτ}, the probability of at least one hit in a stage is the same in
both models. The discrete model differs in two respects: it allows at most one event per stage, and
it places each event at the end of its stage. The probability of two or more hits in one stage is of
order (Λτ)², so the discrete model approximates the continuous one when Λτ is small. The discrete
stages are therefore not a claim that generation proceeds in jumps; they are a sampling of a
continuous process, used in Part II because finite induction makes every proof exact.
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10. CTGP’s Further Requirements (Task T5)
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Cross-frame consistency is not the whole of CTGP’s demands. A QRF-compatible single-outcome model
must also satisfy definite generative ordering and generability. This section checks these and
related requirements.
10.1 Definite Generative Ordering
Proposition 10.1 (Frame changes act within stages and commute with generation). For every admissible
law, every pair of frames, and every event k, the frame-A and frame-B single-step maps G^A_k(ψ) =
F^A_k W^A ψ / ‖F^A_k W^A ψ‖ and G^B_k satisfy V_{A→B} ∘ G^A_k ≐ G^B_k ∘ V_{A→B}. The stage index is
unchanged by every frame change. The same holds in continuous time, with hit times unchanged.
Proof. Lemma 3.7, Definition 3.4, and Proposition 9.2(iv). Frame changes are maps H_A → H_B at a
fixed stage or time, and the stage or time index is a parameter of the law, not a quantum degree of
freedom. □
The linear order of stages, and the order of events within a history, are therefore the same in
every frame: event k_j precedes k_{j+1} in every description, and no frame change in this class
superposes the order of events. This satisfies (C-e) for spatial QRFs. The status of this result
should be stated plainly. The ordering is preserved because the stage or time index is a parameter
on which frame changes do not act. That is an assumption of the class of models, not a derivation of
definite order from more basic principles. Whether a definite order can be maintained when the
reference frame is itself a quantum clock is the principal open problem (§14; §17.3, O3).
10.2 Generability
The kernel of Definition 4.2, and the jump process of Definition 9.1, give the distribution of the
next state and event from the present state alone. They use no data from later stages, no final-
state condition, and no global self-consistency condition, so the model is generable in exactly the
sense of (C-f). The state space on which the law acts is H_phys, the physically instantiated present
state. The truth profile is not added to it (Proposition 4.9), so generability’s restriction on
enlarging the state space is respected. By Corollary 5.14, the generability property holds in every
frame. The model is stochastic, which CTGP – explained, §12.2, explicitly allows.
10.3 Open Future and Asymmetry
At stage σ, L_σ contains no proposition about a later stage (Theorem 4.8(e)), and no proposition
about a later stage is determinate (Definition 5.1). Before a transition there are only kernel
probabilities over possible events; after it there is one realized event and its Had-property. This
matches clause (d) of CTGP’s Definition 17.20 and the claim that the future is ungenerated rather
than merely unknown.
10.4 No Relative Facts
Every determinate proposition has the same truth value in every frame (Proposition 5.2 and Theorem
5.16), so the model contains no frame-relative facts. It contains frame-relative descriptions and
relational quantities, which CTGP does not reject (§2.3).
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PART IV. Evaluation
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11. What CTGP Adds Beyond QRF Covariance
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A skeptic can grant every theorem above and still ask whether CTGP contributes any substantive
constraint beyond the existence of a QRF-covariant, single-outcome stochastic model with an attached
historical truth profile. It does. The test of whether a commitment adds content is whether it
excludes some model that satisfies everything else. Each of the following commitments passes that
test.
CTGP commitment | Formal content in this framework | Excludes
-------------------------+-----------------------------------------+-------------------------------
Generability (C-f) | The law is a Markov kernel on present | Laws depending on past facts
| physical states; Cor. 4.10, Cor. 5.14 | not carried by the present
| | state (Example 11.3)
Single-outcome (C-d) | Exactly one event per transition; | Everettian readings on which
| Theorem 4.8(a) | all branches are realized
Frame-invariant | Determinacy and truth invariant under | Relational interpretations with
determinacy (C-d) | all frame changes; Prop. 5.2, Thm 5.16 | frame-relative facts
No revision (C-b) | Admissible profiles are nested; | Readings on which later events
| Theorem 4.8; Prop. 5.17 | fix earlier facts
Definite ordering (C-e) | Events totally ordered, order invariant | Fundamental indefinite causal
| under frame changes; Prop. 10.1 | order at the level of stages
Lucretian accuracy (C-b) | Profiles are admissible and accurate; | Truth profiles inconsistent
| Theorem 4.8, Prop. 4.11 | with the law
Table 2. Constraints CTGP imposes and what each excludes, even among QRF-covariant models.
Generability needs the most care, because a law that looks future-dependent may have an equivalent
forward formulation. CTGP – explained, §12.2, builds this into its criterion: future dependence
counts against generability only if no empirically equivalent generable formulation exists. Three
cases must therefore be distinguished: conditioning on the future as a mathematical operation, a
final-boundary requirement imposed as a fundamental input, and genuine non-generability relative to
the admissible present physical states. The first two are treated by Proposition 11.1 and Example
11.2, the third by Example 11.3.
Proposition 11.1 (Future conditioning has a forward representation). Let E be an event label and n ≥