Proposition 14.5 proves clock-neutral equivalence for a specified family of instruments, late-record
projective measurements built from commuting projections, together with the offset-coherent
projection of part (e), in an ideal two-clock constrained model. It is the ideal-clock case of
Conjecture 14.4 for this family only. It does not establish equivalence for arbitrary clock-neutral
instruments, for events that occur while interactions are active, or for an indefinitely extended
succession of physically generated records, and it is not a general solution of temporal-QRF
compatibility. In the numerical check (Appendix C.2), a record written by an interaction keyed to A,
with the offset spread over [0.06, 1.64], has probability 0.139627690767 in the clock-neutral
description and in both reductions at late readings. Relative to A, the record probability is
exactly zero before A-time 0, the start of the interaction. Relative to B, it is already 0.021 at
B-time −1.0, because in part of the offset distribution the interaction has begun. For these
instruments, the event, its probability, and its instrument are the same in both descriptions; only
its temporal localization differs. With one interaction keyed to each clock and the offset in an
equal superposition of two wave packets centred at −1.6 and +1.6, the record law is 0.706791827009
in all three descriptions. This equals the equal-weight mixture of the two definite-order values,
0.868488 and 0.545095.
Part (e) answers the question that part (d) leaves open: whether the superposition of orders has
operational consequences. It does, for an observable that measures the clocks’ relative offset in a
coherent basis. In the numerical check, this probability is 0.926776695297 in the clock-neutral
description and in both reductions, equal to ½(1 + Re⟨W_1ψ, W_2ψ⟩), whereas an incoherent mixture of
the two offsets gives exactly ½. The statistic depends jointly on both orders. It is reproduced
neither by one definite order common to both offsets, which would give 1, nor by an order correlated
classically with the offset, which would give ½. This is a calculation within the model, not a
device-independent certification of indefinite causal order, which would require a causal witness in
the sense of Araújo et al. (2015).
For CTGP, parts (d) and (e) have a specific lesson. The order of unitary interactions keyed to
different clocks can be indefinite, with operational consequences, as in the analyses of Castro-Ruiz
et al. (2020) and Baumann and Lock (2026). The realized events, the record outcomes and the outcome
of an offset-coherent measurement, nevertheless have clock-neutral identities, probabilities, and
instruments. CTGP’s definite-ordering commitment (C-e) concerns stages and the realized events
located in them, not unitary couplings, so parts (d) and (e) do not conflict with it. What
Proposition 14.5 does not supply is a generative order of the record events. Their instruments
commute, so every order gives the same law, and designating one order as generative is interpretive
in the sense of §14.6. The proposition is also limited by its idealizations. Ideal clocks have
Hamiltonians unbounded below, and for clocks with bounded Hamiltonians the reductions are no longer
unitary (Höhn, Smith & Lock 2021). Events that occur while interactions are active, and
interventions that do not single out one clock, are not covered.
14.6 The Possible Obstruction
A fully relational quantum theory may treat all clock choices as physically equivalent and deny that
any further, objective generative ordering exists. CTGP could then still be added as an ontological
interpretation, but the added structure would have to be compatible with the theory’s constraints
and gauge symmetries. Two cases must be distinguished. If the added ordering has no physical
effects, it is interpretive rather than a new physical variable. If it affects event probabilities,
it is a physical hypothesis subject to empirical test. Section 13.10 meets the same alternative from
the side of collapse dynamics: cosmological time could fix the local frame and the order of
conflicting collapses covariantly, but only as a dynamical ordering (Remark 13.25).
Recent work sharpens the problem. Baumann and Lock (2026) report that, of the ways of modeling
interventions within relational dynamics, only one that builds the interventions into the constraint
remains causally consistent across multiple clocks, that it requires some temporal delocalization,
and that it naturally describes indefinite causal order. CTGP cannot simply stipulate that every
clock-relative temporal ordering is the same. It must identify which physical ordering is
ontologically fundamental and prove that legitimate temporal descriptions can be related to it. If
indefinite causal order turned out to be fundamental, rather than a feature of clock-relative
descriptions, CTGP’s primitive ordering would require generalization (C-e; CTGP – explained, §6.4).
