What Can Select a Branch? Kernel-Universal Born Martingales, a No-Selection Theorem for Classical Spacetime Noise, and Gravitationally Induced Collapse as a Physical Basis for Local Branch Selection
Local branch selection—a frame-independent postulate under which environmental decoherence einselects a branch structure and a local stochastic process realizes one branch with Born probability—has so far been implemented with continuous spontaneous localization (CSL), whose noise is phenomenological. Here we determine which physical noises can play this role. First we prove a kernel-universality theorem: for any set of commuting Hermitian collapse operators driven by Gaussian noises with an arbitrary positive-semidefinite correlation kernel \( C \), the einselected branch weights are exact martingales, so absorption occurs with Born probability for every kernel; the kernel controls only the selection rate \( \Gamma=\Delta^{T}C\,\Delta \), where \( \Delta \) is the vector of branch eigenvalue differences. For the Diósi–Penrose (DP) kernel \( G/\hbar|\mathbf{x}-\mathbf{x}'| \) acting on mass densities, the rate is \( \Gamma=2EG/\hbar \), with \( EG \) the gravitational self-energy of the branch mass-density difference: Penrose's collapse criterion emerges as the nondegeneracy condition of the Born martingale, and gravitationally induced collapse becomes a parameter-free physical implementation of the selection postulate, with \( \tau_{\rm sel}\sim3\times10^{-16}\, \)s for a milligram pointer. Second, we prove a no-selection theorem: noise coupled as a random Hamiltonian (classical stochastic spacetime fluctuations included) leaves every branch weight exactly constant along every realization while dephasing the ensemble—decoherence without selection—so classical metric-fluctuation dephasing cannot implement the postulate; selection requires measurement-type (norm-changing) noise. Third, for the symmetric EPR configuration we show that gravitational noise cross-correlation between the wings preserves no-signaling exactly (the cross terms of the correlated-kernel Lindblad dissipator vanish under the local partial trace), shifts only the selection rate, and is of relative size \( \sim R\,\delta^{2}/L^{3}\approx4\times10^{-9} \) at one-meter separation; a common-mode stall at perfect correlation is exhibited and verified numerically. Stochastic simulations confirm Born statistics across kernels to sampling precision, machine-zero weight drift for random-unitary noise, and the rate law across the full correlation range. The results upgrade the selection postulate from “CSL-compatible” to “gravitationally implementable,” identify exactly which proposed mechanisms can and cannot supply its randomness, and inherit the active experimental program constraining the DP model.
Local branch selection—a frame-independent postulate under which environmental decoherence einselects a branch structure and a local stochastic process realizes one branch with Born probability—has so far been implemented with continuous spontaneous localization (CSL), whose noise is phenomenological. Here we determine which physical noises can play this role. First we prove a kernel-universality theorem: for any set of commuting Hermitian collapse operators driven by Gaussian noises with an arbitrary positive-semidefinite correlation kernel ( C ), the einselected branch weights are exact martingales, so absorption occurs with Born probability for every kernel; the kernel controls only the selection rate ( \Gamma=\Delta^{T}C,\Delta ), where ( \Delta ) is the vector of branch eigenvalue differences. For the Diósi–Penrose (DP) kernel ( G/\hbar|\mathbf{x}-\mathbf{x}'| ) acting on mass densities, the rate is ( \Gamma=2EG/\hbar ), with ( EG ) the gravitational self-energy of the branch mass-density difference: Penrose's collapse criterion emerges as the nondegeneracy condition of the Born martingale, and gravitationally induced collapse becomes a parameter-free physical implementation of the selection postulate, with ( \tau_{\rm sel}\sim3\times10^{-16}, )s for a milligram pointer. Second, we prove a no-selection theorem: noise coupled as a random Hamiltonian (classical stochastic spacetime fluctuations included) leaves every branch weight exactly constant along every realization while dephasing the ensemble—decoherence without selection—so classical metric-fluctuation dephasing cannot implement the postulate; selection requires measurement-type (norm-changing) noise. Third, for the symmetric EPR configuration we show that gravitational noise cross-correlation between the wings preserves no-signaling exactly (the cross terms of the correlated-kernel Lindblad dissipator vanish under the local partial trace), shifts only the selection rate, and is of relative size ( \sim R,\delta^{2}/L^{3}\approx4\times10^{-9} ) at one-meter separation; a common-mode stall at perfect correlation is exhibited and verified numerically. Stochastic simulations confirm Born statistics across kernels to sampling precision, machine-zero weight drift for random-unitary noise, and the rate law across the full correlation range. The results upgrade the selection postulate from “CSL-compatible” to “gravitationally implementable,” identify exactly which proposed mechanisms can and cannot supply its randomness, and inherit the active experimental program constraining the DP model.
