This record contains the manuscript “Propagation of Chaos after a Frozen Density-Selected Phase Imprint”. Abstract. We study a bosonic mean-field system after a single frozen density-selected phase imprint, with phase profile determined by the pre-imprint density. The main result isolates the exact frozen-density interface at the imprint time and shows that the one-particle defect is transported through the jump without additional loss. On each fixed observation window, this reduces the post-imprint analysis to the standard Hartree propagation-of-chaos framework and yields the usual O(N^{-1}) trace-norm convergence in the regular bounded-integrable kernel regime. The manuscript also records two controlled extensions. First, finite-duration imprints are compared with the same frozen benchmark through area matching, with the dynamic-reference mismatch quantified separately. Second, for prescribed finite kick schedules independent of N, the same bookkeeping yields a conservative fixed-schedule extension on fixed windows. In addition, a hypothesis-dependent weakly singular branch is included under explicit extra spacetime assumptions; this branch is conditional and is stated separately from the unconditional regular result. The paper is intended as a rigorous bridge result: it identifies the exact frozen-density jump law, the induced defect transport, and the subsequent standard Hartree comparison. It does not claim a refreshed-generator theorem, a continuum-drive limit, or a uniform-in-the-number-of-kicks result. Files in this record include the main manuscript and supporting source materials.
This record contains the manuscript “Propagation of Chaos after a Frozen Density-Selected Phase Imprint”. Abstract. We study a bosonic mean-field system after a single frozen density-selected phase imprint, with phase profile determined by the pre-imprint density. The main result isolates the exact frozen-density interface at the imprint time and shows that the one-particle defect is transported through the jump without additional loss. On each fixed observation window, this reduces the post-imprint analysis to the standard Hartree propagation-of-chaos framework and yields the usual O(N^{-1}) trace-norm convergence in the regular bounded-integrable kernel regime. The manuscript also records two controlled extensions. First, finite-duration imprints are compared with the same frozen benchmark through area matching, with the dynamic-reference mismatch quantified separately. Second, for prescribed finite kick schedules independent of N, the same bookkeeping yields a conservative fixed-schedule extension on fixed windows. In addition, a hypothesis-dependent weakly singular branch is included under explicit extra spacetime assumptions; this branch is conditional and is stated separately from the unconditional regular result. The paper is intended as a rigorous bridge result: it identifies the exact frozen-density jump law, the induced defect transport, and the subsequent standard Hartree comparison. It does not claim a refreshed-generator theorem, a continuum-drive limit, or a uniform-in-the-number-of-kicks result. Files in this record include the main manuscript and supporting source materials.
This record contains the manuscript “Frozen Density-Selected Unitaries: Exact Action on Marginals, Operator Regularity, and Controlled Singular-Kernel Extensions”. The paper studies a one-step class of density-selected phase transformations in bosonic quantum dynamics. The transformation is frozen in the sense that the density profile used to generate the phase is fixed externally during the imprint, rather than being updated through a self-consistent feedback loop. The manuscript develops three complementary results. First, it proves the exact action of the frozen product-imprint map on reduced marginals of bosonic many-body states. Second, it establishes regularity properties of the associated operator map in the bounded-kernel regime, including Lipschitz continuity and differentiability with respect to the prescribed input density. Third, it gives a controlled extension to singular kernels, where operator-norm convergence is generally too strong and strong-operator convergence becomes the natural topology. The paper is intended as an operator-theoretic foundation for density-selected phase-imprint models and as a bridge toward later mean-field and Hartree-type analyses. It separates the frozen external-density layer from genuinely self-consistent feedback mechanisms and clarifies which conclusions belong to each regime.
This preprint is Paper IV in the research series “Operator Properties and Mathematical Structure of Nonlocal Morphing”. We study short pulse windows in which the phase generator is refreshed along a self-consistent Hartree reference path and the microscopic pulse is driven by the corresponding Hartree-refreshed field lifted to the many-body level. On the pulse window we construct the refreshed reference branch by a local Picard argument, compare the refreshed and frozen microscopic pulse propagators by Duhamel’s formula, and transfer the resulting mismatch to an explicit one-body endpoint defect. We then formulate an abstract restart mechanism showing that any endpoint defect controlled at time τ can be propagated by the standard post-pulse mean-field closure. In the theorem-clean regular regime, and on every fixed post-pulse window [τ, τ+T], we prove that the one-body defect is bounded by C_T·(τ + N^{-1}), provided the frozen regular-regime endpoint scale is available. The main new contribution is the endpoint-transfer mechanism for short Hartree-refreshed pulses: the pulse-window discrepancy is converted into an explicit one-body endpoint defect, and the post-pulse evolution is then handled by a restarted mean-field closure. Thus the frozen bridge remains the leading-order law for short Hartree-refreshed pulses, while the refreshed pulse enters the post-pulse theory only through an explicit endpoint defect. This record is published as a citable preprint in order to provide a public reference point for the companion mean-field arguments used in the present paper and to make the manuscript openly accessible prior to journal submission.