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15. Quantum Gravity
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15.1 Fundamental Generative Order Versus Classical Cosmological Time
In the classical regime, CTGP identifies the present with a level set of the cosmological time
T_g(p) = sup { L_g(γ) : γ a past-directed causal curve from p }, subject to the regularity
conditions of §13.1. A fundamental quantum theory of gravity may have no single classical metric. A
state of the schematic form α|g_1⟩ + β|g_2⟩, with g_1 and g_2 inducing different cosmological-time
functions, has no unique classical T. The program therefore distinguishes
fundamental generative order ≠ classical cosmological-time function.
The two must agree in the appropriate classical limit, but they need not coincide at the fundamental
level. This develops the distinction between primitive generation and emergent geometry that CTGP
already draws (CTGP – explained, §§10.3–10.4). It also raises a demanding question: what physical
structure carries the objective generative ordering before classical spacetime exists? Naming a
parameter does not answer it.
15.2 Three Pathways
(i) Semiclassical and quantum-geometric emergence. Begin with quantum matter on an approximately
classical metric and progressively allow gravitational degrees of freedom to fluctuate. This is
the most incremental route from the constructions of this preprint, and it tests how much
geometric uncertainty CTGP can accommodate before its definition of a unique present becomes
problematic.
(ii) Canonical quantum gravity. Start from constrained gravitational dynamics, with the
Hamiltonian constraint written schematically as ĤΨ = 0, and ask whether an internal or emergent
time permits a single-history generative interpretation. A formally stationary physical state
does not mean that relational observables are static (Kuchař 1992; Isham 1993), but an objective
succession of unique outcomes requires further interpretation or dynamics. Material reference
systems such as Brown–Kuchař dust (Brown & Kuchař 1995) are candidates for a physically
instantiated time.
(iii) Causal sets. Causal-set approaches make causal order fundamental, and classical sequential
growth models generate a causal set element by element (Bombelli, Lee, Meyer & Sorkin 1987;
Rideout & Sorkin 2000). The labels of the growth sequence are, however, gauge: the physics is
required to be independent of the order in which elements are born. Theorem 13.4 and Proposition
13.7 offer an analogue of this situation: for a past-finite causal order with commuting
operations at unrelated elements, every natural labeling generates the same history measure, so
the choice of generative labeling is empirically invisible (Corollary 13.5, Proposition
13.7(b)). A CTGP reading would then treat one labeling as ontologically generative without
physical effect, which places it in the interpretive case of §14.6, unless an invariant ordering
or an appropriate continuum limit can be found.
The recommended order is to begin with pathway (i), to use pathway (ii) to examine the problem of
time, and to treat pathway (iii) as a possible realization rather than CTGP’s presumed foundation.
15.3 Compatibility Conditions
A candidate quantum-gravity theory would have to answer five questions affirmatively.
(Q1) Forward generation. Does it admit a physically meaningful forward-generative formulation? A
global mathematical description is not disqualifying; an indispensable independent future
boundary input would conflict with generability.
(Q2) One realized history. Can it support one realized physical history? A superposition in a
quantum state is not a contradiction, but CTGP requires a definite factual history rather than
many equally realized branches.
(Q3) Gauge symmetry. Does it respect the relevant gauge symmetries? A proposed generation
parameter must not promote an arbitrary coordinate label to a physical observable.
(Q4) Classical spacetime. Can it recover classical spacetime, reproducing Einstein’s equations
and the cosmological-time construction wherever CTGP’s classical assumptions hold?
(Q5) Phenomenology. Does it preserve acceptable relativistic phenomenology? Any preferred-
ordering dynamics must avoid experimentally excluded effects.
The classical-limit requirement can be written schematically as 𝒬_CTGP → (M, g, Σ_{T_g}) in the
appropriate semiclassical limit, where 𝒬_CTGP denotes a hypothetical completed quantum-generative
theory. The arrow represents a research objective, not an existing derivation.
15.4 What Would, and What Would Not, Rule Out Compatibility
The following would be genuinely problematic for CTGP:
(R1) a fundamental theory that requires independent final boundary conditions which cannot be
replaced by legitimate present-state dynamics;
(R2) a theory in which no invariant notion of realized physical history can be defined;
(R3) a quantum-gravity framework whose physical predictions necessarily depend on mutually
incompatible fundamental generative orderings;
(R4) a demonstrated impossibility of recovering CTGP’s classical cosmic-time structure in the
domain where CTGP claims to apply.
By contrast, indefinite causal order, timeless constraint equations, and quantum superpositions of
geometries are not by themselves sufficient to establish any of these conclusions. They are reasons
to investigate, not refutations.