8.1 Generative Law and Single Outcome
With the law of §7, one stage-to-stage transition samples exactly one event k ∈ 𝒦 with probability
‖F_k W Ψ_σ‖² and updates the state. The realized history is a single sequence of events. Unrealized
sequences are not facts: they have kernel probabilities before the transition and no Had-properties
afterward. This is single-outcome realism (C-d) at the level of the primitive ontology. Macroscopic
outcomes would be patterns of many flashes, as in Bell’s flash ontology; in a three-particle model,
each flash is already an event.
8.2 Weak Non-Supervenience in the Sharp Regime
Proposition 8.1 (Two histories, one present). Take the sharp limit s → 0 and set J = 0, so that W =
exp(−iτ Σ v(r) P_{ij}(r)) is diagonal in the configuration basis and commutes with every P_{ij}(r).
Let N ≥ 4, and let Ψ_{σ_0} be any state with nonzero amplitude on the configuration b−a = 1, c−a =
3. The histories
ω = ((A,B,1), (A,C,3)) and ω* = ((A,C,3), (A,B,1))
both have nonzero probability, and both produce the present state Ψ_{σ_2} ≐ R_A^{−1}|1, 3⟩. The
resulting truth profiles differ: L contains Had(σ_0 → σ_1, (A,B,1)) under ω and Had(σ_0 → σ_1,
(A,C,3)) under ω*.
Proof. In frame-A coordinates the two relevant flash operators are multiples of the projectors
P_{AB}(1) → |x = 1⟩⟨x = 1| and P_{AC}(3) → |y = 3⟩⟨y = 3|. These commute, and their product is the
rank-one projector
P_{AB}(1) P_{AC}(3) = P_{AC}(3) P_{AB}(1) = |1, 3⟩⟨1, 3|.
With J = 0, W = exp(−iτ Σ v(r) P_{ij}(r)) is diagonal in the configuration basis, W |x, y⟩ =
e^{iθ(x,y)} |x, y⟩, so it commutes with both projectors. Along ω the unnormalized final vector is
therefore (λ/3) P_{AC}(3) W P_{AB}(1) W ψ_0 = (λ/3) P_{AC}(3) P_{AB}(1) W² ψ_0 = (λ/3) e^{2iθ(1,3)}
⟨1, 3|ψ_0⟩ |1, 3⟩. Along ω* the projectors are applied in the opposite order, which by the identity
above gives the same vector. Since ⟨1, 3|ψ_0⟩ ≠ 0, both histories have nonzero probability, and both
final states are the ray [|1, 3⟩], that is, [R_A^{−1}|1, 3⟩] in the neutral description. □
In this regime, the sharp flash (A,B,1) also makes the Tier 3 proposition “B is at separation 1 from
A just after σ_1” determinately true, illustrating the special case noted after Proposition 4.4.
8.3 Generic Parameters
For finite width, the situation divides as follows.
(i) Strong non-supervenience holds for all parameters with s > 0 and λ < 1, by Proposition 4.12.
The numerical check (Appendix B, item 11) takes a random present state and two different three-
stage histories, reconstructs the unique initial state for each, and confirms that both
histories end in the given present state with nonzero probability.
(ii) Weak non-supervenience at generic parameters is open. Among all 4,096 three-stage histories
from one random initial state at the generic parameter values of §8.4, no two distinct histories
end in exactly the same present ray. This is evidence that fixed-initial-state coincidences are
non-generic, not a proof.
(iii) Near-coincidences occur. The largest overlap between the present states of two distinct
histories in that sample is 0.985, so by Proposition 5.19(iv) no measurement on the present
state distinguishes them with error probability below 0.414. Distinct histories can thus be
observationally nearly indistinguishable even when their present states are not equal.
8.4 Numerical Verification
The claims above were checked numerically with N = 5, λ = 0.6, s = 0.8, J = 0.7, g = 0.9, and τ =
0.5 (and, for the continuous-time model of §9, hit rate Λ = 1.2 and total time T = 5), using a
random initial frame-A state; the checks of Theorem 13.4, Proposition 13.2, and Proposition 14.2 use
the separate settings stated in Appendix B. The checks of §§13.8–13.11 and §14.5 are reported
separately, in Table C.1 of Appendix C. The numerical checks confirm the analytical results in a
specific instance and guard against sign and ordering errors; they are not independent proofs.
Agreement “to machine precision” means deviations of order 10⁻¹⁶, the rounding error of double-
precision arithmetic. The script, its output, the software versions, and the random seeds are in
Appendix B.