A recently proposed local measurement axiom (Postulate M) replaces the instantaneous global projection of textbook quantum mechanics with branch selection triggered by completed local decoherence, while a companion ψ-ontic analysis identifies continuous spontaneous localization (CSL) as a concrete local mechanism for the selection itself. Here we test this two-stage account quantitatively in the Einstein–Podolsky–Rosen (EPR) scenario. We integrate the nonlinear CSL stochastic Schrödinger equation for an entangled spin pair in which one particle couples to a local spin bath with random couplings, bath self-dynamics, and intra-bath interactions, while the noise field and collapse operator act only at the measurement site. Analytically, we derive an exact law for the maximal Clauser–Horne–Shimony–Holt (CHSH) value under local decoherence, Smax(t)=21+sin2(2θ)∣z(t)∣2, with z(t) the decoherence factor and θ the entanglement angle: the CHSH excess over the classical bound is consumed, never the measurement-basis probabilities, which are exactly conserved. We further show that the selection stage is governed by an exact martingale: the branch weight obeys a scalar Itô equation whose derivation requires only the orthogonality of the branches—supplied by the spin structure, not by the environment—so the Born rule emerges as an absorption probability irrespective of the residual bath coherence. Numerically, the law holds to machine precision (∼ 10−15) for all three bath classes, including interacting environments, and through finite-size recurrences; stochastic trajectories confirm Born statistics to within sampling error, with all realizations fully resolved. The ensemble-averaged reduced state of the distant particle is invariant under the local collapse-operator axis and rate—parameter independence as a structural theorem, verified to machine precision in the unitary sector—and branch amplitudes are invariant across all environmental initial conditions and dynamics (∼ 10−11), confirming that outcome randomness is not of environmental origin. In the symmetric configuration, with independent noises at both wings, the two local noises provably drive a single global branch martingale at the summed rate: selection is one joint event whose absorbing states are the correlated branches—mismatched outcomes are dynamically excluded (zero events in 300 realizations, against ∼ 25% each for a product-state control)—the joint statistics are invariant under swapping which wing completes first (93% vs 9% orderings, distributions equal within sampling error), and sampled outcomes reproduce the singlet’s measurement correlation E(β) = − cos β—with β the angle between Alice’s and Bob’s measurement axes—including a CHSH value |S| = 2.710±0.073, consistent with the Tsirelson bound 2√2 within sampling errors. Decoherence and local stochastic selection jointly reproduce the empirical content of quantum measurement in the EPR scenario with no nonlocal dynamical element: the only nonclassical resource is the nonseparability of the initial state. Nonlocal projection is thus dispensable not only axiomatically and ontologically but operationally.
