This preprint develops a microscopic pair-correlation framework for cutoff-regularized Coulomb-type BBGKY defect channels. A smooth radial cutoff separates the transition shell, an active mesoscopic annulus, and the macroscopic projected-response region. On the active annulus the paper uses an amplitude formulation F_epsilon = u_epsilon^2, where the radial amplitude belongs to a uniform shell-admissible class allowing correlation-hole behavior without imposing a cutoff-independent positive lower bound. The second-hierarchy defect current is decomposed, in a tested distributional sense, into macroscopic projected, shell-pair, pair-residual, three-body, and stress components. The scalar-pair-generated defect class is closed in the calibrated dual norm, while three-body residues outside this class are detected by annihilating test functionals. A model angular example illustrates how non-radial shell components can have positive distance from scalar radial pair corrections. The manuscript formulates basis-dependent leakage conditions preventing shell-scale residuals from contaminating the macroscopic Gram/Riesz certificate and provides sufficient conditions for convergence of the selected defect identity in a fixed tested distributional channel. No cutoff-free Coulomb closure, trace-norm propagation of chaos, full many-body derivation, or complete Bogoliubov fluctuation theorem is claimed.
This preprint develops a microscopic pair-correlation framework for cutoff-regularized Coulomb-type BBGKY defect channels. A smooth radial cutoff separates the transition shell, an active mesoscopic annulus, and the macroscopic projected-response region. On the active annulus the paper uses an amplitude formulation F_epsilon = u_epsilon^2, where the radial amplitude belongs to a uniform shell-admissible class allowing correlation-hole behavior without imposing a cutoff-independent positive lower bound. The second-hierarchy defect current is decomposed, in a tested distributional sense, into macroscopic projected, shell-pair, pair-residual, three-body, and stress components. The scalar-pair-generated defect class is closed in the calibrated dual norm, while three-body residues outside this class are detected by annihilating test functionals. A model angular example illustrates how non-radial shell components can have positive distance from scalar radial pair corrections. The manuscript formulates basis-dependent leakage conditions preventing shell-scale residuals from contaminating the macroscopic Gram/Riesz certificate and provides sufficient conditions for convergence of the selected defect identity in a fixed tested distributional channel. No cutoff-free Coulomb closure, trace-norm propagation of chaos, full many-body derivation, or complete Bogoliubov fluctuation theorem is claimed.
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.
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 preprint develops a projected Fock-space channel framework for finite-rank tensor correctors. The selected two-particle fluctuation channel is realized inside the condensate-orthogonal bosonic Fock space through a symmetric Gram metric with exchange terms. The manuscript proves a correspondence isometry, formulates the projected Bogoliubov comparison as a Duhamel leakage identity, and separates the projected-channel error into resolved leakage, three-body leakage, and finite-dimensional model discrepancy. The paper also provides sufficient spectral-tail conditions for resolved leakage, a regular smoothed Fourier benchmark on the three-dimensional torus, a bounded three-body leakage functional linked to microscopic obstruction diagnostics, and a metric-conditioning transfer estimate for the symmetric two-particle Gram matrix. The results are finite-dimensional and conditional: no full many-body derivation, trace-norm propagation of chaos, automatic N^{-1/2} rate, or cutoff-free Coulomb closure is claimed. This version v0.8 incorporates audit-correction and submission-readiness changes, including metric-consistent symmetric lifting, complex test-density convention, form-domain Duhamel clarification, explicit Fourier lattice-tail constants, and expanded conditioning estimates.
We study stationary quantum fluctuations around a mean field limit in trapped, dilute atomic gases of repulsively interacting bosons at zero temperature. Our goal is to describe quantum-mechanically the lowest macroscopic many-body bound state consistent with a microscopic Hamiltonian that accounts for inhomogeneous particle scattering processes. In the mean field limit, the wave function of the condensate (macroscopic quantum state) satisfies a defocusing cubic nonlinear Schrödinger-type equation, the Gross–Pitaevskii equation. We include consequences of pair excitation, i.e., the scattering of particles in pairs from the condensate to other states, proposed in [T. T. Wu, J. Math. Phys., 2 (1961), pp. 105–123]. Our derivations rely on an uncontrolled yet physically motivated assumption for the many-body wave function. By relaxing mathematical rigor, from a particle Hamiltonian with a spatially varying interaction strength we derive via heuristics an integro–partial differential equation for the pair collision kernel, K, under a stationary condensate wave function, Φ. For a scattering length with periodic microstructure of subscale ϵ, we formally describe via classical homogenization the lowest many-body bound state in terms of Φ and K up to second order in ϵ. If the external potential is slowly varying, we solve the homogenized equations via boundary layer theory. As an application, we describe the partial depletion of the condensate.
We formulate a conditional projected-channel renormalization theorem forcutoff-regularized Coulomb BBGKY residuals. The starting point is the spectralobstruction that a similarity transform Z−1W Z cannot attenuate a non-zeroHermitian matrix witness in full operator norm. The replacement mechanismstudied here is a non-similarity observation channel: in the minimal model, aPauli-diagonal completely positive trace-preserving channel acts on an M2(C)internal layer and damps declared contrast observables while preserving thetrace sector. The main result is a conditional dynamic gate theorem: if theresidual source, comparison defect, derivative budget, interaction-transportcommutator, and dual-orbit attenuation assumptions are supplied by a concretecutoff model or by an upstream higher-correlation gate, then the projectedwitness component is controlled with an explicit error budget. The theoremis deliberately conditional; it is not a cutoff-free Coulomb closure, not apropagation-of-chaos theorem in trace norm, and not a derivation of thesource budget from the full BBGKY hierarchy. Its contribution is to isolate theexact price of an E6-type operator-valued renormalization: contrast damping,observable-comparison loss, derivative compatibility, and source-budget controlmust all be paid separately.
This preprint is written in Ukrainian. It develops a toy-model theorem note for the E4/E6 escape gates in a post-obstruction BBGKY/Coulomb closure program. The manuscript proves a spectral obstruction to full witness attenuation by invertible similarity renormalization and formulates projected, quotient, and non-similarity channel alternatives with explicit comparison budgets. The work is a toy-gate impossibility and certification note; it does not claim cutoff-free Coulomb BBGKY closure. Препринт написано українською мовою. У роботі розроблено toy-model theorem note для шлюзів E4/E6 у пост-обструкційній програмі BBGKY/Coulomb closure. Основний результат — спектральна перешкода для повного поглинання witness через інвертовану similarity-ренормалізацію, а також projected/quotient/non-similarity channel альтернативи з явними comparison budgets. Робота не стверджує cutoff-free Coulomb BBGKY closure.