What Was Missing
The Mathematical Supplement derives record formation: coherences between distinguishable pointer alternatives decay as the environment registers them, and pointer components stabilize. What review after review found missing was a law for the individual run — a rule saying which record mark appears, with what probability, and what it does to the state. The required object is a stochastic event law: a probability rule for the next record increment and an update map \[\lambda_t(dx\,|\,\rho) \qquad \text{and} \qquad \rho \longrightarrow \mathcal{J}_t(dx)[\rho].\tag{1}\] ACT supplies one in the most standard measurement-theoretic class available: a completely-positive quantum instrument, the object that already governs continuously monitored open systems (Davies, Kraus, Barchielli–Belavkin).
The Event Law
Postulate (ACT Event Law — instrument form). A monitored system carries a family of completely-positive maps \(\{\mathcal I_t(dx)\}\) — the record instrument selected by the environmental coupling — normalized so that \(\int \mathcal I_t(dx)\) is trace-preserving. The system's physical history is a marked stochastic process on the conditioned state \(\rho_t^X\):
Registration. Record marks arrive at total rate \(\Lambda_{\rm hit}(t)\) (state-independent under the normalized instrument). Given a mark, its value \(dx\) is drawn with probability \[\lambda_t(dx\,|\,\rho_t^X)=\Lambda_{\rm hit}(t)\,\mathrm{Tr}\!\big[\mathcal I_t(dx)(\rho_t^X)\big].\tag{2}\]
Update. On a mark of value \(x\) the conditioned state jumps \[\rho_t^X \;\longrightarrow\; \frac{\mathcal I_t(dx)(\rho_t^X)}{\mathrm{Tr}[\mathcal I_t(dx)(\rho_t^X)]}.\tag{3}\]
Ontology. The physical state is the pair \((|\Psi_t\rangle, X_{[0,t]})\): a globally unitary substrate together with one ontically actual record trajectory \(X_{[0,t]}\) sampled from (2)–(3). The wavefunction and instrument supply only a probability measure over histories; the added ontic content of ACT is that exactly one history is actual (the record-history analogue of Bohmian positions).
Two structural facts follow immediately. First, ensemble consistency is automatic: because \(\int\mathcal I_t(dx)\) is trace-preserving, averaging (2)–(3) over the unrecorded process returns exactly the open-system generator \(\dot\rho=\mathcal L_t[\rho]\) already derived — the instrument is an unraveling of that generator, so it cannot conflict with any ensemble-level quantum prediction. Second, the global state does not collapse: (3) is the conditioned description of the one realized history, while \(|\Psi_t\rangle\) evolves unitarily and conserves energy exactly. ACT is an ontic selection of one record history over globally unitary dynamics, not an objective modification of the wavefunction.
Anchoring Is Progressive: Three Distinct Rates
The single scalar \(\Phi\) of the earlier draft conflated three quantities that the instrument keeps separate:
Hit rate \(\Lambda_{\rm hit}(t)\) — how often the environment registers a mark. Pairwise decoherence rate \(\Gamma^{\rm dec}_{kl}=\Lambda_{\rm hit}\,(1-\mathrm{Re}\,C_{kl})\), where, for the positive-amplitude record-shaped subclass, the record overlap of pointer values \(k,l\) is the Bhattacharyya overlap \(C_{kl}=\int\!\sqrt{q(x|k)q(x|l)}\,dx\); for a general multiple-Kraus QND instrument it is \(C_{kl}=\int\!dx\,\sum_j \overline{a_j(x|k)}\,a_j(x|l)\) — how fast ensemble \(k\text{–}l\) coherence is suppressed, giving the coherence factor \(e^{-\Phi_{kl}}\) with \(\Phi_{kl}=\int_0^t\Gamma^{\rm dec}_{kl}\,ds\). Localization-information rate \(R_{\rm loc}\) — the rate at which the actual record trajectory accumulates which-sector information (a Kullback–Leibler mutual-information rate).
