Two clarifications fix the status of the results. First, QRF covariance is not discovered here;
within the perspective-neutral framework it is automatic for any law defined on physical states
(Remark 3.9). What is shown is that a CTGP interpretation can be placed on such dynamics
consistently, and what follows when it is. Second, the question whether CTGP contributes anything
beyond “a QRF-covariant single-outcome model with an attached historical truth profile” has a
definite answer, given in §11: CTGP’s generability, single-outcome, frame-invariant determinacy, no-
revision, and definite-ordering requirements each exclude models that are QRF-covariant, and, within
the perspective-neutral construction, the requirement that every realized event carry physical
states to physical states makes the collapse operators act as relational operators.
In brief, the answers are as follows. Determinacy is not frame-relative (§5.1). Two frame
descriptions describe the same present and the same history exactly when they belong to one orbit of
frame descriptions of a single complete state, which is equivalent to an explicit criterion on
stage, state, and truth profile (§5.2). Matching states is not sufficient, because a present state
does not fix its past; what settles identity of history is the truth profile (§5.2, §5.5). Valid
frame changes intertwine the generative law, so the probability measure over histories, the
conditional independence of the future from the past, and the truth of every determinate proposition
are the same in every frame (§5.3–5.4). And L sits beside the quantum state without driving it: it
is not a function of the present state, the law never takes it as input, and given the law it is
accurate to the state it accompanies (§4.6, §5.6). Part III shows that an explicit dynamics, in
discrete-stage and continuous-time form, has all of these properties.
Part V looks beyond the finite model. Relativistic quantum field theory, temporal quantum reference
frames, and quantum gravity each put CTGP’s commitments under new pressure, and none of them can be
settled by the constructions of Parts II–IV. Part V therefore turns the commitments into an
admissibility criterion and proves what can be proved now. In finite form, this includes a realized-
history consistency theorem under which, in causally local finite models, the choice of generative
foliation is empirically invisible. In relativistic form, it includes networks of gapless detectors
on a free field with operational no-signaling in Sorkin’s configuration, a regulated spontaneous-
collapse network built from Gaussian collapses of smeared fields together with the exact energy cost
of such collapses, a no-signaling criterion that does not depend on free fields, and collapses at
Poisson-distributed locations, conditionally on causally orderable configurations, together with the
obstructions to a fully covariant version and a candidate completion using cosmological time that is
developed but not adopted. For quantum clocks, it includes a construction with two ideal clocks in
which the interventions are part of the constraint. Part V also states the remaining obligations and
possible obstructions precisely, so that each direction can be pursued as a sequence of mathematical
results.
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2. Background
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2.1 Quantum Reference Frames and the Perspective-Neutral Framework
In standard quantum mechanics, states are written relative to an external classical reference frame:
positions are measured from a fixed origin. If the universe is treated as closed, there is no
external frame, and only relations between subsystems are physical. The perspective-neutral
framework (Vanrietvelde et al. 2020; de la Hamette & Galley 2020; Krumm, Höhn & Müller 2021)
implements this by imposing invariance under a symmetry group G acting on all subsystems at once.
The kinematical Hilbert space H_kin carries a unitary representation U of G, and the physical
(perspective-neutral) Hilbert space H_phys is the subspace of states invariant under every U_g.
These states make no reference to any external frame. Choosing a subsystem as reference frame
defines a reduction map from H_phys to a frame-relative Hilbert space describing the remaining
subsystems relative to that frame, and changes of quantum reference frame are compositions of one
reduction map with the inverse of another. This construction reproduces the frame changes of
Giacomini, Castro-Ruiz and Brukner (2019), and Krumm, Höhn and Müller (2021) show that QRF
transformations are canonically defined on the invariant sector. Related ideas appear in the
literature on reference frames and superselection (Aharonov & Kaufherr 1984; Bartlett, Rudolph &
Spekkens 2007) and on relational quantities (Loveridge, Miyadera & Busch 2018).
Two features matter for CTGP. First, frame changes in this framework are not choices of “whose
facts” to consider; they are changes of description of one invariant state, which is the reading
CTGP needs. Second, operators that are not invariant under G do not in general map H_phys to itself,
so a law written in absolute, non-relational terms is not well defined on physical states. This
constraint does substantive work in §7.4.