Version 2.8 — submission manuscript. Preprint prepared for submission to Foundations of Physics. Supersedes v2.6, v2.5, v2.3, v2.2 and v1. A gravitational collapse model in which spacetime geometry is postulated to be irreducibly classical and superpositions of massive systems decay under a stochastic dynamics driven by a physical temporal medium: one stationary noise field combining the Diósi–Penrose spatial kernel with a coloured temporal kernel of finite universal bandwidth ω_u — the single new parameter of the primary model. The decoherence envelope is derived exactly (Gaussian onset, DP exponential tail), and two universality propositions are proved: the DP tail is fixed by the kernel's zero-frequency weight alone, and a Gaussian onset is generic for any finite-bandwidth medium — making the crossover, not any particular kernel, the experimental signature. A selectivity theorem shows each configuration draws its collapse dominantly from noise modes near its own gravitational frequency ΔE_g/ℏ. The parameter-free tuning of earlier versions is retained as named Conjecture C1, with a proof that no single stationary spectrum realises it for all configurations at once. The coloured kernel suppresses collapse-induced spontaneous radiation, reconciling the model with underground null results; heating bounds survive at full strength. Predictions include a closed-form optimal measurement window (separation 0.330 at t = 0.837 τ_dec under C1) and a mass-and-time ceiling on Bell violations, with five falsification conditions. The BMV/GIE treatment is current with the 2024–2026 literature. Changes in v2.8 (Appendix B, B11–B13): universal-bandwidth primary model with derived (ω_u, R_0) window; C1 demoted to conjecture with proved obstruction; no-signalling corrected and branch-conditioned; non-Markovian trajectory literature engaged, Born-rule theorem declared open; one prediction and the "present moment" identification withdrawn with reasons; GIE dispute disentangled; references verified. This paper builds on v1 of the companion collapse-model work (10.5281/zenodo.19960071). The lowest-energy principle assumed here was withdrawn in v2.2 of that paper and does not appear in the current version (v2.8); the framework presented here has not been revised accordingly. Retained as a record of the earlier line of enquiry All revisions, withdrawn claims, and corrected errors are documented in Appendix B; open problems in Section 8. AI tools (Claude, Anthropic) assisted with formalisation and editing; the conceptual framework originated with the author, who takes full responsibility. Full declaration in the manuscript.
Generalized trace dynamics (GTD) proposes that quantum theory is the thermodynamicequilibrium of a deterministic matrix dynamics, and that the collapse of the wave function iswhat controlled departures from that equilibrium produce. In the same framework the StandardModel and gravitation are sought as emergent structures, so that one programme aims atunification and at the measurement problem together. We ask, carefully, what such a programmemust deliver to explain \emph{both} state-vector reduction \emph{and} the Born rule---and weargue that the right target is collapse-with-Born statistics, not the specific phenomenologicallaw of continuous spontaneous localization (CSL), which may or may not be the correct law. Wefirst set out the four ingredients any dynamical account of collapse requires, recall the sense inwhich CSL's stochastic structure is forced by its premises, and locate the premise a foundationaltheory most naturally questions. We then identify what GTD already supplies---an intrinsicallyfermionic, anti-self-adjoint reduction channel---and what it does not: a derived noise, aspatial collapse operator, and the Born weights. On the simplest GTD-relevant model we establishseveral facts. A static distribution over a state-independent drift variablecannot reproduce the Born rule; the randomness must be a state-coupled process. For continuousreduction that completes, the Born rule is the statement that the branch-weight process is adriftless It\^o martingale. Because the collapse noise in GTD is emergent from a\emph{deterministic} substrate (hence smooth), its simplest white-noise limit is\emph{Stratonovich}, which carries a spurious drift; we show this both spoils the martingale atorder one (independently of the noise correlation time) and prevents the collapse from completing.This is not a claim that non-white noise generically spoils Born---it does not---but that a\emph{bare} smooth-noise branch-weight equation, lacking a response counterterm, fails. The Bornrule is recovered only if a dissipative drift---a candidate being the response part of the\emph{projected} Adler--Millard charge fluctuation---cancels this Stratonovich drift; we presentthis as a precise, conjectural acceptance test, not a derived identity. We further show that thisrequired cancellation is equivalent to linearity of the noise-averaged dynamics, and hence tono-signalling, so the Born condition acquires three equivalent faces; in generalized trace dynamics,however, no-signalling is itself an emergent, off-equilibrium property rather than a fundamentalinput, so this sharpens the open problem rather than resolving it. Ordinaryfluctuation--dissipation relations are exact at thermal equilibrium; \emph{if} the projectedAdler--Millard equilibrium relation induces the specific nonlinear counterterm required, thenthe Born rule is exact at equipartition and violations are controlled by the mismatch from thatbalance---a quantity distinct from the overall strength of collapse, and small near equilibrium.We separate throughout what is established from what is conjectural, and identify the singlecalculation---the interacting, projected Adler--Millard fluctuation propagator---on which theprogramme now turns.
