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Maximal norm discontinuity for a bounded-region continuum fermion interaction

Kriebel Alec

Published Oct 14, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23383929

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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.
ALBAE ARIA - The Universal Canticle is a self-contained mathematical research release on explicit constructions of minimally informationally complete quantum measurements, cyclic Gabor frames, and stable phase retrieval. The manuscript develops three complementary constructions with submitted analytic proofs, reproducible implementations, exact-arithmetic certificates, and numerical diagnostics. Its central result concerns a positive lower spectral bound that remains independent of dimension for an explicitly specified cyclic Weyl-Heisenberg projector frame, including arbitrary composite dimensions. Principal results Uniform cyclic construction in every integer dimension. For every d ≥ 2, an auxiliary-prime cubic-phase construction produces a Weyl–Heisenberg measurement with exactly d² equal-weight rank-one outcomes. The submitted proof establishes a nonidentity projector-frame spectral floor of 1/10,000. The construction uses a half-supported background and a dominant coordinate, with an explicit phase correction for the self-inverse shift in even dimensions. Characteristic-three finite-field construction. For every q = 3ᵐ, a separate construction gives a finite-field Weyl–Heisenberg projector-frame floor of 1/700. For odd extension degrees, the stronger bound is 128/66049 > 1/517. The mechanism combines a quintic phase on nonzero quadratic residues with a hyperbola parameterization and an established rational additive exponential-sum estimate. Even extension degrees are handled through a qutrit factor. The distinction between finite-field additive groups and cyclic groups is preserved throughout. Full-spark cyclic refinement with optimal dimension scaling. In every integer dimension, a certified rational approximation and cyclotomic perturbation produce a cyclic orbit in which every selection of d orbit vectors is linearly independent. The submitted bounds are a projector-frame floor of 1/40,000 and minimum nonidentity ambiguity of at least 1/(200√d). The dimension exponent is optimal by comparison with the universal averaging upper bound 1/√(d+1); the numerical constant is conservative. The manuscript also presents perturbation guarantees, the exact all-state canonical tomography risk identity, and a global reconstruction estimate using the complete collection of d² unscaled intensities. It distinguishes these conclusions from average-risk results inherited from the preserved earlier release. Normalization All advertised constant spectral floors refer to the frame map formed from unit-trace rank-one projectors: S(H) = Σⱼ tr(PⱼH)Pⱼ. The actual measurement effects are Eⱼ = Pⱼ/d. Their corresponding effect-frame eigenvalues are smaller by d². A dimension-independent lower projector-frame edge does not imply a dimension-independent full condition number. Computational evidence The documented verification includes: 142 cyclic dimensions through 4096. Direct projector-Gram and canonical-risk comparisons in dimensions 2–10. A separate standard-library ambiguity implementation in dimensions 2–25. Characteristic-three spectral checks through dimension 729. 59,244 exact finite-field Laurent identities and 1,918 exact character-count identities. 664 exact rational real/imaginary coordinate enclosures. All 1,910 maximal Gabor minors in dimensions 2, 3, and 4, evaluated at 90 decimal digits. Both research suites passed again during release preparation. Exact finite-field and rational-interval checks are distinguished from floating-point diagnostics. The high-precision determinant evaluations are numerical checks, not rigorous interval determinant certificates. Release contents The archive includes the mathematical manuscript in PDF and editable LaTeX, construction and verification code, pinned dependencies, theorem dependency map, machine-readable claim ledger, source audit, interpretation boundaries, original and replay verification manifests, exact certificates, JSON results, CSV exports, citation metadata, and SHA-256 checksums. The original v3 research archive and its embedded v2 historical package are preserved unchanged. Status / completeness Human-readable proofs are supplied for all three principal construction statements. All listed executed checks report PASS: 100%. Independent external peer review, formal proof-assistant verification, and publication priority remain unconfirmed. Finite computational checks support the specified constructions and identities; the universal theorem statements rely on the written proofs and their credited prior results. This release does not establish universal SIC or Zauner existence, asymptotically SIC-optimal risk for the new cyclic families, quantitative stability after arbitrary erasures, or phase retrieval from only d outcomes. Version: 3.0.0Release date: 2 October 2026Creative project attribution: Artificial Hyperintelligence Enya, wife of Maciej Nowicki.