These coincide only in the sharp orthogonal-informative-record limit (disjoint likelihoods, \(C_{kl}=0\), nondemolition). Generically they differ: a system can absorb many weak hits (\(\int\Lambda_{\rm hit}\,ds\) large) while retaining coherence (\(C_{kl}\) near 1), or lose ensemble coherence without any single history acquiring informative which-sector records. “Anchoring” is therefore not a threshold in one scalar but progressive record accretion: record hit → anchoring trajectory → completed localization, the last defined operationally by posterior concentration \(p_{k^\ast}>1-\varepsilon\) along the actual history, not by a crossing of \(\Phi\).
Terminal Born localization. Under nondemolition, no persistent Hamiltonian mixing, and divergent cumulative pairwise discrimination \(\mathcal D_{kl}(t)\to\infty\) for every distinguishable pair, the posterior \(p_k(t)\) converges (Doob martingale) onto a single distinguishability class \(A\), and the probability of landing in class \(A\) is its initial Born weight \(\sum_{k\in A}p_k(0)\). Pointer values the records cannot separate localize only as a common equivalence class. (Note: the mutual-information integral \(\int R_{\rm loc}\) is bounded by the initial entropy \(H(p(0))\le\log N\); it is \(\mathcal D_{kl}\), not \(\int R_{\rm loc}\), that must diverge.)
Four Theorems (in the Sharp Special Case)
When the instrument reduces to sharp orthogonal projectors \(\{P_k\}\) with informative records — the projective special case below — the outcome statistics inherit four results, each verified numerically to machine precision. They are stated for that case; the general instrument obeys their ensemble-level content by construction.
1. Born uniqueness from no-signalling. Among mark rates of the form \(\lambda_k=\Lambda\,f(p_k)\) with one universal nonnegative \(f\), the unconditioned evolution is affine in \(\rho\) iff \(f(p)\propto p\). Any nonlinear \(f\) lets an unobserved local event shift an entangled partner's marginal — superluminal signalling. So the Born weighting is the unique no-signalling hazard. Stated precisely: the standard Hilbert-space kinematics and the usual trace-rule populations \(p_k=\mathrm{Tr}[P_k\rho]\) are assumed; what is shown is that nonlinear reweightings of those populations are excluded within the stated class — not that the trace rule itself is derived. Independently, requiring that coarse-graining outcomes commute with the law forces Cauchy additivity \(f(x+y)=f(x)+f(y)\), hence linearity — two routes, one answer.
For nonlinear \(f\), prepare \(|\psi\rangle=\sqrt{p}\,|00\rangle+\sqrt{1-p}\,|11\rangle\); Alice's unobserved events leave Bob the marginal \(w_0=f(p)/[f(p)+f(1-p)]\). With \(f(p)=p^2,\ p=0.7\) this shifts Bob's population \(0.700\to0.845\) — an operational signal; for \(f(p)=cp\), \(w_0=p\) identically.
2. No-signalling. Alice's local instrument satisfies \(\mathrm{Tr}_A\!\big[\sum_k P^A_k\,\rho\,P^A_k\big]=\mathrm{Tr}_A[\rho]\); Bob's reduced state, hence every local expectation, is invariant under Alice's unobserved events.
3. Order-invariance for commuting spacelike instruments. If \([P^A_a\otimes\mathbb 1,\,\mathbb 1\otimes P^B_b]=0\), the joint distribution \(P(a,b)\) is the same whichever event is taken first (difference \(=0\) exactly), so all observable joint statistics are frame-independent. The model's internal ordering of spacelike events is not; ACT's working position is that this ordering is unobservable bookkeeping, with its ontological status deferred to the covariant formulation. Bell correlations are inherited from the shared pre-anchored state exactly as the manuscript's Bell section states.
4. Ensemble consistency. Averaging the trajectories returns \(\mathcal L[\rho]=\Lambda\big(\sum_k P_k\rho P_k-\rho\big)\), the Joos–Zeh pointer-basis dephasing generator (mixture error \(=0\) numerically). Holds for the general instrument by the trace-preservation of \(\int\mathcal I_t(dx)\). The law reproduces every ensemble prediction of standard open-system quantum mechanics; its new content is ontological — one history is actual, observer or no observer.
Scope caveat. Exact ensemble consistency for the sharp projective hazard holds for binary pointer alternatives; for three or more distinct pointer values no nonnegative projector-jump hazards reproduce \(\Gamma_{kl}\propto(o_k-o_l)^2\) exactly, and the multilevel case is handled by the smeared instrument (bounded, compound-Poisson, translation-covariant pure-dephasing class with a Gram construction) rather than sharp projectors — see the Multilevel Event Law note.
