# Quantum Reference Frames and Single-Outcome Dynamics — A Compatibility Framework for Generative Presentism Part 21 of 40 (1,561 words) What the obstructions teach. Randomizing the locations does not by itself yield a covariant spontaneous-collapse law. The construction holds exactly on every admissible configuration (Theorem 13.21). It fails in two precise places. First, admissibility is lost in large volumes (Proposition 13.22), and the order of interfering collapses then has physical consequences (Proposition 13.23). Second, no Lorentz-invariant law for the smearing of a free field at a centre exists (Proposition 13.24). A point process with a covariant law therefore does not supply covariant extended events. Each available repair adds structure at each event, and the structure should be named. Remark 13.25 (Repairs and their costs). (i) A background frame. Translates of a fixed smearing are covariant under translations, and under whatever rotations leave the smearing invariant, but not under boosts. By Campbell’s formula, the expected energy–momentum injected by the collapses centred in D is then λ Vol(D) ΔP(h). It is finite, and it is a fixed future-directed four-vector whose rest frame is a preferred frame for the energy injection. (ii) Conditioning on admissibility. By Proposition 13.22 this conditions on an event whose probability tends to zero as the domain grows, so it has no infinite-volume version without further work. On a finite domain the conditioned law has a forward form (Proposition 11.1), but that form depends on the extent of the domain still to be generated. (iii) Ordering conflicting collapses by a rule. Ordering them by the times of their centres in a chosen frame makes the law frame-dependent. In the numerical check, 234 of 300 sampled non- admissible configurations receive different laws in two frames of relative rapidity 0.6 (Appendix C.1, item C3(f)). A thinning rule applied in generative order, which keeps a candidate centre only if its region does not conflict with one already kept, is generable and produces admissible configurations by construction. But which collapses occur then depends on the generative order. (iv) Structure carried by realized events. In Tumulka’s rGRWf, the law for each flash refers to hyperboloids centred on the same particle’s previous flash (Tumulka 2006). The local frame is then supplied covariantly by an earlier realized event rather than by a background. For CTGP the law would depend on the location of a past event, so causal inheritance (C-g) requires that location to be physically instantiated in the present state; whether rGRWf admits such a representation is the open question noted in §13.12. (v) Cosmological structure. Where cosmological time T is regular, it is defined by the metric alone, so T ∘ ψ = T for every time-orientation-preserving isometry ψ. A smearing rule built only from T and g is then covariant under the entire isometry group. In an FLRW spacetime, for example, h_x can be the product of a function of T(p) − T(x) and a function of the distance from the comoving worldline through x, measured within the level set of p, with support small enough for that function to be smooth (Candidate extension 13.27). In FLRW spacetimes with a big-bang singularity the isometries fixing a point form a compact group, the rotations about the comoving worldline through it, so the obstruction of Proposition 13.24 does not arise. By symmetry, the stress–energy injected by a collapse then singles out the cosmic rest frame. Cosmological time could also fix the order of conflicting collapses covariantly. (vi) A mediating field. Bedingham (2011) obtains a Lorentz-covariant collapse dynamics free of the divergences of point-like collapse by letting the collapse act through an additional relativistic field. Whether a construction of this kind can be represented generably on CTGP’s hypersurfaces is open. (vii) State-dependent smearings would make the unread dynamics nonlinear in the state. Theorem 13.17, which concerns linear maps, would then no longer apply, and such rules are not pursued here. The CTGP-specific conclusion should be stated carefully. Route (v) uses only structure that CTGP already posits, and §13.11 shows that it addresses both obstructions consistently: the resulting dynamics is complete and normalized on every domain of finite four-volume, it can be generated forward along cosmological time in its free-field representation, and it admits no signaling between causally separated regions for the operations it covers. But it gives cosmological time a dynamical role. The collapse law, and in principle its observable consequences, would then refer to