(17) Generative succession, clock readings, and clock-conditional descriptions are distinct;
with an external generative order, imperfect clocks can repeat readings, delocalize events, and
invert their recorded order without affecting event identity or probability (Proposition 14.2).
(18) The unread Gaussian collapse channel is a Gaussian mixture of unitary kicks. Position
collapse of a particle injects exactly ħ²/(8mσ²), whatever the potential, and Gaussian collapse
of a smeared free field in 3 + 1 dimensions injects a state-independent, future-timelike
energy–momentum equal to that of the classical solution Eh divided by 4σ². On a curved
background it changes the renormalized stress–energy of a Hadamard state by the classical
stress–energy of Eh divided by 4σ² (Propositions 13.15 and 13.16).
(19) If the operations performed between a tested operation and the targets preserve an algebra
on which the tested operation acts trivially, the tested operation cannot signal to the targets,
whatever the field theory; microcausality supplies only the second condition (Theorem 13.17,
Corollary 13.18).
(20) For Gaussian collapses at Poisson-distributed locations with a covariant law, histories are
foliation-independent and non-signaling conditionally on causally orderable configurations,
which is not yet an unconditional collapse law (Theorem 13.21). Orderability fails in large
volumes, the order of interfering collapses then matters physically, and no Lorentz-invariant
law for point-centred smearings exists (Propositions 13.22–13.24). Covariant repairs require
additional structure at each event, and those that use cosmological time would give the
generative order a dynamical role (Remark 13.25).
(21) For two ideal clocks with the interventions in the constraint, a family of late-record
projective instruments has clock-neutral identities, probabilities, and instruments, which
intertwine with both reductions, while temporal localization differs between the clocks. When
the order of interactions depends on the clock offset, an offset-coherent observable exhibits
interference between the orders (Proposition 14.5).
(22) A candidate extension, not adopted, in which cosmological time fixes the smearing and the
order of collapses, is complete and normalized on every domain of finite four-volume, and on
infinite histories when cosmological-time slabs have finite volume; it can be generated forward
along cosmological time in its free-field representation and admits no signaling between
causally separated regions for the operations it covers. It agrees with the causal-order
dynamics wherever that is defined in the flat local model. It injects energy at rest in the
cosmic frame and makes the correlations among collapse outcomes depend on that frame, which is
distinct from superluminal signaling and not yet shown to be operationally detectable (Lemma
13.26, Proposition 13.28). Ordering the collapses by the completion of their regions gives the
same law, with forward generation from the generated past (Proposition 13.29). Realizing each
collapse as the reading of a gapless pointer coupled locally to the field derives the collapse
statistics and gives a variant with no ordering rule: its evolution is foliation-independent on
every finite configuration, it admits no signaling by the modeled pointers and classical sources
outside the causal past of the targets, it agrees with sequential collapse on causally orderable
configurations, it has a law of outcome records on every FLRW spacetime of the class, and it
injects the same energy–momentum, while its outcome law differs from the candidate’s where
collapses interfere (Lemma 13.30, Proposition 13.31). Its forward generation is established for
finite configurations, and its definite readings require a primitive single-outcome postulate,
which the coupling does not supply. The energy–momentum injection is a theoretical prediction;
an experimental signature would require a collapse law for matter and an account of how an
apparatus records the effect.
17.3 Open Problems
(O1) Continuous groups. Physical QRFs use translations, rotations, and boosts, which form
continuous groups (Giacomini et al. 2019). For non-compact groups, invariant states are not
normalizable in H_kin, and H_phys must be built by group averaging with a physical inner
product. Reduction maps remain unitary in the perspective-neutral framework (Vanrietvelde et al.
2020), so Part II is expected to carry over, but the flash dynamics needs a relational version
of the GRW or CSL smearing in continuous space, and its well-definedness must be proved.
(O2) Relativistic quantum fields. The ring model is non-relativistic, and Theorem 13.4 and
Proposition 13.7 are discrete analogues; the assumptions a relativistic extension must supply
are listed in §13.5 as (E1)–(E6). Theorem 13.9 settles the measurement-model case for finite
networks of gapless detectors on a free field. Theorem 13.13 with Corollary 13.14 gives a
regulated spontaneous-collapse network for Gaussian collapse of smeared fields on a fixed family
of regions, and Theorem 13.21 extends it to Poisson-distributed locations on causally orderable
configurations. Open are a covariant law for extended collapse events, for which Propositions
13.22–13.24 identify the obstructions, Remark 13.25 the available repairs, and §13.11 a
candidate completion that uses cosmological time, together with a gapless-pointer variant that
needs no ordering rule (Proposition 13.31), neither of which is adopted; a physical account of
the pointers and of their definite readings, which the variant postulates; a forward evolution
of the present state with infinitely many pointers; a local probe field in place of single-
degree-of-freedom pointers; an operational prediction for the predicted preferred-frame energy
injection; collapse of matter density rather than of linear field observables; detectors with
internal dynamics; interacting fields, for which Theorem 13.17 gives a sufficient condition and
Target 13.19 a first test; and continuously distributed events. The relativistic realized-
history consistency result (Target 13.6) is open: it requires an event-history measure for a
regulated relativistic single-outcome model, a measurement framework free of Sorkin-type
signaling, and, for CTGP, a generable representation on cosmological-time hypersurfaces that
would derive the identification postulated in §9.4. CTGP’s ensemble-level results (CTGP –
explained, Proposition 17.16 and Theorem 17.17) are conditional and do not by themselves supply
a covariant stochastic ontology (Proposition 13.2).
