The question

The multilevel note constructed the event process from smeared Kraus operators and showed it reproduces the environmental dephasing generator exactly; the correct objection was that the same generator admits inequivalent representations, so \(\widetilde m=\sqrt{\widetilde C}\) is a canonical factorization, not a derivation of which unraveling is physical. The required repair: an explicit dilation identifying which probe observable the world registers, whether those registrations are redundant, and what conditioning on them induces. This note supplies that analysis at the strength its equations support.

The dilation and the two conditionings

Let the system carry pointer observable \(\hat O\) and let one environmental probe, prepared in wavepacket \(\varphi\), acquire the pointer value through the translation-covariant interaction (units \(\hbar=1\), unit coupling) \[U \;=\; e^{-i\hat O\otimes \hat p_E}, \qquad \Bigl(\textstyle\sum_k c_k|o_k\rangle\Bigr)\otimes|\varphi\rangle \;\longmapsto\; \sum_k c_k\,|o_k\rangle\otimes|\varphi_{o_k}\rangle, \quad \varphi_{o_k}(y)=\varphi(y-o_k). \label{eq:dilation}\] Tracing the probe gives the collisional dephasing factor \[\langle\varphi_{o_l}|\varphi_{o_k}\rangle=C(o_k-o_l), \qquad C(\Delta)\equiv\int dy\;\varphi^*(y)\,\varphi(y+\Delta), \qquad \widetilde C=|\widetilde\varphi|^2, \label{eq:kernel}\] the informational structure of the collisional kernel, with the momentum-transfer distribution as its spectrum . (This is a standard von Neumann dilation of translation-covariant pure dephasing; a microscopic scattering derivation with recoil, energy exchange, and finite interaction duration is listed open.) (Verified: overlaps match \(e^{-\Delta^2/4\sigma^2}\) to four decimals in the worked model.)

Condition on the outgoing probe. Two complementary registrations, two exact identities: \[\begin{aligned} \textbf{probe position } y:&\qquad K_y \;=\; \langle y|\,e^{-i\hat O\hat p_E}\,|\varphi\rangle \;=\; \varphi(y-\hat O) \;\;=\;\; M_y , \label{eq:hitcond}\\[4pt] \textbf{probe momentum } q:&\qquad K_q \;=\; \langle q|\,e^{-i\hat O\hat p_E}\,|\varphi\rangle \;=\; \widetilde\varphi(q)\,e^{-iq\hat O} \;\;=\;\; L_q . \label{eq:kickcond}\end{aligned}\] The rival unravelings are not rival dynamics; they are the two complementary questions one can ask the same outgoing probe. The hit profile in [eq:hitcond] is the probe wavepacket itself: the real-symmetric square root of the multilevel note is recovered when \(\varphi\) is real-symmetric, and chirped or complex packets carry their phase into the profile with \(\widetilde C=|\widetilde\varphi|^2\) unchanged. Which unraveling is physical is therefore the question: which probe observable does the world register?

The selection theorem

The probe meets its own environment. Model this third tier as a controlled position-diagonal fanout onto fragments \(F_1,\dots,F_N\) by an isometry \[V \;=\; \int dy\;|y\rangle\langle y|\otimes V_y , \qquad V_y^\dagger V_y=\mathbb 1, \label{eq:qnd}\] and display the physical regime it represents: a rapid, effectively quantum-nondemolition position monitoring, completed before probe dispersion acts. A global wavepacket-spreading condition is \(\tau_{\rm rec}\ll t_{\rm disp}\), with \(t_{\rm disp}\sim m_E\sigma^2/\hbar\) up to width convention; for a particular momentum component \(q\), a more directly operational sufficient condition is \[\frac{\hbar|q|\,\tau_{\rm rec}}{m_E}\;\ll\;\delta y_{\rm rec}. \label{eq:regime}\] Equation [eq:qnd] is stronger than generic locality: a local interaction Hamiltonian commutes with \(\hat y\), but the probe’s kinetic term does not, and free flight converts a momentum displacement into a later position displacement \(\Delta y=\hbar q\,t/m_E\). Outside the regime [eq:regime], a downstream position-local environment can record momentum—through time of flight, deflection direction, or recoil position; that is how momentum transfer is ordinarily measured—and the induced instrument then follows the recorded variable. The theorem below is a statement about the fast-record regime, stated as such.

Lemma (Phase-only momentum information is absent from fragment-local records). [lem:unrecorded] Under [eq:qnd], every fragment reduced state depends on the probe only through the position-diagonal data \(\langle y|\rho_P|y\rangle\). A phase-only momentum relabeling \(\varphi(y)\to e^{iqy}\varphi(y)\) leaves this data invariant, so fragment-only records cannot distinguish such relabelings or instantiate the coherent kick instrument \(\{L_q\}\). (Proof: \(V\) acts within \(\hat y\) eigenspaces; \(|e^{iqy}\varphi(y)|^2=|\varphi(y)|^2\). Verified: fragment-state change under \(q=2\) relabeling \(=4\times10^{-17}\).) This does not exclude statistical correlation between a fragment result and a subsequently performed complementary momentum measurement on the probe; it excludes fragment-only realization of coherent kick conditioning. Nor does [eq:qnd] imply that records form at all (\(V=\mathbb 1\) satisfies it); it constrains what any record that does form can contain.