Check | Status | Result
-----------------------------------------------+-------------+--------------------------------------
R_A, R_B, R_C unitary onto frame spaces | Prop. 3.3 | Confirmed (machine precision)
V_{A→B} equals |x,y⟩ ↦ |−x, y−x⟩ | §6.2 | Confirmed
Groupoid law V_{B→C} V_{A→B} = V_{A→C} | Prop. 3.5 | Confirmed
Instrument complete; W and all F_k invariant | §7.1–7.2 | Confirmed
V F^A_k V^{−1} = F^B_k and V W^A V^{−1} = W^B | Lemma 3.7 | Confirmed for all 16 events
Measures over 3 stages (16³ = 4096 histories) | Thm 5.12(a) | max |p_A − p_B| = 2.8 × 10⁻¹⁷
Frame A vs neutral description | Thm 5.12(a) | max |p_A − p_neutral| = 1.4 × 10⁻¹⁷
States along every history | Thm 5.12(b) | max (1 − |⟨Vψ^A, ψ^B⟩|) = 6.7 × 10⁻¹⁶
Same-form law in frame B | Cor. 5.13 | Total-variation distance 0.180
Two histories, one present (sharp, J = 0) | Prop. 8.1 | Same present ray; different L
Product state (frame A) → entangled (frame B) | §6.3 | 0 bits → 1.000 bit
Absolute collapse leakage | §7.4 | 0.894 = √(1 − 1/5)
Continuous time: 200 trajectories, 1158 events | Prop. 9.2 | Identical event records in A, B
Continuous time: final states | Prop. 9.2 | max (1 − |⟨Vψ^A, ψ^B⟩|) = 6.7 × 10⁻¹⁶
Generic parameters: exact coincidences | §8.3(ii) | None among 4096 histories
Generic parameters: largest overlap | §8.3(iii) | 0.985; Helstrom error ≥ 0.414
Every history reaches a given present | Prop. 4.12 | Confirmed for two 3-stage histories
History-dependent law, across initial states | Prop. 11.4a | Same present ray; TV = 0.1000
History-dependent law, fixed initial state | Prop. 11.4b | Same present ray; TV = 0.1000
Doob kernel vs future-conditioned measure | Prop. 11.1 | max diff = 8.3 × 10⁻¹⁷
Foliation independence, causally local | Thm 13.4(a) | All 136 foliations agree (7 × 10⁻¹⁸)
No dependence outside the predecessor set | Thm 13.4(c) | max marginal change 8 × 10⁻¹⁶
Order-ideal law vs marginal of full law | Prop. 13.7 | max diff 1.1 × 10⁻¹⁶
Gapless detectors, three-region test | Thm 13.9 | No shift (< 10⁻¹⁵); formula (c) met
Unsharp Lüders instrument at C | Ex. 13.10 | Shifts −0.0215 and −0.0466
Gaussian field collapse at C | Cor. 13.14 | No shift (< 10⁻¹³); formula met
Non-commuting “spacelike” pair | §13.4 | Orders differ: TV = 0.165
Same channel, different history measures | Prop. 13.2 | Mutual information 0 vs 1 bit
Drifting quantum clock, recorded readings | Prop. 14.2 | P(backward) = 0.104 (exact, ring)
Table 1. Numerical checks of the concrete model (N = 5). The “Status” column names the analytical
result that each check instantiates.
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9. Continuous-Time Embedding and the Status of Generative Stages
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9.1 The Question
CTGP describes generation as a continuous causal-temporal progression (C-h). Part II uses discrete
stages σ_0 < σ_1 < … with one stochastic transition per stage. A discrete-stage model can represent
selected stages of a continuous process, but it does not by itself establish compatibility with a
continuous generative ontology. This section gives a continuous-time version of the model, proves
that it has the properties of Part II, and states precisely what remains a postulate.
9.2 A Relational Jump Process
Definition 9.1 (Continuous-time relational flash process). Let H be the invariant Hamiltonian of
§7.1, let Λ > 0 be a hit rate, and define jump operators
J_{(i,j,d)} = √(Λ/3) · Σ_r √(w(r−d)) P_{ij}(r), so that Σ_k J_k† J_k = Λ · 1.
The process runs as follows. Hit times t_1 < t_2 < … form a Poisson process of rate Λ. Between hits
the state evolves by exp(−iH(t − t_j)). At a hit time, event k is drawn with probability ‖J_k ψ‖² /
Λ, and the state jumps to J_k ψ / ‖J_k ψ‖.
This is the standard quantum-jump unravelling of a Lindblad equation. Because Σ_k J_k† J_k is
proportional to the identity, the effective non-Hermitian Hamiltonian H_eff = H − (i/2) Σ_k J_k† J_k
= H − (iΛ/2) · 1 differs from H by a constant, so the evolution between hits is unitary up to a
uniform decay of norm, and the hit times are independent of the state. This is precisely the
structure of the original GRW model, here with relational jump operators.
Proposition 9.2 (Well-definedness, invariance, and covariance). (i) The process of Definition 9.1 is
a well-defined piecewise-deterministic Markov process on the unit sphere of H_phys, with almost
surely finitely many hits in any bounded time interval. (ii) Every trajectory remains in H_phys.
(iii) Its ensemble average ρ(t) obeys the Lindblad equation
dρ/dt = −i [H, ρ] + Σ_k J_k ρ J_k† − Λ ρ,
which preserves the set of density operators supported on H_phys. (iv) The frame-A and frame-B
processes, built from the frame forms H^A, J^A_k and H^B, J^B_k, assign the same probabilities to
every finite sequence of (hit time, event) pairs, and along every trajectory their states satisfy
ψ^B(t) ≐ V_{A→B} ψ^A(t) for all t.
Proof. (i) In finite dimension, with bounded H and state-independent Poisson hit times, the
construction is explicit: a Poisson process of finite rate has almost surely finitely many points in
any bounded interval, and between them the evolution is a fixed unitary flow. (ii) H and every J_k
are invariant, so exp(−iHt) and the normalized jumps map H_phys to itself. (iii) The Lindblad form
follows from averaging the jump process, which is standard; the term −Λρ is −(1/2){Σ_k J_k† J_k, ρ}
with Σ_k J_k† J_k = Λ · 1. Invariance of H and J_k implies that the generator commutes with
conjugation by U_h, so density operators supported on H_phys remain so. (iv) By Lemma 3.7, ‖J^A_k
exp(−iH^A t) R_A Ψ‖ = ‖J_k exp(−iHt) Ψ‖ and likewise for frame B; hit times do not depend on the
state; so the joint law of (hit time, event) sequences is the same in every frame, and induction
over hits gives the state relation, as in Theorem 5.12. □