5.2 Cross-Frame Identification (Task T2)
Definition 5.3 (Frame description). A frame-A description at stage σ is a triple D^A_σ = (σ, ψ^A_σ,
L^A_σ), where ψ^A_σ is a ray in H_A and L^A_σ is a set of frame-A past-tensed propositions of the
forms Had(σ″, [φ]) with φ ∈ H_A and Had(σ″ → σ‴, k, E^A_k), where E^A_k = (F^A_k)† F^A_k is the
frame-A form of the effect of event k.
Definition 5.4 (Translations). The frame-A translation τ_A maps neutral propositions to frame-A
propositions:
τ_A Had(σ″, [Ψ]) = Had(σ″, [R_A Ψ]), τ_A Had(σ″ → σ‴, k) = Had(σ″ → σ‴, k, E^A_k).
Because R_A is unitary and E^A_k is determined by k, τ_A is a bijection. The frame-change
translation is τ_{A→B} = τ_B ∘ τ_A^{−1}; it replaces each state [φ] by [V_{A→B} φ] and each effect
E^A_k by V_{A→B} E^A_k V_{A→B}^{−1} = E^B_k, keeping the stage indices and the event label.
Criterion 5.5 (Cross-frame identification). Frame descriptions D^A_σ and D^B_τ describe the same
present world, with the same realized history, iff
(C1) Same stage: σ = τ.
(C2) State correspondence: ψ^B ≐ V_{A→B} ψ^A.
(C3) History correspondence: L^B = τ_{A→B}(L^A).
Theorem 5.6 (Common perspective-neutral pre-image). D^A_σ and D^B_σ satisfy (C1)–(C3) iff there is a
single neutral complete state Ω_σ = (S_σ, L_σ) with ψ^A ≐ R_A Ψ_σ, ψ^B ≐ R_B Ψ_σ, L^A = τ_A(L_σ),
and L^B = τ_B(L_σ).
Proof. (⇐) Given Ω_σ, ψ^B ≐ R_B Ψ_σ = R_B R_A^{−1} R_A Ψ_σ ≐ V_{A→B} ψ^A, which is (C2), and L^B =
τ_B(L_σ) = τ_B τ_A^{−1} τ_A(L_σ) = τ_{A→B}(L^A), which is (C3). (⇒) Given (C1)–(C3), define Ψ_σ =
R_A^{−1} ψ^A and L_σ = τ_A^{−1}(L^A). Then R_A Ψ_σ = ψ^A by construction, R_B Ψ_σ = V_{A→B} ψ^A ≐
ψ^B by (C2), τ_A(L_σ) = L^A, and τ_B(L_σ) = τ_{A→B}(L^A) = L^B by (C3). □
The content of Theorem 5.6 is clearest when stated in terms of orbits.
Definition 5.7 (Description orbit). Let 𝔉 be the set of admissible frames. For a complete state Ω_σ
= (S_σ, L_σ), its description orbit is
𝒟(Ω_σ) = { (σ, [R_A Ψ_σ], τ_A(L_σ)) : A ∈ 𝔉 }.
Proposition 5.8 (Orbit identification). (i) Two frame descriptions satisfy Criterion 5.5 iff they
belong to the same description orbit. (ii) Distinct complete states have disjoint orbits, so the
orbits partition the set of frame descriptions. (iii) Consequently, satisfying Criterion 5.5 is an
equivalence relation. (iv) If Ω_σ is admissible (Ω_σ ∈ 𝔒_n), every member of its orbit describes
that admissible complete state.
Proof. (i) is Theorem 5.6 restated in terms of orbits. (ii) Each frame description belongs to the
description space of exactly one frame A. If two complete states Ω and Ω* share an orbit member,
that member is (σ, [R_A Ψ], τ_A(L)) = (σ, [R_A Ψ*], τ_A(L*)) for a single frame A, and injectivity
of R_A and τ_A gives [Ψ] = [Ψ*] and L = L*. (iii) follows from (i) and (ii). (iv) holds by
definition of the orbit. □
Proposition 5.9 ((C3) is not redundant). There exist frame descriptions satisfying (C1) and (C2) but
not (C3); they describe different realized histories with the same present state. More precisely:
(a) For complete histories, in which the initial condition counts as part of the past, such
pairs exist for every invertible law.
(b) Within a single space ℋ_n with a fixed initial state, such pairs exist in the special regime
of Proposition 4.13. Whether they exist for generic finite-width parameters is open (§8.3).
Proof. (a) By Proposition 4.12, for any present ray [Φ] and two distinct event sequences there are
initial rays from which each sequence ends at [Φ]. The two complete histories have different truth
profiles, since their transition propositions differ. Take the frame-A description of one and the
frame-B description of the other: (C1) and (C2) hold, and (C3) fails. (b) The same argument applies
to the two histories of Proposition 4.13 (Proposition 8.1), which share an initial state. □
Case (a) compares descriptions drawn from admissible spaces with different initial conditions. Under
presentism the initial condition is a past fact like any other, so this is the comparison relevant
to CTGP; case (b) shows that the phenomenon also occurs, in a special regime, with the initial
condition held fixed.
Three levels of identity can now be distinguished, and the point at which interpretation enters can
be located exactly.
Representational equivalence. Two descriptions are related by a valid QRF transformation: (C1)
and (C2) hold, so the two quantum states are reductions of one perspective-neutral state. This
is a mathematical relation between representations.
Dynamical equivalence. The two frames’ laws are related by the frame-change unitary (Definition
5.11), so their probability laws and event records transform consistently (Theorem 5.12). This
is a mathematical relation between laws.