We formulate a regular-regime theorem for continuum density-selected phase drives refreshed along a Hartree reference path. In contrast to a prescribed continuum drive, the density path is generated by the reference Hartree solution itself, namely ρt = |φt|2 . The microscopic comparison is nevertheless made against the corresponding time-dependent Hartree-refreshed reference field, not against a fully microscopic feedback law depending on the evolving reduced density of the same N-body state. In this regular finite-window setting we isolate three mechanisms: construction of the Hartree-refreshed reference branch, convergence of finite refreshed phase schedules to an abelian multiplication product integral, and a Pickl-type propagation-of-condensation estimate for the associated many-body reference dynamics.
Finite-resolution density imaging is a central diagnostic in ultracold atomic gases and quantum simulators. In density-selected phase-imprint protocols, an unknown interaction or response kernel is encoded into a phase kick, whose subsequent short-time dynamics produces a measurable density contrast. This work develops a scaling-law theory for the resulting many-body-to-pixel inverse-readout problem, from microscopic density data to elliptic phase reconstruction and finite-band kernel recovery. The paper addresses a protocol-level question: when can finite particle number, finite frame averaging, and finite optical resolution still support certified phase or kernel recovery? Rather than reproving a propagation-of-chaos theorem, the analysis treats a controlled many-body interface as input and derives the certifiability and design theory that follows. The resulting bounds identify no-go regimes, optimal readout times, many-body-limited saturation thresholds, and a conservative pilot-calibrated rule for Richardson order selection. A finite-band kernel recovery theorem adds a bandwidth-conditioning trade-off for stable kernel recovery. This is the submitted preprint version of a manuscript submitted to The European Physical Journal D.
We use probabilistic and stochastic-analysis methods to prove trace-norm propagation of chaos for finite-dimensional quantum mean-field systems governed by Belavkin equations. The particles are density matrices, interact through a mean-field Hamiltonian, and are continuously monitored through independent diffusive observation channels. The limiting dynamics is a nonlinear matrix-valued McKean-Vlasov diffusion, random through its local observation record and coupled through the deterministic averaged state. The main result treats arbitrary one-particle density matrices, including mixed states, and both perfect and inefficient measurement regimes. Under strong tensorization of the initial data, every fixed marginal converges uniformly on compact time intervals to the tensor product of the nonlinear limiting filters, with an explicit quantitative bound. The proof combines purification, fully observed dilation, conditional expectation, relative entropy, and uniform stability of the associated Zakai equations. In the skew-adjoint measurement case, exterior observation noises disappear from the marginal equations and a stochastic BBGKY hierarchy is recovered. Under only marginal chaoticity of permutation-invariant initial states, we prove convergence of fixed marginals by an iteration of this hierarchy.
We give a self-contained explicit counterexample for the existential unregularized bounded-region fermion question on printed page 371 of Oberwolfach Report 7/2025, with pointwise operator-norm topology distinguished from strong-operator topology on page 365. In dimension three, one fixed smooth nonnegative radial compactly supported pair potential and one fixed interaction ball give a fixed smooth normalized annihilation mode whose Heisenberg norm differences tend to the maximal value 2. Explicit packed finite-particle Slaters and times pi/(2N+1-kappa) witness the same vector-norm limit, a stronger vector asymptotic, and changing normal-state values tending to 4. Both diagonal-density and normal-ordered interactions are covered with ordered-pair coefficient 1 and no ultraviolet smearing. A fixed smooth-cutoff commutator, Slater second moment and two-sector Duhamel comparison give error O(N^(-1/6)+N^(-1/3)). The kinetic energy acts on all of R³. Continuity on every fixed vector and in every fixed normal state remains valid. Siebert's arXiv:2609.34402v1, submitted 28 September 2026 before the original 7 October candidate, supplies the earlier high-density cluster and localized phase mechanism. Its stated theorem concerns full-space difference potentials, but its proof directly adapts to this bounded symmetric cutoff kernel and yields the qualitative existential conclusion. Its Proposition 2.6 explicitly gives maximal norm 2 for the full-space constant-number reference. The present note gives an explicit free-Slater transfer to the fixed spatial cutoff. A separately checked audit-time energy bootstrap also recovers the cutoff maximal limit through the earlier localized phase mechanism. These cutoff implications are deductions made in the audit, not literal cutoff theorems attributed to the prior paper. No first solution, first quantitative refinement, historically certified novelty, or continued open status through October 2026 is claimed. The bounded primary-source priority audit records access and indexing limitations. This five-page unrefereed research note is accompanied by its identical authored PDF/LaTeX source, unchanged 26-file historical candidate and independently authored analytic, operator-algebra, computational, source/model, priority and priority-mapping audits. The fresh computational family has 2,668 exact finite predicates and 558 floating diagnostics; finite tests supplement the universal written proof, and numerical diagnostics are not rigorous interval enclosures or formal proof certificates. Historical optimized-assertion and inventory weaknesses are repaired by externally pinned closed regular-file validation and isolated nonoptimized checker execution, while all original bytes and the original 1/5 candidate count are preserved. Historical result outputs are not claimed byte-identical to current replays, and runtime source-byte checks are explicitly omitted. AI tools were used extensively in development, verification, adversarial review and writing. Independent AI-assisted review is not human or journal peer review; neither is claimed. No copied third-party source bodies are distributed. No all-potential, all-region, dimension-one/two, fixed-density/energy, thermodynamic, fixed-state-discontinuity or CAR-invariance result is asserted.