The problem

In the einselected pointer basis, physical dephasing is not the uniform-rate law of the flat projector unraveling but carries pair structure. For position-type observables with environmental correlation length \(\xi\), the reduced dynamics is \[\dot\rho_{kl} = -\Gamma_{kl}\,\rho_{kl}, \qquad \Gamma_{kl} = \Lambda\,\bigl[1 - C(o_k-o_l)\bigr], \label{eq:physkernel}\] with \(C\) the normalized environmental correlator: \(C(0)=1\), \(C(\Delta)\to0\) for \(|\Delta|\gg\xi\), and \(\Gamma_{kl}\approx \gamma\,(o_k-o_l)^2\) for \(|\Delta|\ll\xi\) (with \(\gamma = -\tfrac{\Lambda}{2}C''(0)\)). This is the exact form of collisional decoherence , of which the Joos–Zeh quadratic law is the small-separation limit. The event-law paper proved that projector jumps \(L_k=\sqrt{\Lambda_k}P_k\) cannot reproduce quadratic pair rates with nonnegative hazards, and posed hierarchical partial resolutions as the candidate repair. We first show the candidate fails—and then what succeeds.

The obstruction, hardened

Consider the most general static architecture built from projective partial resolutions: a family of coarse-grainings \(\{\Pi^{(c)}\}\), where resolution channel \(c\) partitions the outcome set into cells and fires as a jump into one cell, \(\rho\to \Pi_a^{(c)}\rho\,\Pi_a^{(c)}/p_a\), with channel weight \(w_c\geq0\). This includes the flat law (singleton cells), any fixed binary tree of coarse-grainings, and arbitrary collections of bipartitions. Since a partition is a union of bipartitions with common damping action, it suffices to consider bipartitions (“cuts”) \(A_c\,|\,B_c\). The unconditioned generator of channel \(c\) is \[\mathcal D_c[\rho] = w_c\bigl(P_{A_c}\rho P_{A_c} + P_{B_c}\rho P_{B_c} - \rho\bigr), \qquad \mathcal D_c[\rho]_{kl} = \begin{cases} 0, & k,l \text{ on the same side of } c,\\[2pt] -\,w_c\,\rho_{kl}, & k,l \text{ separated by } c, \end{cases}\] so the total pair rates are \[\Gamma_{kl} \;=\; \sum_c w_c\,\delta_c(k,l), \qquad \delta_c(k,l)=\mathbb 1[\,c \text{ separates } k,l\,]. \label{eq:cutcombo}\]

Theorem (Cut-cone obstruction). [thm:cutcone] The pair rates achievable by any static family of coarse projective jump channels form the cut cone: the set of nonnegative combinations [eq:cutcombo] of cut pseudometrics, i.e. exactly the \(\ell^1\)-embeddable pseudometrics . Every element obeys the triangle inequality. But any kernel of the form [eq:physkernel] with \(C''(0)<0\) violates the triangle inequality at small separations: for collinear points with gaps \(u,u\), \[\Gamma(2u)\;\simeq\;4\gamma u^2\;>\;2\gamma u^2\;=\;\Gamma(u)+\Gamma(u),\qquad \Gamma(2u)-2\Gamma(u)\;\to\;2\gamma u^2\;>\;0.\] Hence no static family of fixed projective pointer coarse-grainings—flat, hierarchical, or otherwise—reproduces such kernels; adaptive, history-dependent, or noncommuting-intermediate projective processes are not covered by this statement. Kernels with linear small-separation onset, e.g. \(C=e^{-|\Delta|/\xi}\) with distance \(1-e^{-|\Delta|/\xi}\) satisfying the triangle inequality via \(d(a)+d(b)-d(a{+}b)=(1-e^{-a/\xi})(1-e^{-b/\xi})\geq0\), are cut-cone-realizable and constitute the exception. (Verified: for three points with gaps \(a,b\) the unique cut weights require \(w_2=-2\gamma ab<0\), recovering the obstruction proposition as the \(n{=}3\) singleton-cut case; and numerically, \(\Gamma(2u)>2\Gamma(u)\) for the saturating kernel at \(u=0.2\xi\): \(0.0392>0.0199\).)

The obstruction is therefore not about the flat law’s lack of structure; it is geometric. Cut combinations are additive along a line, while which-path information grows quadratically in separation—distinguishability is not additive over intermediate stations. No architecture of sharp partial resolutions can be assembled into a quadratic.

