Which properties survive. Measured against the six parts of Proposition 13.28, the three
implementations compare as follows.
(a) Completeness. Centre-time ordering: proved. Completion-time ordering: unchanged, with the
same proof. Gapless pointers: holds for every finite configuration, with a separate proof
(Proposition 13.31(a)–(b)), and needs no ordering rule.
(b) Agreement with the causal-order dynamics. Centre-time ordering: proved in the flat local
model. Completion-time ordering: unchanged for a uniform half-width; a separate proof would be
needed if the half-width varied. Gapless pointers: holds on every admissible configuration of
any globally hyperbolic background, with a separate proof (Proposition 13.31(d)), provided that
the pointers are pre-chirped; without the pre-chirp it fails by the self-terms of Proposition
13.31(b).
(c) No signaling. Centre-time and completion-time ordering: proved for the covered operations.
Gapless pointers: holds, with a separate proof, for every pointer or classical source whose
region does not meet the causal past of the targets, on every configuration (Proposition
13.31(e)); it is not established for arbitrary interventions. The second limit above does not
arise for such operations, since no rule orders them.
(d) Forward generation. Centre-time ordering: forward computability from present Cauchy data,
not a mechanism. Completion-time ordering: strengthened to computability from the generated past
(Proposition 13.29(c)), still without a mechanism. Gapless pointers: replaced, for finite
configurations and given postulate (R), by forward generation through a local coupling, each
collapse being the reading of a pointer whose coupling has ended (Proposition 13.31(g)); on
spatially flat or open FLRW spacetimes, a forward evolution of the present state is not
constructed.
(e) Preferred-frame structure. Centre-time and completion-time ordering: proved. Gapless
pointers: the energy–momentum injection survives unchanged, because the unread channel is the
same (Proposition 13.31(c)); the cosmic-frame dependence of the outcome correlations that comes
from the ordering rule fails: the law of Proposition 13.31(b) contains no ordering rule, and its
correlations depend on the cosmic frame only through the smearings.
(f) Infinite histories. Centre-time and completion-time ordering: proved when the past regions
have finite four-volume, and open otherwise. Gapless pointers: a law of the pointer records
exists and is unique on every FLRW spacetime of the class, including spatially infinite ones,
with a separate proof (Lemma 13.30, Proposition 13.31(f)); it is a law of records, not a global
present state containing infinitely many pointers, and it does not establish the generation of
those records in an infinite volume.
What fails is the identification of the pointer law with the law of Candidate extension 13.27. On
non-admissible configurations the two differ, so the gapless-pointer construction is a different
candidate law, not a physical implementation of rule (iii). The gapless pointer, the natural local
realization of a Gaussian collapse, does not reproduce centre-time ordering: it yields a law with no
ordering rule.
The pointer realization has further costs. First, as explained above, it adds physical degrees of
freedom, one pointer at each Poisson centre prepared in a fixed state that depends on the smearing
through Γ_kk, together with postulate (R). Second, cosmological time still enters the law, through
the smearing law of Candidate extension 13.27(ii), which Proposition 13.24 makes necessary for
point-centred smearings; only the ordering rule is removed. Third, the coupling is a prescribed
function on spacetime, so the energy of Proposition 13.31(c) is supplied by the prescribed coupling,
and no dynamical account of the energy budget is given. Fourth, each pointer is a single degree of
freedom coupled throughout an extended region. It is a regulated probe, not a local probe field in
the sense of Fewster and Verch (2020), and the results depend on the free field and on the absence
of pointer dynamics. Fifth, the readings localize smeared field amplitudes, not matter (L2, D7). In
Bedingham’s (2011) model, a mediating relativistic field attaches a quantized degree of freedom to
each spacetime point; the pointers here attach one degree of freedom to each Poisson centre. The
pointer realization is recorded as a further candidate, alongside Candidate extension 13.27, and is
not adopted.
