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.