The resolution: record-shaped events

Drop sharpness, not discreteness. Let the event operators be smeared registrations of the pointer observable \(\hat O\), \[M_x = m(\hat O - x), \qquad \int dx\, M_x^\dagger M_x = \mathbb 1 \;\;\Bigl(\textstyle\int m^2 = 1\Bigr), \label{eq:kraus}\] with events at hazard density \(\lambda(x)=\Lambda\,\langle M_x^\dagger M_x\rangle\) (total hazard \(\Lambda\), state-independent, so the no-jump attenuation is scalar and the conditional state is unchanged between events) and jump \(\rho\to M_x\rho M_x^\dagger/\mathrm{Tr}[\cdot]\). The unconditioned generator is \[\mathcal D[\rho] = \Lambda\Bigl(\int dx\, M_x\rho M_x^\dagger - \rho\Bigr), \qquad \mathcal D[\rho]_{kl} = -\Lambda\bigl[1-(m\!\star\!m)(o_k-o_l)\bigr]\rho_{kl}, \label{eq:gen}\] where \((m\!\star\!m)(\Delta)=\int m(u)\,m(u+\Delta)\,du\) is the autocorrelation of the hit profile. Populations are exactly conserved.

Theorem (Exact realization; canonical factorization with no independent scale). [thm:construction] Let \(C\) be a real, even, normalized positive-definite correlator with an absolutely continuous nonnegative spectral density \(\widetilde C(q)\) (real and even so that \(\widetilde C\) is itself real and even and \(\widetilde m=\sqrt{\widetilde C}\) admits a real, even branch; Bochner guarantees a positive spectral measure; the density assumption, satisfied by ordinary collisional and thermal kernels, is used here), and restrict to the bounded, compound-Poisson, translation-covariant pure-dephasing class—general Lévy–Khintchine generators with Gaussian-diffusion or infinite-activity components lie outside it. For a finite pointer set the density assumption can be dropped: any positive-semidefinite \(C_{kl}\) admits a Gram factorization \(C_{kl}=\langle v_k,v_l\rangle\), giving the finite-\(n\) analogue directly. Choose the hit profile \[\widetilde m(q) \;=\; \sqrt{\widetilde C(q)}\,. \label{eq:derived}\] Then \((m\!\star\!m)=C\) exactly, and the event process [eq:kraus]–[eq:gen] reproduces the physical dephasing [eq:physkernel] exactly, for all separations and all \(n\). For a Gaussian correlator \(C(\Delta)=e^{-\Delta^2/4\sigma^2}\) the hits are Gaussian with width \(\sigma\) and \[\Gamma_{kl} = \Lambda\bigl(1-e^{-(o_k-o_l)^2/4\sigma^2}\bigr) \;\xrightarrow{\;|\Delta|\ll\sigma\;}\; \frac{\Lambda}{4\sigma^2}(o_k-o_l)^2 ,\] the quadratic law with \(\gamma=\Lambda/4\sigma^2\).

Three remarks. (i) No independent scale is introduced—but one selection is postulated. The width is the environmental correlation length and the rate the environmental interaction rate, both read off the decoherence kernel. What is postulated is which unraveling is physical: the same generator admits the momentum-kick representation \(L_q=\sqrt{\widetilde C(q)}\,e^{iq\hat O}\) (and Kraus rotations thereof), and \(\widetilde m=\sqrt{\widetilde C}\) is the canonical localization-type factorization among them. The ACT reading—events shaped like the records—is the Record Condition applied here: pointer-position records are claimed to be the redundantly accessible environmental data. A companion note resolves this by an explicit translation-covariant scattering dilation: conditioning the outgoing environmental probe on its position reproduces \(M_x\) exactly, conditioning on its momentum reproduces the kick family exactly, and within the class of position-locally coupled secondary environments—the same locality assumption underlying einselection itself—the momentum-kick label is proven exactly unrecorded, so the Record Condition selects the localization unraveling as a theorem, not a postulate, within that class. The open problem narrows to extending the theorem beyond that class, in particular to the quantum-Brownian-motion/coherent-state regime. (ii) These are GRW’s hit operators with zero new dynamical constants (and one selection postulate). GRW postulates Gaussian hits with two new constants of nature (\(\lambda_{\rm GRW}\sim10^{-16}\) s\(^{-1}\), \(r_C\sim10^{-7}\) m) acting universally; here the same mathematical objects act at the environmentally derived rate and width, add no parameters, and modify no ensemble prediction. (iii) For collisional decoherence the identification is physical: the Gallis–Fleming/Hornberger–Sipe kernel is \(\Lambda(1-\int|f(q)|^2e^{iq\Delta})\) with \(f\) the scattering amplitude—each environmental scattering supplies the interaction underlying one step of the channel, with \(|f|\) playing the role of \(\widetilde m\); identifying the scattering’s ontic conditional update with the localization unraveling—rather than a momentum-kick or rotated representation—requires the Record Condition and the record-basis dilation analysis above.

The strictly unbounded quadratic law (\(\Gamma=\gamma\Delta^2\) for all \(\Delta\)) is recovered only in the limit \(\sigma\to\infty\), \(\Lambda=4\gamma\sigma^2\to\infty\): infinitely frequent, infinitely uninformative hits—the diffusive limit, outside the discrete-event class. Theorem [thm:cutcone] and this limit together dissolve the original obstruction physically within the discrete-event class: the only law discrete events cannot realize is the strictly unbounded quadratic kernel, which belongs to the diffusive or infinite-activity limit rather than the finite-rate discrete-hit class considered here—itself a perfectly physical effective description in some regimes, just not one built from finite-rate discrete events; ordinary finite-correlation-length collisional and thermal environments instead produce saturating kernels realized exactly by the process above.

