Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel $C_{\rm FB}=\mathbb E_ω[ψ_Fψ_B]$ resolves an organized connected sector, with weight $ρ_{\rm BSM}=-C_{\rm FB}(x,x)$ on the anticorrelated branch.
Here we investigate how this sector forms, localizes, and propagates under Bohm/Fisher feedback. We retain the branching covariance on the prescribed background as the bare noise kernel and truncate the nonlinear interaction to Bohm/Fisher drift vertices. A Martin--Siggia--Rose--Janssen--de Dominicis formulation and a causal two-loop two-particle-irreducible (2PI) closure determine response and correlation functions self-consistently. While the free connected theory exhibits secular growth and ultraviolet accumulation, the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of the localized sector, and a self-similar Fisher construction defines a saturated information velocity $c_\star$.
A Born--Oppenheimer separation connects this internal organization to collective propagation. The instantaneous adiabatic kernel has two pole families at fixed internal momentum: a gapless difference branch and a gapped sum branch. Projecting both inverse response kernels onto the same localized internal profile defines their collective coefficients. If both propagation speeds match $c_\star$ and the projected gap matches $mc_\star^2/\hbar$, the gapped branch takes the infrared Klein--Gordon form. These matching conditions define a candidate relativistic fixed point whose dynamical realization remains to be tested.
Branching stochastic mechanics (BSM) provides a reciprocal branching representation of the Schrödinger--Nagasawa pair. Its centered forward--backward kernel CFB=Eω[ψFψB] resolves an organized connected sector, with weight ρBSM=−CFB(x,x) on the anticorrelated branch.
Here we investigate how this sector forms, localizes, and propagates under Bohm/Fisher feedback. We retain the branching covariance on the prescribed background as the bare noise kernel and truncate the nonlinear interaction to Bohm/Fisher drift vertices. A Martin--Siggia--Rose--Janssen--de Dominicis formulation and a causal two-loop two-particle-irreducible (2PI) closure determine response and correlation functions self-consistently. While the free connected theory exhibits secular growth and ultraviolet accumulation, the dressed theory develops a finite relative screening length. A reduced numerical evolution shows bounded formation of the localized sector, and a self-similar Fisher construction defines a saturated information velocity c⋆.
A Born--Oppenheimer separation connects this internal organization to collective propagation. The instantaneous adiabatic kernel has two pole families at fixed internal momentum: a gapless difference branch and a gapped sum branch. Projecting both inverse response kernels onto the same localized internal profile defines their collective coefficients. If both propagation speeds match c⋆ and the projected gap matches mc⋆2/ℏ, the gapped branch takes the infrared Klein--Gordon form. These matching conditions define a candidate relativistic fixed point whose dynamical realization remains to be tested.
Can finite-range correlations hide in the statistics of an extended quantum state? The Schrödinger-Nagasawa transform represents the wave function by positive forward and backward diffusion fields whose product is the Born density. We promote them to branching superprocesses, ΦF and ΦB: diffusion samples stochastic paths, whereas Bohm/Fisher-controlled branching generates genealogies of alternative continuations. With rates evaluated on the prescribed Born density, their means reproduce Schrödinger dynamics exactly. The connected sector exhibits supercritical, critical, and subcritical clustering in confinement and has critical dimension dc=2 in free space. Stationary eigenmodes remain extended; subcritical clusters acquire the reduced de~Broglie scale. We then let the branching rate respond to the fluctuating product ΦFΦB. In the reciprocal basis, the fields equal a smooth reference R plus centered fluctuations ψF,ψB, defining the signed kernel CFB(x,y)=Eω[ψF(x)ψB(y)]. On the anticorrelated branch, ρBSM(x)=−CFB(x,x) is the positive organized connected weight. Stationary matching of this weight to the Born profile is imposed as a closure condition, while off-diagonal decay defines the correlation range. The pair equation splits into collective and relative sectors: spectral cancellation selects the Born collective mode, while suppression of the leading density fluctuation selects the anticorrelated source channel. With relative diffusivity Deff and positive relaxation rate μFB, correlations have screening length ξFB=Deff/μFB. Within this reciprocal closure, an extended Born-shaped connected weight and a finite correlation range can coexist in one stochastic kernel.
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.