(O3) Temporal QRFs. This is the most consequential open problem; §14 sets out the distinctions,
a worked model, a proposed identification criterion for overlapping descriptions (Definition
14.3), its proof for late-record instruments with ideal clocks and the interventions in the
constraint (Proposition 14.5), and a target for imperfect clocks (Conjecture 14.4). The
construction here assumes a shared stage or time index, and Proposition 10.1 preserves it
because frame changes act only on spatial relational degrees of freedom. Temporal QRFs, in which
a quantum clock is the frame (Page & Wootters 1983; Höhn, Smith & Lock 2021), can assign
superposed readings to one clock relative to another and can make the temporal localization of
events indefinite (Castro-Ruiz, Giacomini, Belenchia & Brukner 2020). Two structures must then
be distinguished: the ontological generation order, which CTGP requires to be definite, and
quantum-clock relational time, the readings of physical clock subsystems relative to one
another. CTGP does not require quantum clocks to reveal a preferred cosmic time. It requires an
account of how relational clock descriptions correspond to an underlying generative order, if
there is one. The research question is whether a temporally relational model can be constructed
with a definite underlying order of events that remains invariant under every admissible clock-
frame transformation, without that order being assumed and then merely preserved. One candidate
route is to add a quantum clock subsystem to the ring model: clock-relative descriptions would
assign superpositions of readings to events, while the order of events in the record is fixed by
generation. A second is a material reference system of the Brown–Kuchař dust type (Brown &
Kuchař 1995), whose proper time could serve as a physically instantiated cosmic time akin to
CTGP’s matter-frame cosmological time. Both are conjectures, and the result of Baumann and Lock
(2026) on causal consistency across clocks (§14.6) indicates that temporal delocalization, and
possibly indefinite causal order, must be expected in clock-relative descriptions.
(O4) Quantum gravity and indefinite causal order. As CTGP – explained notes (§6.4), if quantum
gravity requires indefinite causal order at the level of spacetime itself, CTGP’s primitive
ordering would require generalization. Proposition 10.1 holds for the class of models here but
says nothing about such theories. Section 15 distinguishes fundamental generative order from
classical cosmological time, sets out three pathways, and lists the conditions a quantum-gravity
theory would have to meet (Q1–Q5) and the findings that would rule out compatibility (R1–R4),
and §15.5 states three finite test problems.
(O5) Records. The model has no record subsystems; evidence about the past is whatever the
present state encodes. Adding pointer or environment systems would allow a formal treatment of
how records serve as evidence for L across frames, of how records degrade while L does not (CTGP
– explained, §14.6), and of observers with access to records rather than to the full present
state.
(O6) Weak non-supervenience at generic parameters. Whether two distinct histories from a fixed
initial state can end in exactly the same present state for generic finite-width parameters is
open (§8.3). It is not needed for CTGP, which requires only strong non-supervenience
(Proposition 4.12).
(O7) Empirical parameters. The parameters used here are illustrative. Real collapse models are
tightly constrained by experiment (Bassi et al. 2013), and a relational collapse model must
respect the same bounds.
----------------------------------------------------------------------------------------------------
18. Answers to the Original Questions
----------------------------------------------------------------------------------------------------
Question | Answer in this preprint
---------------------------------------------+------------------------------------------------------
(T1) Which propositions are determinate? | Three tiers (Def. 5.1): events, states, sharp
| properties; events are not sharp values (Prop. 4.4).
| Determinacy is frame-independent (Prop. 5.2).
(T2) How are facts identified across QRFs? | Criterion 5.5 ⇔ one description orbit (Thm 5.6,
| Prop. 5.8); identity of the world is Thesis 5.10.
| State alone is insufficient (Prop. 5.9).
(T3) Do valid QRF changes preserve truth? | Yes (Thm 5.16), and preservation persists under
| generation (Thm 5.12).
(T4) Where does L sit relative to the state? | Generated, admissible, not a function of S, accurate,
| frame-covariant, dynamically idle (Thm 5.20).
(T5) Generative ordering and generability? | Order shared by all frames (Prop. 10.1); generable
| in every frame (Cor. 5.14); continuous form (§9).
Same history, or merely assumed? | Exact criterion (C3) that can fail and is preserved
| by the dynamics; checkable only through records,
| and possibly unknowable (Prop. 5.19).
Does CTGP add anything beyond covariance? | Yes: it excludes otherwise QRF-covariant models,
| e.g. history-dependent laws (Prop. 11.4), and,
| within the construction, requires collapse
| operators that act relationally on physical
| states (§7.4, §11).
Why a primitive profile, not an encoding? | Formally interchangeable (Prop. 5.21); given CTGP’s
| truthmaker principle and the non-existence of past
| events, the profile supplies the present ground for
| past-tense truths (§5.7).
Concrete single-outcome model? | Three particles on ℤ_N, relational flashes (§§6–9).
Relativistic quantum field theory? | Admissibility criterion (Def. 12.1); ensembles do
| not fix histories (Prop. 13.2); finite realized-
| history consistency (Thm 13.4, Prop. 13.7); free-
| field detectors with no-signaling (Thm 13.9);
| Gaussian collapse on fixed regions (Cor. 13.14),
| its energy cost (Prop. 13.16), and at Poisson
| locations, conditionally (Thm 13.21), with the
| obstructions to full covariance (Props 13.22–