Born statistics for arbitrary \(n\)

Theorem (Martingale Born rule). [thm:born] Under the event process [eq:kraus], the pointer populations are a bounded martingale: at each event, \[\mathbb E\bigl[p_k'\bigr] = \int dx\; \frac{m_k^2(x)\,p_k}{\sum_j m_j^2(x)\,p_j}\;\Bigl(\sum_j m_j^2(x)\,p_j\Bigr) = p_k \int dx\, m_k^2(x) = p_k .\] For distinct pointer values the likelihood ratios \(m_k^2/m_l^2\) are nondegenerate, so trajectories localize almost surely onto single outcomes ; by optional stopping, \(P(k)=p_k(0)\): the Born rule, for any \(n\), any spacing, any correlator—for a completed measurement, defined by divergent cumulative pairwise discrimination \(\mathcal D_{kl}\to\infty\) for every distinguishable pair (the correct condition — the mutual-information integral is bounded by the initial entropy and cannot diverge; indistinguishable pointer values localize only as a common equivalence class); at finite total hazard the same process is an incomplete weak measurement with a residual unlocalized possibility. (Verified: \(n=4\), \(40{,}000\) trajectories, outcome frequencies \([0.406,0.298,0.200,0.096]\) against \(p_0=[0.4,0.3,0.2,0.1]\); per-event martingale defect \(-0.0025\pm0.0015\).)

The unconditioned map is linear by construction, so no-signalling is automatic; the binary linearity-uniqueness argument carries over verbatim to the hazard density (affinity of the unconditioned map forces \(\lambda(x)\propto\langle M_x^\dagger M_x\rangle\)), stated here as carried over rather than re-proved.

The emergent cascade and the binary limit

Per event, each coherence is damped by the factor \(C(o_k-o_l)\): an event distinguishes \(k\) from \(l\) exactly to the extent the environmental record does. Two regimes follow from one law. For \(|\Delta|\gg\sigma\), \(C\approx0\): a single event fully resolves the pair, the resolution rate is \(\Lambda(1-C)\approx\Lambda=\Gamma_{kl}\), and the process reduces to the binary event law of the companion paper in the orthogonal-record limit \(C(\Delta)\to0\), to exponential accuracy for \(|\Delta|\gg\sigma\), with its rate identification \(\Lambda_{\rm binary}=\Gamma_{12}\) intact. (Verified: \(\Delta=6\sigma\): \(\Gamma_{12}/\Lambda=0.9999\); \(99.98\%\) of trajectories fully resolve at the first event; \(P(\text{outcome }1)=0.696\) vs \(0.7\).) For \(|\Delta|\lesssim\sigma\), many weak events resolve the pair progressively at rate \(\gamma\Delta^2\). Widely separated alternatives therefore resolve first and finer structure later—the coarse-to-fine cascade the multilevel problem seemed to demand—but the hierarchy is emergent from a single homogeneous event family, not an architecture imposed on it. The pairwise survival reading generalizes gracefully: the coherence factor equals the expected product of per-event overlaps, \(e^{-\Gamma_{kl}t}=\mathbb E\bigl[\prod_{\rm events}C(\Delta_{kl})\bigr]\), exact as a no-resolving-event survival probability in the separated regime and an expected residual coherence in general.

Numerical verification

Gaussian correlator, \(\sigma=1\), \(\Lambda=1\), \(o=(0,0.7,1.6,3.0)\), \(p_0=(0.4,0.3,0.2,0.1)\), \(40{,}000\) trajectories, seed 42; scripts to be posted with the note.

Ledger

Constructed here: the multilevel event law for the bounded compound-Poisson pure-dephasing kernel class (with the finite-\(n\) Gram extension); the canonical event-operator factorization with no independent scale; Born statistics for arbitrary \(n\) (martingale); the binary law as the separated limit; the emergent coarse-to-fine cascade; and the hardened obstruction (Theorem [thm:cutcone]) showing why every projective architecture had to fail. Postulated: the ontic status of events; that the physical secondary environment couples position-locally, the input class of a companion selection theorem that otherwise derives the localization-type factorization over momentum-kick and rotated representations—extending the theorem beyond that class, not the selection mechanism itself, is now the open item; hazard linearity (inherited from the binary case, restated not re-proved); the real, even branch choice in \(\widetilde m=\sqrt{\widetilde C}\), now a stated hypothesis of Theorem [thm:construction]. Open, unchanged: the noncommuting-Hamiltonian regime and per-event energy accounting with a bath-conditioned map; the non-Markovian (recohering) extension via the recorded/unrecorded generator split; the covariant formulation; degenerate and multidimensional pointer observables (the construction extends componentwise; stated, not worked). This note has not been externally reviewed; its claims should be audited with the same severity the binary law received before any surface beyond this note asserts them.