2.2 Single-Outcome Dynamics as Quantum Instruments and Jump Processes
A stochastic quantum dynamics with definite outcomes can be written as a sequence of quantum
instruments (Davies & Lewis 1970; Kraus 1983). An instrument is a family of operators {F_k}, indexed
by outcomes k, satisfying the completeness relation Σ_k F_k† F_k = 1. Given a normalized state ψ,
outcome k occurs with probability ‖F_k ψ‖², and the state is updated to F_k ψ / ‖F_k ψ‖. In
continuous time, the corresponding object is a piecewise-deterministic jump process: the state
evolves smoothly between random jump times and jumps by normalized jump operators, and its ensemble
average obeys a Lindblad master equation (Lindblad 1976; Gorini, Kossakowski & Sudarshan 1976;
Dalibard, Castin & Mølmer 1992; Breuer & Petruccione 2002).
Objective-collapse models have this form. In the GRW model, a spontaneous “hit” multiplies the wave
function by a Gaussian of fixed width centred at a random point, with the centre drawn according to
the Born rule, and hits occur at the times of a Poisson process. Bell (1987) proposed taking the hit
events themselves, the “flashes”, as the primitive ontology: the material world consists of the
flashes, and the wave function governs their probabilities. Tumulka (2006) gave a relativistic
version (rGRWf). Allori, Goldstein, Tumulka and Zanghì (2008) analyze the common structure of such
primitive-ontology theories.
The instrument picture gives the generative law a precise meaning. One stage-to-stage transition
applies the deterministic part of the dynamics, then samples an outcome k from the instrument and
updates the state. The pair (new state, outcome) is drawn from a probability kernel that depends
only on the current state, which is the form of law required by CTGP’s generability constraint (CTGP
– explained, §12.2): a law fixing transition probabilities P(S_{σ′} | S_σ) without reference to data
on later stages.
2.3 The CTGP Commitments Used Here
The following CTGP commitments are used below, cited to CTGP – explained and More Info.
(C-a) Ontological primacy of the present (Postulate 1). Only the current stage exists.
(C-b) Lucretian truthmakers (§5.2.1; Definition 17.20). “At σ′ it was the case that φ” is true
at σ iff the present world W_σ instantiates the primitive property Had(σ′, φ). The accuracy
condition requires that W_σ instantiates Had(σ′, φ) iff W_σ′ ⊨ φ, and past-tensed properties,
once acquired, are retained (Proposition 17.22, No Revision of the Past).
(C-c) Non-supervenience. Past-tensed properties do not supervene on the present’s categorical
physical configuration: two present states could agree in every categorical respect while
differing in their pasts (§5.2.1).
(C-d) Single-outcome realism (§10.1). At each stage, each measurement has exactly one realized
macroscopic outcome. Relational interpretations that make facts relative to observers are not
available to CTGP.
(C-e) Definite generative ordering (§6.4). Of two causally related stages, one precedes the
other.
(C-f) Generability (§12.2). There is a state S_σ on each stage and a law fixing P(S_{σ′} | S_σ)
for σ′ > σ without reference to later data. Enlarging the state space is legitimate only if the
added variables are physically instantiated on the present stage and have dynamics of their own.
(C-g) Causal inheritance (More Info, §11). Whatever part of history can still affect the future
must already be physically instantiated in the present; two present states that agree in every
physically instantiated respect have futures governed identically by the laws, whatever their
histories.
(C-h) Continuous generation (CTGP – explained, Abstract and §5.4). Generation is a continuous
causal-temporal progression, not a succession of disconnected states.
(C-i) Truth versus recoverability (More Info, §§4, 9–10). Whether something happened is settled
by the present world’s past-tensed profile; whether anyone can establish that it happened
depends on surviving records and available signals. Some truths may be structurally unknowable.
(C-j) Relational descriptive freedom. Different valid reference frames, clocks, and formulations
may describe the same physical history differently; CTGP requires one history, not one
description of it.
(C-k) Emergent classical cosmic time (CTGP – explained, §6). In the general-relativistic regime,
where the structure exists, the generative present is identified with the level sets of a
regular cosmological time function. This commitment is conditional on the classical spacetime
admitting that structure; §13.1 states the regularity results precisely.
(C-l) No density-weighted generation (CTGP – explained, §11). Generation laws on which regions
become real at rates set by local matter density fail CTGP’s consistency analysis; the
generation rate may vary with cosmic epoch but not across space.
One terminological point prevents a misunderstanding. The model below uses relational quantities:
the transition events concern separations between particles, not positions relative to an external
origin. This is not a relational interpretation in the sense rejected by (C-d). Whether a given
event occurred is an absolute, frame-independent fact; what is relational is the quantity the event
concerns, in the same way that “the distance between two stones is one metre” is an absolute fact
about a relational quantity.
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PART II. The Formalism
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3. Kinematics: Perspective-Neutral States and Frame Reductions
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