Abstract This paper proposes a parameter-free objective reduction theory where wavefunction collapse is induced by gravitational self-energy differences between mass distributions. Unlike Continuous Spontaneous Localization (CSL) models, which rely on phenomenological parameters, the collapse rate in this model (Γ = Eg/ħ) is uniquely determined by universal constants G and ħ. Key Highlights: Dynamical Derivation of the Born Rule: The model demonstrates that the Born probability rule emerges naturally from the martingale property of the stochastic dynamics, rather than being postulated. Experimental Consistency: The theory is shown to be consistent with current laboratory constraints, including double-slit interference experiments with large molecules (mcrit ≈ 10⁻¹⁴ kg) and loophole-free Bell inequality tests. Astrophysical Stability: Analysis confirms the stability of compact objects like neutron stars and the preservation of primordial fluctuations in the Cosmic Microwave Background (CMB). Relativistic Extension: A Lorentz-covariant field-theoretic generalization is provided, using the energy-momentum tensor to source collapse noise while respecting microcausality. The results suggest a self-contained, falsifiable framework for the transition from quantum to classical spacetime without the need for adjustable parameters.
We study idealized levitated protocols that create spatial superpositions of massive test particles. For each protocol, we compare the dimensionless contrast-loss exponent of mass-proportional continuous spontaneous localization (CSL) with the Diósi--Penrose (DP) self-energy exponent EGτ/ℏ. We first prove that the point-particle CSL separation kernel has an exact random-unitary realization: Gaussian momentum kicks arriving at Poisson-distributed times produce the same unconditional decay of spatial coherence, although a pure state conditioned on the complete kick record remains pure. The separation kernel alone therefore specifies an operational decoherence law, not the occurrence of objective collapse. We then show that the ratio of the CSL and DP exponents is independent of particle mass and interrogation time. In the point-particle model it depends only on branch separation and an effective distance; for the standard GRW reference parameters, its resolved-superposition crossover is x∗≈1.91nm. For rigid spherical bodies with an arbitrary normalized radial mass profile, total mass, overall density scale, and interrogation time again cancel, leaving a dimensionless geometry factor. The results distinguish three requirements for a decisive experiment: detectable absolute effects, a controlled comparison of CSL and DP scales, and observables capable of discriminating physically different dynamics that share the same ensemble decoherence kernel.
The textbook projection postulate of quantum mechanics stands in tension with the symmetries of relativistic spacetime: it is stated as an update applied at a single instant to the entire composite state, which presupposes a preferred simultaneity surface, and its description of the distant partner of an entangled pair invites the reading that measurement physically changes that partner. We argue that this tension arises from one removable ingredient of the axiom, and we make precise what that ingredient is and is not. Separating the postulate into three logically independent claims—definite-outcome realization, Born-rule probability, and a globally-timed projection—we show that the third can be replaced by a locally triggered rule: branch selection upon completed local decoherence, in the einselected pointer basis (Postulate M). The measurement operator itself is not modified: the update at a site A is of the local form PiA⊗IB under both the standard postulate and Postulate M, and the two rules yield identical post-measurement states and probabilities, the standard rule being recovered as the vanishing-decoherence-time limit. Postulate M is therefore a reinterpretation of standard quantum mechanics rather than a modification of it; what it changes is the trigger of the update, from a global instant to a local decoherence event, so that the axiom can be stated without reference to any foliation. Parameter independence holds exactly, as it does in the standard theory; the final branch structure and joint outcome statistics are foliation-independent; and the exchange symmetry of the entangled pair is respected. We illustrate the framework in the Einstein–Podolsky–Rosen (EPR) scenario with one and with both particles coupled to local spin environments. The Born-rule probabilities in the pointer basis are exactly conserved throughout decoherence, while the maximal CHSH (Clauser–Horne–Shimony–Holt) value, obtained from the Horodecki criterion, is Smax=21+∣z(t)∣2 in terms of the decoherence factor z(t), interpolating between the Tsirelson bound and the classical bound; this corrects the constant-22 claim of Ref. [10]. The single symmetry that quantum mechanics does break, outcome independence, is located entirely in the nonseparability of the global state, never in the measurement operator: it transmits no signal and selects no frame. The single-outcome question—which branch is realized—is left open.