Comparison with a covariant alternative. Routes (iv) and (vi) of Remark 13.25 avoid a preferred
frame, and the two kinds of construction differ point by point. In frame, the candidate refers every
collapse to the cosmic rest frame, a structure CTGP already posits, whereas rGRWf refers each flash
to an earlier flash and Bedingham’s model works through an additional relativistic field. In scope,
the candidate is defined for a free field on a fixed background, on every domain of finite four-
volume and, when cosmological-time slabs have finite volume, on infinite histories (Proposition
13.28(f)), and in its gapless-pointer form, as a law of outcome records, on every FLRW spacetime of
the class (Proposition 13.31(f)), whereas rGRWf is defined for a fixed number of non-interacting
particles from given initial data. In generability, the candidate can be generated forward along
cosmological time in its free-field representation (Proposition 13.28(d)), and in its gapless-
pointer form, for finite configurations and given its reading postulate, through a local coupling
(Proposition 13.31(g)), whereas for rGRWf causal inheritance requires the earlier flash locations to
be physically instantiated in the present state, which is open (§13.12). In phenomenology, the
candidate predicts, as theoretical consequences of the free-field model, energy injected at rest in
the cosmic frame and, at the scales τ and ℓ, order effects for interfering interventions and cosmic-
frame-dependent correlations among collapse outcomes. The last two arise from the ordering rule and
are absent in the gapless-pointer form, whose outcome correlations depend on the cosmic frame only
through the smearings. None of these effects has yet been turned into an operational prediction for
an apparatus; any that become observable must respect the experimental bounds on collapse effects
(Bassi et al. 2013) and on preferred-frame effects (Kostelecký & Russell 2011), whereas a fully
covariant model predicts no such frame dependence. In localization, both leave open how matter,
rather than a field amplitude, is localized (L2, D7).
Adopting the candidate would change CTGP in one respect only: cosmological time would enter the
collapse law, and the generative foliation would become empirically accessible in principle. In the
gapless-pointer form it would enter only through the smearing law. That is a substantive commitment,
whose consequences become testable once its predicted preferred-frame effects are turned into
operational predictions. It is recorded here as a candidate, with its law, its variants, and its
predicted signatures stated, and it is not adopted.
13.12 Candidate Relativistic Realizations
Relativistic flash model. Tumulka’s relativistic GRW model (rGRWf) treats discrete spacetime flashes
as objectively realized events, with a Lorentz-invariant law for their joint distribution, for a
fixed number of non-interacting distinguishable particles (Tumulka 2006). Its law refers to
hyperboloids centred on each particle’s previous flash, which is one way of supplying the local
frame that, by Proposition 13.24, a point-centred law needs. It is the natural first test case. A
continuous generative present can contain a locally finite set of discrete flash events, so discrete
flashes do not imply discrete ontological time. The open questions are whether the flash law can be
represented as a generable process on CTGP’s cosmic-time hypersurfaces, with a physically
instantiated present state on each, and whether the construction extends to interacting fields,
particle creation, renormalization, and curved backgrounds. A realistic interacting relativistic
quantum field theory is a much harder target than a non-interacting model.
Spontaneous events. The detector networks of §13.6 are a measurement model: their events are
triggered by specified couplings. A spontaneous relativistic model would replace them by events
prescribed by the law itself, with a conditional intensity λ(x | S_Σ) defined from the present
state, together with the state update associated with each event. Such a law must be covariant
across admissible hypersurface descriptions, well defined on a regulated field state space, causally
admissible in the sense of (E3), normalized and free of pathological accumulation of events, and
generable from the physically instantiated present state. This is the relativistic objective-
collapse realization that CTGP contemplates. Section 13.7 constructs a regulated version, Gaussian
collapse of smeared fields on a fixed family of regions (Corollary 13.14), and §13.10 lets a
covariant Poisson law choose the locations (Theorem 13.21) and identifies the obstructions to a
fully covariant version (Propositions 13.22–13.24). A covariant version that localizes matter,
rather than linear field observables, remains open.
Bohmian dynamics. A second route keeps a definite configuration rather than collapsing the quantum
state: the present state is S_Σ = (Ψ_Σ, Q_Σ), with Q_Σ a particle or field configuration on the
hypersurface, and the dynamics is a guidance law. Such a theory already contains one actual
configuration and therefore one realized history. Relativistic Bohmian theories typically require a
distinguished foliation (Dürr, Goldstein, Münch-Berndl & Zanghì 1999), and the foliation may itself
be determined covariantly, for instance by the wave function (Dürr, Goldstein, Norsen, Struyve &
Zanghì 2014). CTGP’s cosmological-time foliation is a natural candidate. It must be shown that the
resulting predictions remain compatible with relativistic observations.
Success in either route would mean that at least one mathematically consistent single-outcome
relativistic quantum model can be equipped with CTGP’s ontology and its generability requirement
without unacceptable observational effects. It would not mean that every relativistic quantum field
theory or every interpretation is compatible with CTGP.