# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism Part 27 of 40 (1,518 words) 13.13 Eight Distinctions Any Extension Must Respect (D1) Geometric covariance versus physical foliation independence. A foliation defined covariantly from the metric is not arbitrary, but a quantum dynamics built around it may still produce physically detectable preferred-frame effects. Theorem 13.4 identifies, for finite event structures, one sufficient condition under which it does not: causal locality of the realized- event dynamics, relative to an order that must itself be shown to be the operationally relevant causal order. (D2) Ensemble consistency versus realized-history consistency. A covariant master equation does not by itself supply a covariant stochastic ontology (Proposition 13.2). (D3) Clock neutrality versus absence of ontological ordering. A constrained quantum theory can lack a preferred clock variable while still admitting an additional generative interpretation; whether that interpretation is consistent must be proved (§14). (D4) Conditional compatibility propositions versus completed existence theorems. The Tomonaga–Schwinger and continuum-limit results of CTGP – explained (Proposition 17.16 and Theorem 17.17) are conditional, and their operator-theoretic and stochastic assumptions require further justification before they can count as existence theorems. (D5) Selective versus unread updates. Operational no-signaling concerns the unconditioned statistics of target outcomes, with the outcomes of all other operations summed over. Conditioning on an intermediate outcome is a different question, and Theorem 13.17 does not address it. (D6) Covariance of a law versus invariance of a realization. A Poisson law for collapse centres can be covariant under every isometry, although no realized configuration is invariant, and a covariant law for locations does not by itself supply covariant extended events (Theorem 13.21(d), Proposition 13.24). Likewise, events triggered by the law rather than by an experimenter are spontaneous whether or not their locations have a covariant law. (D7) Localization of a field amplitude versus localization of matter. A collapse that localizes a smeared linear field observable does not thereby localize matter, and macroscopic localization is an independent obligation of physical adequacy (L2). (D8) Lorentz invariance, covariance with background structure, preferred-frame effects, and superluminal signaling. A law can be covariant under all isometries while depending on background structure such as cosmological time; such structure can produce preferred-frame effects, which are measurable only if an operational prediction shows how they reach a record; and a theory with preferred-frame effects need not permit superluminal signaling (§13.11). Each of the four must be established or excluded separately. ---------------------------------------------------------------------------------------------------- 14. Temporal Quantum Reference Frames ---------------------------------------------------------------------------------------------------- 14.1 Clock-Neutral Descriptions A temporal quantum reference frame is not merely a coordinate clock. It is a quantum subsystem whose correlations with the rest define the evolution of another subsystem. In a constrained system, physical states satisfy Ĉ Ψ_phys = 0, and such a state is clock-neutral: it is not yet a description relative to any particular clock. Choosing clock A yields a conditional description ψ^{(A)}(τ_A) of the remaining degrees of freedom; choosing clock B yields ψ^{(B)}(τ_B). Höhn, Smith and Lock (2021) show, for a class of constrained systems whose constraint is a sum of clock and system Hamiltonians, that three formulations of relational dynamics are equivalent (clock-neutral relational observables, Page–Wootters conditional states, and quantum deparametrization), and they develop transformations between temporal frames. The spatial construction of Part II transfers only in part. Temporal reduction maps R_A and R_B can be defined, and where their inverses exist on the relevant sectors one can form V_{A→B} = R_B R_A^{−1}. For realistic clocks, however, these maps involve conditional states, clock POVMs, and sector restrictions, and they are not in general global unitaries of the kind used in Proposition 3.3. For ideal clocks they are unitary (Proposition 14.5). 14.2 Generation, Clock Readings, and Clock-Conditional Descriptions Definition 14.1 (Three temporal notions). In a CTGP model with clock subsystems: (i) The generative succession is the order of stages, or of event slots, in which the history is generated. (ii) A clock reading is the value of a clock observable of a physical subsystem at a stage. It is a Tier 3 proposition (§5.1) unless a recording event makes it a Tier 1 fact. (iii) A clock-conditional description is a description of the rest of the world conditioned on the readings of a chosen clock. CTGP identifies the ontological order of existence with (i) alone. A clock reading is physically instantiated information about a process, in the sense of the signal-based epistemology discussed in More Info, not automatically the fundamental ordering of existence. 