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ACT Working Note · Instrument formulation (v3.1) · July 2026

The ACT Event Law: The Instrument Formulation

Status — amended August 2026. This note states the event law in its single-condition form, $h=\dot\Phi_{\rm irr}$. That has been superseded by the two-condition law of the August 2026 manuscript, $h(t)=g_\mathcal{I}(t)\,\dot\Phi_{\rm irr}(t)$, in which an admissible record architecture (approximate QND coupling in the pointer variable plus fragment readability) is a separate eligibility condition and is not implied by $\dot\Phi_{\rm irr}>0$. The law is additionally scoped to CP-divisible record-forming regimes, and the hazard is now read as an upper-bound approximation to genuine record-production timing. What follows is retained as the derivation of the rate factor. Within that scope it remains the current statement of ACT's event law. It supersedes the June 2026 projective/hazard draft (piecewise-deterministic pointer jumps at \(\lambda_k=\Lambda p_k\) with survival \(e^{-\Phi}\)), which survives here only as the sharp orthogonal-informative-record limit of the instrument — recorded in §The Projective Special Case. Canonical companion: the site's Claim Ledger and Mathematical Supplement §5.

One ontic history, one completely-positive instrument

Amended August 2026 — rate factor current; see the two-condition law above.

Kelly Sonderegger · Independent Researcher, Santaquin, Utah · ksondere@gmail.com

Abstract

Every reviewer of ACT converged on one missing object: an explicit event law — a rule for the occurrence of a record event and a map \(\rho\to\mathcal{J}[\rho]\) stating what an event physically does. ACT's canonical answer is a completely-positive (CP) quantum instrument. At registration rate \(\Lambda_{\rm hit}(t)\) a record mark \(dx\) is drawn with probability \(\lambda_t(dx\,|\,\rho)=\Lambda_{\rm hit}\,\mathrm{Tr}[\mathcal I_t(dx)(\rho)]\) and the conditioned state updates as \(\rho\to\mathcal I_t(dx)(\rho)/\mathrm{Tr}[\cdot]\), with \(\int\mathcal I_t(dx)\) trace-preserving so the ensemble average is exactly the standard open-system generator. Anchoring is not a threshold crossing but progressive record accretion: successive marks concentrate the conditioned state onto one pointer sector along the single actual record history. The physical state is the pair \((|\Psi_t\rangle, X_{[0,t]})\) — a globally unitary substrate plus one ontically actual record trajectory; the wavefunction and instrument supply only a probability measure, and making one history actual is the added ontic variable. Three rates that the old scalar \(\Phi\) conflated are now distinct: the hit rate \(\Lambda_{\rm hit}\), the pairwise decoherence rate \(\Gamma^{\rm dec}_{kl}=\Lambda_{\rm hit}(1-\mathrm{Re}\,C_{kl})\), and the localization-information rate \(R_{\rm loc}\); they coincide only in the sharp orthogonal-informative-record limit, where the instrument reduces to the June projective law. Within that special case four results still hold, each verified numerically: Born weighting is the unique no-signalling hazard; local event maps leave spacelike marginals invariant; commuting spacelike instruments give order-independent joint statistics; and the ensemble average reproduces the Joos–Zeh dephasing master equation. The note closes with an honest ledger: one postulate (one record history is ontically actual), four conditional theorems, and the open items — a fully covariant field-theoretic instrument chief among them.

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 \(\rho_t^X\):

  1. 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 \[\lambda_t(dx\,|\,\rho_t^X)=\Lambda_{\rm hit}(t)\,\mathrm{Tr}\!\big[\mathcal I_t(dx)(\rho_t^X)\big].\tag{2}\]

  2. Update. On a mark of value \(x\) the conditioned state jumps \[\rho_t^X \;\longrightarrow\; \frac{\mathcal I_t(dx)(\rho_t^X)}{\mathrm{Tr}[\mathcal I_t(dx)(\rho_t^X)]}.\tag{3}\]

  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.