the generative foliation, which would change from empirically invisible, as in Corollary 13.5, to a physical hypothesis subject to test, the dynamical case of §14.6. The same holds for thinning in generative order in route (iii). Whether CTGP should accept such a role is a decision about the framework, not a result. Section 13.11 therefore develops route (v) as an explicitly labelled candidate extension, so that what it would assert and predict can be assessed, and does not adopt it. Without it, the remaining covariant options are routes (iv) and (vi) and a formulation of local collapse that does not attach an extended region to a point. None of these has yet been given for fields in a form that CTGP can adopt. The numerical checks (Appendix C.1, item C3) use a massless field in 1 + 1 dimensions, with causal diamonds of half-size 0.5 as regions. For these smearings the causal propagator can be computed exactly, and it vanishes exactly for causally separated regions. Poisson counts have mean 96.00 and variance 96.35 at expected value 96. At intensity 1.5, the fraction of admissible configurations falls from 0.599 to 0.015 as the box side grows from 1 to 2.5, below the bound of Proposition 13.22 throughout. Over 300 admissible configurations, with 1265 collapses and 1176 distinct linear extensions, the outcome covariances agree to 1.8 × 10⁻¹⁵. In direct Hilbert-space computations for three-collapse configurations, the history laws of different linear extensions agree to 4.8 × 10⁻¹⁶, and the characteristic function matches the closed form to 1.8 × 10⁻¹⁴. For two overlapping regions, the variance shift is E(h_1, h_2)²/(4σ_1²) to ten digits, and the two orders differ in total variation by 0.057. 13.11 A Candidate Extension: Collapse Guided by Cosmological Time This subsection develops route (v) of Remark 13.25 as a candidate extension of CTGP. It is not adopted. It gives cosmological time a role in the collapse law, which CTGP’s postulates do not contain, and accepting that role would be a decision about the framework (§14.6). The purpose is to state exactly what the extension would assert, what can be proved about it, what it would predict, and how far it can be realized by a physical coupling. Lemma 13.26 (Cosmological-time order and causal order). Let T be a time function, and suppose that every collapse region lies in a slab of cosmological time around its centre, O_x ⊆ { p : |T(p) − T(x)| ≤ τ }. (a) If T(x_j) − T(x_i) > 2τ, then O_i does not meet the causal future of O_j. Ordering collapses by the cosmological time of their centres therefore agrees with causal precedence for every pair whose centres differ in cosmological time by more than 2τ. (b) In Minkowski space with T = t, suppose the regions are translates of one compact convex region that is symmetric under time reflection about its centre. If O_j meets the causal future of O_i but O_i does not meet the causal future of O_j, then t(x_i) < t(x_j). Consequently, on every admissible configuration, ordering by the times of the centres is a linear extension of the causal order. Proof. (a) T increases strictly along future-directed causal curves, and every point of O_i has T ≤ T(x_i) + τ < T(x_j) − τ, which is at most the value of T at every point of O_j. (b) Write vectors as a = (a⁰, a^s), with a^s the spatial part, and let O_k = O + x_k, d = x_j − x_i, and D = O − O. Then O_j meets the causal future of O_i exactly when d ∈ J⁺ + D, with J⁺ the closed future cone, and O_i meets the causal future of O_j exactly when −d ∈ J⁺ + D. The set D is convex and invariant under a ↦ −a, and, because O is symmetric under time reflection, it is also invariant under (a⁰, a^s) ↦ (−a⁰, a^s). Suppose d⁰ ≤ 0 and d = k + w with k ∈ J⁺ and w ∈ D. Then w⁰ = d⁰ − k⁰ ≤ d⁰ ≤ 0. The set D contains (−w⁰, −w^s) and (w⁰, −w^s), and hence, by convexity, w′ = (w⁰ − 2d⁰, −w^s), because w⁰ ≤ w⁰ − 2d⁰ ≤ −w⁰. Then −d − w′ = (k⁰, −k^s), which lies in J⁺, so −d ∈ J⁺ + D and the relation is mutual. A one-way relation therefore forces d⁰ > 0. On an admissible configuration every relation between regions is one-way, and the causal order is the transitive closure of these relations, so it increases the time of the centres. □ Candidate extension 13.27 (Collapse guided by cosmological time; not adopted). Let (M, g) be an FLRW spacetime with a big-bang singularity, so that its cosmological time T is regular, coincides with the FLRW time, is smooth, and has Cauchy surfaces as level sets (§13.1), and let φ be the free field of §13.6. The candidate is stated for this class only. For other spacetimes with regular cosmological time, a smooth smearing law built from T and g alone would first have to be specified.