14.3 A Worked Model With Imperfect Clocks Proposition 14.2 (Imperfect clocks in a generative model). Consider a model of the kind defined in Part II whose physical state includes one or more clock subsystems. (a) Clock-independence of event identity and probability. The measure over histories of Tier 1 events is fixed by the law and the complete initial state. Describing those events relative to one clock rather than another changes no event label, no stage, and no probability. (b) Periodic clocks. A clock of period M takes the same reading at stages σ_n and σ_{n+M}, so equal readings do not imply equal stages: τ_A(σ_1) = τ_A(σ_2) is compatible with σ_1 ≠ σ_2. A repeated reading means that the clock fails to distinguish two stages, not that generation has reversed. (c) Temporal delocalization. If a clock is in a superposition of readings at the stage of an event, the Tier 3 proposition “the clock read r when the event occurred” has no determinate value, although the event and its stage are determinate. For example, a counter prepared in an equal superposition of the readings 0 and 4 and advancing by one per stage reads n or n + 4, each with probability 1/2, at stage n. (d) Clock-relative order can disagree with generative order. If a non-ideal clock’s readings are recorded at successive events, the later event can carry an earlier reading. Consider a drifting quantum-walk clock on a ring of 32 sites, which advances one site per stage on average under the one-step propagator S exp(−i θ (S + S†)) with θ = 0.9. Because a ring has no intrinsic linear order, comparing two readings requires a convention: the displacement between consecutive readings r_1 and r_2 is the representative m of r_2 − r_1 modulo 32 in {−15, …, 16}, and the second reading counts as lower when m < 0 and as equal when m = 0. Assume that each event records the clock’s reading by a projective measurement of its position, so that the first recording leaves the clock in a definite position state, and that exactly one stage transition separates two consecutive recordings. Then, conditional on the first recorded reading, the exact probabilities on the 32-site ring are P(m < 0) = 0.104042655 and P(m = 0) = 0.338161965, to nine decimal places. They agree to better than 10⁻¹² with the infinite-lattice expressions (1 − J_0(1.8)² − 2 J_1(1.8)²)/2 and J_1(1.8)², where J_n are Bessel functions. By translation invariance these values do not depend on the first reading. Displacements of eight or more sites have total probability below 10⁻⁸, so any comparison convention that agrees with this one on displacements of fewer than eight sites gives the same values to that accuracy. The values do depend on the preparation and recording procedure: a clock not recorded at the first event, an unsharp recording, or a comparison across more than one transition gives different values. Proof. (a) Clock subsystems are part of the physical state, and the law takes the complete present state as input; the choice of a subsystem as clock for purposes of description is not an input to the law. If a clock interacts with the system, event probabilities depend on the clock’s physical state, but that is a dependence on the present physical state, not on the choice of description. (b) and (c) follow from the definitions. (d) The first recording projects the clock onto a position eigenstate |r⟩. During the single intervening transition the clock evolves by the stated propagator. By the Jacobi–Anger expansion, exp(−i θ (S + S†)) = Σ_{k∈ℤ} (−i)^k J_k(2θ) S^k, so on the infinite lattice the amplitude to move from r to r + m is, up to a phase, J_{m−1}(2θ). On the ring S^{32} = 1, and the amplitude for displacement m is the sum of the lattice amplitudes for all displacements congruent to m modulo 32. For m ∈ {−15, …, 16} the additional terms involve J_k(1.8) with |k| ≥ 16, each of magnitude below 10⁻¹⁴, so the ring probabilities differ from the lattice values J_{m−1}(1.8)² by less than 10⁻¹² in total. Summing the lattice values over m ≤ −1 gives Σ_{k≥2} J_k(2θ)² = (1 − J_0(2θ)² − 2 J_1(2θ)²)/2, using Σ_k J_k² = 1 and J_{−k}² = J_k², and m = 0 gives J_{−1}(2θ)² = J_1(2θ)². The exact ring values were computed directly from the 32-site propagator. The computation is item 16 of Appendix B. □ Proposition 14.2 has an external generative order built in: the stage index is a parameter of the model. It therefore shows that the three-way distinction of Definition 14.1 is consistent, and that clock-relative localization and ordering can differ from generative order without contradiction. It does not derive a definite generative order from clock-neutral data. That is the task of §§14.4–14.6.