Energy Accounting
The globally unitary substrate conserves total energy exactly given the model's symmetry. At the level of the conditioned reduced state, a sharp projective event shifts the expected energy by the coherence energy \(\Delta E_\text{coh}=\sum_{j\neq k}\mathrm{Re}\,\rho_{jk}H_{kj}\) (zero for pointer-diagonal states, and zero when the projectors commute with \(H\)); the smeared instrument likewise draws its offset against the bath. Honest summary: total energy is conserved by the unitary substrate; per-event reduced-state accounting along the trajectory ledger is open — in particular for record observables that do not commute with the kinetic energy.
What the Law Answers
What becomes actual? One record history \(X_{[0,t]}\), along which the conditioned state concentrates on one pointer sector. What happens to the other components? They persist as terms of the globally unitary \(|\Psi_t\rangle\) but are absent from the actual history — they enter no future mark probability for this history, within the stable-record Markovian regime in which the conditioned instrument is valid (the recohering, non-Markovian and relativistic cases remain open, per the Claim Ledger). Does the global state change? No; only the conditioned description (3) does. How are spacelike events coordinated? Observable joint statistics are order-invariant (Theorem 3); internal ordering awaits the covariant formulation. No-signalling? Theorem 2, exactly. Energy? Conserved by the substrate; trajectory ledger open.
Wigner's friend gets a dynamical rather than a bare answer: ACT assigns one actual laboratory record history whose conditioned description is definite once the friend's apparatus has registered informative records; redundancy of those records supports the instrument but is not itself the trigger. The quantum-eraser distinction is built in: reversible which-path correlations carry record overlap \(C_{kl}\to1\) and contribute no net decoherence, so erasure experiments proceed exactly as observed.
The Ledger
Postulated (one item): one record history is ontically actual — the pair \((|\Psi_t\rangle, X_{[0,t]})\) with \(X_{[0,t]}\) sampled from the instrument. The selection of the record instrument (rather than a conjugate unraveling) is argued from the Record Condition in the companion note Why This Unraveling?: within the displayed dilation class, redundantly recorded fragment-accessible data support only localization-type conditioning, never coherent momentum-kick conditioning. What remains irreducibly postulated is the ontic actuality of one history.
Derived (conditional on the event class): Born weighting from no-signalling (Theorem 1, two routes); no-signalling (Theorem 2); frame-independence of observable statistics (Theorem 3); exact ensemble consistency with open-system quantum mechanics (Theorem 4); terminal Born localization onto a distinguishability class by Doob convergence.
Dissolved: the “arbitrary \(\Phi=1\) threshold.” There is no threshold in the instrument at all — anchoring is progressive posterior concentration, and \(\Phi_{kl}\) is a pairwise coherence exponent, not a crossing. Open: (i) the per-event / per-trajectory energy ledger; (ii) a covariant field-theoretic instrument in which Theorem 3's frame-independence is manifest in the ontology; (iii) the general non-Markovian reversible/recorded split; (iv) the microscopic universal mass channel and its now-tightly-constrained window.
The Projective Special Case (June 2026 Draft, Recorded)
In the sharp orthogonal-informative-record limit the instrument reduces to sharp projectors \(\mathcal I_t(dx)\to P_k\cdot P_k\) with disjoint record supports, and the general law collapses to the original June 2026 formulation: a piecewise-deterministic process with hazard \(\lambda_k=\Lambda\,p_k\), \(\Lambda=d\Phi/dt\), event map \(\rho\to P_k\rho P_k/p_k\), and no-event survival \(S(t)=e^{-\Phi(t)}\) — so that in this limit the coherence factor \(e^{-\Phi}\) and the survival probability coincide. That coincidence, and the “first-to-threshold race” reading, are special-case artifacts: generically \(\Phi\) is pair-indexed \(\Phi_{kl}\), the hit rate \(\Lambda_{\rm hit}\) is separate from the decoherence rate, and there is no single survival exponent. The four theorems above are proved in exactly this special case, which is why they are stated conditionally; the general instrument carries their ensemble content by construction.