We formulate a conservative, [Formula: see text]-invariant, Hamiltonian extension of Schrödinger/Gross–Pitaevskii hydrodynamics in the rest frame of an effective coherent medium in which the complex matter field [Formula: see text] couples to a real, dimensionless phase-stiffness field [Formula: see text]. The auxiliary field dynamically modulates the Bohm quantum potential through a coupling proportional to the local Fisher-information density, while itself evolving through an inertial equation driven by amplitude gradients. The variationally consistent amplitude-gradient coefficient is [Formula: see text], so positive excursions of [Formula: see text] produce an adaptive softening of quantum pressure. From a single variational principle we obtain coupled evolution equations and a norm-conserving energy functional that is lower bounded in the positive-stiffness sector. Standard Schrödinger/Gross–Pitaevskii dynamics is recovered when [Formula: see text], whereas the classical Hamilton–Jacobi limit requires both [Formula: see text] and the disappearance of the auxiliary excitation. Linear analysis reveals an independent gapped stiffness mode in addition to the Bogoliubov branch, whose zero-momentum gap and quadratic dispersion coefficient determine the parameter ratios [Formula: see text] and [Formula: see text] governing restoration, smoothing and inertia. In the nonlinear regime the same feedback produces an oscillatory auxiliary-field response and a numerically converged, parameter-dependent reduction of wave-packet spreading. The packet remains dispersive rather than forming an interaction-free self-bound state. The framework further predicts a conditional logarithmic visibility relation under a calibrated interferometric readout and reversible finite-rate response loops under cyclic driving. These loops quantify conservative inertial lag rather than dissipated energy or intrinsic hysteretic memory. These results establish the phase-field duality as a minimal and falsifiable Hamiltonian extension in which coherence behaves as an elastic medium.
We formulate minimal and dissipative Continuous Spontaneous Localization (CSL) dynamics in the functional Schrödinger representation for a non-relativistic bosonic field. In this framework, the Fock-space state is encoded in a wave functional, and fixed particle-number wave functions are obtained by sector projection. For the minimal CSL coupling to the smeared mass density, this projection gives the standard nonlinear stochastic dynamics in each N-particle sector, with the collapse operator acting on the total smeared density of the configuration. This makes the amplification mechanism transparent and allows us to discuss sector superpositions, local probability balance, and the status of Bohmian equivariance at the wave-function level. We then consider a dissipative extension in which the collapse operator includes a smeared current contribution. The one-particle sector reproduces the expected dissipative CSL energy balance, while fixed many-body sectors contain additional collective momentum shifts and pair-mixing terms that are not reducible, in general, to independent one-particle contributions. Within a leading compact closure, the collective pair friction produces a non-extensive stationary mean kinetic energy: in three dimensions and for weak dissipation, TNcomp≃2Tβ/N, whereas the corresponding dilute energy remains extensive.
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 Γ=ΔTCΔ, where Δ is the vector of branch eigenvalue differences. For the Diósi–Penrose (DP) kernel G/ℏ∣x−x′∣ acting on mass densities, the rate is Γ=2EG/ℏ, 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 τsel∼3×10−16s 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 ∼Rδ2/L3≈4×10−9 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.
The exact multi-resolvent hierarchy of dense many-body spectra encodes higher-order spectral-weight correlations, but closing it predictively for interacting Hamiltonians remains a challenge. We introduce the multi-resolvent self-consistency (MRSC) equations, which separate an exact algebraic hierarchy from an approximate finite-parameter closure. The exact Feshbach geometric identity organizes r-resolvent cumulants into a ladder over irreducible vertices Gamma^(r), and the projected hierarchy expresses those vertices as functionals of the cumulants. The closure works in finite parameters: a connectivity basis organizes a general Hamiltonian's seed into class amplitudes, and a causal r-fold Stieltjes representation with a moment-field identity reduce moment matching to a finite-dimensional problem closed at a predictive fixed point. On the free-fermion benchmark the tested exact identities are reproduced and the Faddeeva closure is recovered at r=1. In a blind test on the interacting Ising chain, with the exact two-resolvent data withheld, the Hamiltonian-derived closure predicts the tested probes to 0.35%-0.97%. At the observable level the exact two-resolvent sector supplies a sign-correct, energy-resolved correction to the single-resolvent baseline for the off-diagonal eigenstate-thermalization-hypothesis (ETH) variance, of order one relative to it. Closure errors organize hierarchically: a level-by-level deficit decomposition, anchor measurements of the isolatable terms, and an explicit propagation bound. The same moments further yield a diagnostic chain for the window conditions of local rotational invariance -- an exact drift-kernel identity, a proven single-state identifiability boundary, and a transferability census over couplings and realizations -- a finite-order framework, not a proof of universal emergence.