Ontological identity. The two descriptions concern one realized present and one realized
history. Within the model, this is membership in one description orbit (Proposition 5.8).
The first two levels are theorems. The step from orbit membership to ontological identity is not a
theorem of quantum mechanics. It is the following interpretive thesis, which is what CTGP adds.
Thesis 5.10 (Orbit realism). Each description orbit of an admissible complete state represents
exactly one realized present together with exactly one realized history. Frame descriptions in one
orbit are descriptions of the same world; descriptions in different orbits are descriptions of
different possible worlds, at most one of which is actual.
Thesis 5.10 gives the precise answer to the question how one can establish that two descriptions
refer to the same realized history rather than assuming it. Four observations make up the answer.
First, given the thesis, identity of history is not an extra posit laid over frame descriptions; it
is a definite condition, (C3), that can fail. Second, it cannot be settled by the quantum state
alone: Proposition 5.9 shows that state correspondence is compatible with different histories, so
identity of history is carried by L, which is a direct argument for including the Lucretian profile
in the complete ontology. Third, correspondence is preserved by the dynamics (Theorem 5.12(c)), so
frames cannot drift into describing different histories, and correspondence is not maintained by
stipulation at each stage. Fourth, whether two descriptions do in fact correspond is an ontological
question, and whether anyone can verify that they do is an epistemic question; §5.5 shows that these
come apart and that Criterion 5.5 is an exact criterion of identity, not in general an
experimentally decidable one.
5.3 Valid Transformations of the Law
Definition 5.11 (Frame-relative laws describing one world). Frame-relative laws (W^A, {F^A_k}) on
H_A and (W^B, {F^B_k}) on H_B describe one world iff W^B = V_{A→B} W^A V_{A→B}^{−1} and F^B_k =
V_{A→B} F^A_k V_{A→B}^{−1} for every k. Equivalently, by Remark 3.8, both are frame forms of one
admissible neutral law.
Theorem 5.12 (Covariance of generation). Let (W, {F_k}) be an admissible neutral law, with frame
forms (W^A, {F^A_k}) and (W^B, {F^B_k}). Let Ψ_{σ_0} ∈ H_phys, ψ^A_{σ_0} = R_A Ψ_{σ_0}, and
ψ^B_{σ_0} = R_B Ψ_{σ_0}. Then:
(a) For every finite sequence ω = (k_1, …, k_n), the probability of ω is the same whether
computed in frame A, in frame B, or in the neutral description.
(b) Along every history ω of nonzero probability, the frame-evolved states satisfy ψ^B_{σ_j} ≐
V_{A→B} ψ^A_{σ_j} for j = 0, …, n.
(c) Consequently the probability measure over histories is frame-independent, the sets ℋ_n of
admissible histories coincide in all frames, and if D^A_{σ_0} and D^B_{σ_0} correspond, then for
every realized history the descriptions D^A_{σ_n} and D^B_{σ_n} generated from them correspond.
Proof. By induction on n. Suppose ψ^A_{σ_j} ≐ R_A Ψ_{σ_j} and ψ^B_{σ_j} ≐ R_B Ψ_{σ_j}. By Lemma 3.7,
Pr^A(k) = ‖F^A_k W^A R_A Ψ_{σ_j}‖² = ‖R_A F_k W Ψ_{σ_j}‖² = ‖F_k W Ψ_{σ_j}‖², and likewise for frame
B. The updated frame states are R_A Ψ_{σ_{j+1}} and R_B Ψ_{σ_{j+1}} after normalization, which gives
(b). The probability of a sequence is the product of conditional probabilities, which gives (a) and
the equality of the sets ℋ_n. For (c), the truth-profile updates in the two frames add τ_A(ΔL) and
τ_B(ΔL) for the same neutral ΔL, so (C3) is preserved; (C1) is preserved because both frames advance
the same stage index; and (C2) is (b). □
Corollary 5.13 (Failure of same-form laws). If frame B adopts a law with the same matrices as frame
A’s law but written in B’s coordinates, that is, W^B = W^A and F^B_k = F^A_k as matrices, then in
general the two frames assign different probabilities to the same histories and do not describe one
world.
Proof. Such a pair satisfies Definition 5.11 only if W^A and every F^A_k commute with V_{A→B}, which
fails in general. Section 8 gives an instance with total-variation distance 0.180 between the two
measures over histories. □
Corollary 5.13 shows what “valid transformation” excludes: it is not enough for each frame to use a
law that looks natural from its own point of view; the laws must be related by the frame-change
unitary. This is the QRF analogue of requiring that a law be written in covariant form.
Corollary 5.14 (Covariant generability). In every frame A, the future is conditionally independent
of the past given the present:
Pr^A(𝔉⁺_σ | ψ^A_σ, L^A_σ) = Pr^A(𝔉⁺_σ | ψ^A_σ),
and this conditional distribution is the image under the frame change of the corresponding
distribution in any other frame B, with event labels unchanged and states mapped by V_{A→B}.
Proof. Apply Corollary 4.10 to the frame-A law, which is an admissible law on H_A because W^A is
unitary and the F^A_k form an instrument (Lemma 3.7). The relation between frames follows from
Theorem 5.12(a)–(b), applied from stage σ onward with initial states ψ^A_σ and ψ^B_σ ≐ V_{A→B}
ψ^A_σ. □
Corollary 5.14 connects CTGP’s generability and causal-inheritance commitments directly to the QRF
formalism. It is not merely that the operators transform covariantly: the very property that makes
the law generable from the present, and that makes the past causally idle beyond what the present
carries, holds in every frame and is preserved by every valid frame change.