15.5 First Test Problems
Three finite problems would test the quantum-gravity conditions without attempting a complete
theory. Each is a target, not a result.
(G1) Label-invariant realized causal-set histories. Define a stochastic growth law on labeled
causal sets, in which each step adds one element to the future of some of the existing ones,
with a quantum-instrument version in which operations at unrelated elements commute. Impose a
discrete form of general covariance and prove that the probabilities of unlabeled completed
causal sets do not depend on the natural labeling used to compute them. For a fixed causal order
this is Theorem 13.4 and Proposition 13.7. For growth models it adapts the discrete general
covariance of classical sequential growth (Rideout & Sorkin 2000), and it should be presented as
an adaptation rather than as a new theorem. CTGP’s contribution would be ontological: whether
one birth order can be real while its labeling is unobservable. That reading faces a genuine
tension. If the birth orders of unrelated elements are pure gauge, asserting that one of them is
physically real adds structure that the invariant theory cannot detect. That may be acceptable
as empirically inaccessible ontology, the interpretive case of §14.6, but it requires a
principled defense.
(G2) A finite quantum-geometry model. Take finitely many geometric configurations q, a physical
state Σ_q c_q |q⟩, and an instrument representing geometry-related events. Prove that each
update maps physical states to physical states, that outcome probabilities are gauge-invariant,
that the law generates one realized history without a future boundary condition, and that the
history measure is consistent under admissible changes of description. Such a model would be a
controlled finite analogue, not compatibility with an established theory of quantum gravity.
(G3) Two semiclassical branches. Take a state concentrated on two classical geometries whose
cosmological-time functions T_1 and T_2 differ, without assuming that any cosmological time
exists on the superposition. Ask whether one branch-independent ordering of events can be
defined that reduces in each branch i to F_i(T_i), with F_i increasing, wherever T_i is regular.
The possible outcomes are that a common ordering exists, that it exists only approximately, that
an additional physical clock or reference field such as Brown–Kuchař dust is needed, or that the
construction fails. A failure would be informative, since it would identify a genuine
incompatibility between a particular formulation of CTGP and a particular model of quantum
geometry.
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16. Roadmap
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Stage | Goal and deliverable | Status in this preprint
--------------------------+--------------------------------------+-------------------------------
1. Formal specification | Definitions, conditions, exclusions; | Done for the finite domain
| L and S related precisely | (Parts II–IV, §12)
2. Relativistic toy model | Regulated single-outcome model on a | Discrete analogues (Thm 13.4,
| fixed background: normalization, | Prop. 13.7); free-field
| history consistency, covariance | detectors (Thm 13.9); fixed-
| | region Gaussian collapse (Cor.
| | 13.14) and its energy cost
| | (Prop. 13.16); Poisson
| | locations, conditional (Thm
| | 13.21); obstructions to full
| | covariance (Props 13.22–13.24);
| | candidate cosmic-time
| | completion and its gapless-
| | pointer variant (§13.11), not
| | adopted
3. Temporal QRFs | Clock reductions from a neutral | Distinctions and Prop. 14.2;
| description; event identity across | late-record instruments with
| clock changes | ideal clocks (Prop. 14.5);
| | Conjecture 14.4 open
4. Interacting, curved | Interactions, regularization, | Sufficient condition (Thm
quantum field theory | continuum limits, curved backgrounds | 13.17); probe test (Target
| | 13.19) open
5. Quantum geometry | Constrained or pre-geometric model; | Test problems G1–G3 (§15.5);
| event ontology; semiclassical limit | open
Table 4. A staged program, with the status of each stage.
Each stage should be independently publishable if it yields a substantive result. Failure at a stage
would also be informative: it would identify either a needed refinement of CTGP or a genuine
incompatibility, rather than leaving the question open. CTGP does not need to abandon its continuous
global generation, its rejection of density-weighted generation, or its unique factual past to
pursue this program. It does need to keep a clear line between its foundational ontology and any
particular quantum implementation.
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PART VI. Results and Outlook
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17. Scope, Results, and Open Problems
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17.1 What Kind of Result This Is
The preprint establishes an existence result, conditional on the CTGP interpretation: at least one
well-defined model, within an explicitly specified class, has a structure that jointly satisfies the
formal counterparts of CTGP’s quantum commitments. The mathematics establishes the structure;
reading that structure as one realized present grounded in primitive past-tensed properties is
CTGP’s interpretation (Ontological Reading 4.3, Thesis 5.10, §5.7), and mathematical consistency is
not metaphysical vindication. It does not establish general compatibility, which would require
showing that every physically admissible QRF theory can accommodate those commitments. It does not
establish physical adequacy, which would require reproducing the relevant features of established
quantum physics, including relativistic quantum field theory, and of general relativity. Existence
results of this kind are a standard and useful form of progress: they show that a set of commitments
is consistent and provide a concrete object on which objections can be tested. In the terms of Table
3, the finite model establishes mathematical compatibility on its own domain. Part V establishes
finite-dimensional analogues of the relativistic and temporal results it requires, and states the
rest as targets.