Theorem (Fragment-record selection under position-diagonal fanout). [thm:selection] Let the system–probe interaction be [eq:dilation] and the probe–fragment isometry the controlled form [eq:qnd]. Suppose an outcome \(b\) is obtained by a POVM \(\{E_b\}\) on any collection of fragments, while the outgoing probe and unobserved fragments are traced out. Then the induced system operation is \[\mathcal J_b(\rho)\;=\;\int dy\;w_b(y)\,M_y\,\rho\,M_y^\dagger, \qquad w_b(y)=\mathrm{Tr}\!\big[E_b\,\mathrm{Tr}_{\bar F}(V_y\rho_F V_y^\dagger)\big]\geq0. \label{eq:postproc}\] Every fragment-only conditioned operation is thus a classical post-processing of the localization instrument \(\{M_y\}\); no coherent superposition of distinct \(y\)-conditionings, including the kick instrument \(\{L_q\}\), can be implemented by fragment records. What the fanout supports is the commutative record algebra generated by \(\hat y\), whose maximally refined parent instrument is \(\{M_y\}\) (up to relabeling and measure-zero equivalence).

Proof. After [eq:dilation] the system–probe state carries branch amplitudes \(c_k\) on \(|o_k\rangle|\varphi_{o_k}\rangle\); equivalently, conditioned on probe position \(y\), the system operator is \(M_y=\varphi(y-\hat O)\). The fanout [eq:qnd] maps \(|y\rangle|\text{frag}\rangle\to|y\rangle V_y|\text{frag}\rangle\), so a fragment POVM element \(E_b\) has expectation \(w_b(y)\) within each \(\hat y\)-eigenspace. Because the probe is traced out, \(\langle y'|y\rangle=\delta(y-y')\) eliminates all coherent \(y\neq y'\) cross-terms, leaving \(\mathcal J_b(\rho)=\int dy\,w_b(y)\,M_y\rho M_y^\dagger\). A coherent joint measurement involving the probe itself would reinstate the cross-terms and can recover complementary instruments; the restriction is to fragment-only records. \(\square\)

Corollary (Record Condition selection, given redundancy). [cor:redundancy] If multiple disjoint fragments independently carry sufficiently distinguishable \(y\)-dependent records—redundancy above the Record Condition’s threshold, a separate condition, quantified below—then the Record Condition selects the \(y\)-record algebra and, up to classical resolution, its parent localization instrument within this dilation class. The selection postulate of the multilevel note is, within the class and regime displayed and to the extent redundancy holds, a conditional theorem.

Two structural remarks. For real \(\varphi\), fragment conditioning introduces no continuously varying complex phase; for real nonnegative \(\varphi\) (an unchirped Gaussian), the coherence multipliers in [eq:postproc] are nonnegative and produce no phase reversal either—the sharp operational contrast with kick-type conditioning, which rotates the \((k,l)\) coherence by \(q(o_k-o_l)\). And fragment conditioning is a Bayesian update of the pointer populations: individual outcomes need not concentrate the posterior, while the average does (martingale property); in the worked model the mean posterior peak rises from \(0.500\) to \(0.755\).

The worked model and its redundancy

System with \(o=(0,\,1.2,\,3.0)\sigma\); a well-separated two-branch control \(o=(0,\,3.0)\sigma\), amplitudes \(\sqrt{(0.7,0.3)}\); Gaussian probe (\(\sigma=1\), real nonnegative); fanout onto \(N=8\) qubits by threshold rotations (angle \(2.4\)) at positions spanning the pointer range—an explicit instance of [eq:qnd]. \(H_2=0.881\) and \(H_3=1.485\) bits are the Shannon entropies of the two- and three-branch pointer distributions. Scripts, parameters, seed, and every number below will accompany the archival version.

The redundancy is reported honestly: the position information is genuinely fragment-distributed—four disjoint fragments each carry at least half the pointer entropy, so it is not localized in a single fragment—but it is peaked on the mid-threshold fragments and does not yet reach the deep quantum-Darwinism plateau (\(R_{0.2}=1\)). Deep redundancy in this fanout would require either more finely spaced thresholds or a larger fragment count; the model establishes fragment-distributed position records with partial redundancy, not the saturated Darwinism regime, and the theorem’s Corollary is conditional on that redundancy accordingly. The three-branch numbers show the cascade of the multilevel note operating through the records: the overlapping pair (\(\Delta=1.2\sigma\), \(C=0.70\)) is only partially resolvable by any single-probe measurement—its full position-record capacity is \(1.11\) of \(1.485\) bits—and completes over repeated collisions.

What this changes, and what it does not

Before this note, ACT had a valid localization unraveling and no physical derivation selecting it over momentum kicks. After it: within the displayed class and fast-record regime, every fragment-only record induces only a classical coarse-graining of the localization instrument, and coherent kick conditioning requires access to complementary, non-recorded probe degrees of freedom. The selection postulate of the multilevel note becomes a conditional theorem; the residual conditionality is displayed, not hidden: (i) the QND fanout and its dispersion regime [eq:regime]; (ii) actual redundancy, which the worked model achieves only partially; (iii) identification of the ontic process with the maximally refined record instrument; (iv) the ontic status of events, unchanged—this note determines which unraveling records support, not that one unraveling is ontic.

Ledger and the next construction

Derived here: the two-conditionings identity; the physical hit profile \(m=\varphi\); the absence of phase-only kick information from fragment records; the post-processing structure [eq:postproc] with proof; the record-algebra selection theorem. Displayed inputs: the controlled fanout [eq:qnd] with its dispersion regime [eq:regime]; redundancy as a separate condition, only partially met in the worked model. Open: deep redundancy in a realistic fragment architecture; the theorem beyond the QND class—the dispersive regime, where recorded time-of-flight interpolates the conditioning between \(M_y\) and \(L_q\); a microscopic collisional dilation with recoil and energy exchange; and the construction both critique threads now point at: the QBM dilation, treating position, momentum, dissipation, and energy in one exactly solvable bath, which would address the preferred-unraveling generalization and the trajectory-level energy ledger together. Non-Markovian and covariant formulations unchanged. Status: v3 revised following technical critique; has not undergone independent expert or peer review.