17.2 Established
Within the class of models of Part II (finite symmetry group, regular representations, invariant
unitary-plus-instrument dynamics) and its continuous-time form (§9), and in the explicit instance of
Part III:
(1) Two frame descriptions describe one complete state iff they satisfy Criterion 5.5, iff they
lie in one description orbit; the orbits partition the frame descriptions (Theorem 5.6,
Proposition 5.8). Ontological identity of the represented world is the further interpretive
Thesis 5.10.
(2) Admissible truth profiles are characterized exactly (Theorem 4.8).
(3) State correspondence does not settle identity of history (Proposition 5.9); for invertible
laws, every present ray is reachable along every finite event sequence from a suitably chosen
initial ray (Proposition 4.12).
(4) Determinacy is frame-independent, with a three-tier criterion in which primitive events,
distinguished from sharp values, carry single-outcome realism (Definition 5.1, Propositions 4.4
and 5.2).
(5) Valid frame changes intertwine the generative law: the measure over histories, the
conditional independence of future from past given the present, and the truth of every
determinate proposition are frame-invariant and preserved by generation (Theorems 5.12 and 5.16,
Corollary 5.14). Laws that merely look alike in different frames do not describe one world
(Corollary 5.13).
(6) Ontological identity, physical equivalence, and observational indistinguishability are
distinct relations, ordered by strict implication (Proposition 5.19).
(7) L is generated, admissible, not reducible to S, accurate, frame-covariant, and dynamically
idle (Theorem 5.20).
(8) The model has a continuous-time form, with histories as marked point processes, for which
admissibility, reconstruction, dynamical idleness, cross-frame correspondence, and truth
preservation are proved (Propositions 9.2–9.9, Theorems 9.7 and 9.10); the identification of its
time parameter with cosmic time is stated as a postulate (§9.4).
(9) Within the class, generative order is shared by all frames, the law is generable, the future
is open, and there are no relative facts (§10).
(10) The truth profile is formally interchangeable with the event history (Proposition 5.21);
its independent role is ontological, as the present ground for past-tense truths required by
CTGP’s truthmaker principle, and the choice between this reading and an eternalist flash
ontology rests on CTGP’s independent arguments (§5.7).
(11) CTGP’s commitments exclude models that are otherwise QRF-covariant (Table 2); in
particular, a QRF-covariant law that depends on past facts not carried by the present state is
not generable (Proposition 11.4), whereas conditioning on a future event has an equivalent
forward formulation and is not by itself a failure of generability (Proposition 11.1, Example
11.2). Within the perspective-neutral construction, events that carry physical states to
physical states require collapse operators whose action on physical states agrees with that of
invariant, relational operators; this does not single out the particular flash ontology used
here (§7.4).
(12) The finite model is CTGP-admissible on its domain under the proposed criterion of
Definition 12.1 (Proposition 12.2).
(13) The level sets of regular cosmological time are future Cauchy surfaces, and a level set is
a Cauchy surface exactly when every future-inextendible causal curve passes beyond it
(Proposition 13.1).
(14) Agreement of density operators on every hypersurface does not fix the probabilities of
realized histories (Proposition 13.2). For causally local instruments on a finite causal order,
realized-history probabilities are foliation-independent and outcome marginals do not depend on
instruments outside the designated predecessor set, so within such finite models the choice of
generative foliation is empirically invisible (Theorem 13.4, Corollary 13.5); the result extends
to countably many events with finite pasts (Proposition 13.7), and identifying the predecessor
set with the relativistic causal past requires the further assumptions (E1)–(E6).
(15) For finite networks of gapless detectors on a free relativistic field, realized-history
probabilities are foliation-independent, detector statistics are given explicitly, and there is
no operational signaling between causally separated regions, including in Sorkin’s three-region
configuration (Theorem 13.9). Localizing the Kraus operators of an operation does not by itself
secure this (Example 13.10).
(16) The same conclusions hold for every finite, causally orderable network of Weyl-diagonal
instruments on a free field (Theorem 13.13). Gaussian collapse of a smeared field is Weyl-
diagonal (Proposition 13.12), which yields a regulated spontaneous-collapse network with
foliation-independent histories and no signaling between causally separated regions (Corollary
13.14), subject to the limitations (L1)–(L4).