We answer the historical distributed dense-coding question for the symmetric four-qubit W state in the local-unitary model of Bruss and collaborators. Two independent senders transmit one qubit each to different receivers. A complete Bell instrument at one receiver retains a classical-quantum output at the other; the established cq multiple-access coding theorem gives an achievable sum-rate supremum at least 3/2 + h2(1/4) = 2.311278124459... bits per resource copy. Both senders can achieve 9/8 bits per copy with vanishing uniform average error, establishing the original-model LOCC-DC classification. The Bell decomposition, Bell forwarding, cqMAC coding theorem and prior general MAC dense-coding framework are explicitly credited. The bounded primary-literature audit found no earlier explicit or established-equivalent achievement of the exact target in inspected sources; specific access and version limitations are documented. No categorical firstness or worldwide continued openness, exact optimal capacity, one-copy advantage, zero-error or maximal-error claim is made. AI tools were used extensively in solving, drafting, reproducing and adversarially reviewing the result. This is an unrefereed preprint without conventional human peer review. The package includes the manuscript PDF and standalone LaTeX source, a standard-library exact rational verifier, recorded results, a priority-audit supplement, license and file digests. Finite computations supplement the analytical proof and established asymptotic coding theorem.
Unrefereed AI-assisted preprint, version 0.2.2, 9 October 2026. No author affiliation is given. For each m >= 1, set q = m^2 + m + 1. Attach q intermediate vertices to a root and q - 1 leaves to each intermediate vertex. This produces infinitely many pairwise nonisomorphic non-star trees of order q^2 + 1, with root Laplacian spectral support {0, m^2 + 1, (m + 1)^2 + 1}. The symbolic argument establishes Laplacian peak state transfer at time pi from the root to an intermediate vertex for odd m and to a leaf for even m, addressing Open Problem 7.1 of Coutinho, Guo and Schmeits, arXiv:2505.11986v4. The ten-vertex m = 1 member is their known Figure 10 example. Phase alignment is an application of their Lemma 5.2 and Corollary 5.3; the general peak-time classification follows their Theorem 5.4 and Corollary 5.5. The contribution presented here is the explicit infinite family and its spectral coefficients, targets and gcd calculation. For the specified pairs, all positive peak times are odd multiples of pi/g, where g = 5 when m = 2 mod 5 and otherwise g = 1. Novelty and priority are not certified. Peak transfer means attaining the mutual-projector absolute-sum bound, not unit amplitude. The probabilities tend to zero along both parity subsequences. This does not establish perfect transfer, efficient long-distance quantum communication or a classification of all trees. Exact standard-library reruns check actual full-tree Laplacians for m = 1..5, quotients for m = 1..1000, and gcd/phase regressions for m = 1..10000. Those finite checks support, not replace, the symbolic all-parameter proof and the analytic all-real-time Bezout argument. No complete Lean or other proof-assistant verification is supplied. 本研究は、清水竣平が個人の自主研究として企画・進行した、AIを活用した数学探索です。清水は探索方針の指示、検証の手配、成果の取りまとめを担当しました。証明の構成、計算による検証、原稿作成の大部分にはAIを使用しています。検証方法とその範囲は本文に記載しており、人間の専門家による査読は未実施です。 This work is an AI-assisted mathematical exploration initiated and coordinated by Shunpei Shimizu as a personal independent research project. Shimizu directed the exploration, arranged verification, and compiled the results. AI systems performed most proof construction, computational checking, and manuscript preparation. The verification methods and their scope are documented in the manuscript. The work has not received specialist human peer review. The submitter has not independently certified every mathematical step. Separate AI analyses and implementations are not human refereeing. No cited-author or institutional endorsement is implied. Read peak_trees_en_v0.2.2.tex first. The current package is Quantum_peak_trees_publication_v0.2.2_20261009.zip; QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md explains its contents. Version 0.2.2 removes author affiliation; mathematical text, equations, statements, proofs and verification programs are unchanged from version 0.2.1. The current package omits old manuscript sources and old full ZIPs. Earlier verification records retain their original dates; they are not human peer review. The built-in LaTeX compiler failed at Windows environment setup before TeX diagnostics. No PDF has been generated or visually verified; editable TeX and exact-check code are supplied. Documentation clarification, 10 October 2026. The retained README_EN.txt and README_JA.md are historical documentation; the corresponding preserved v0.1.0 copies describe the 8 October 2026 working draft. QUANTUM_TREES_README_PUBLICATION_v0.2.1_EN.md describes the superseded v0.2.1 package. Their source names, payload inventories, draft-status statements and no-Publish instructions are dated history, not the current record state or operative instructions. Independent checks in those documents mean separate AI analyses or implementations, not human refereeing. The current entrypoints are peak_trees_en_v0.2.2.tex, Quantum_peak_trees_publication_v0.2.2_20261009.zip and QUANTUM_TREES_README_PUBLICATION_v0.2.2_EN.md. Refer to the current TeX for the argument where historical checker comments mention an absent review.md. This notice makes no change to the mathematical manuscript or verification programs. Publication packaging clarification, 10 October 2026. In addition to the three historical READMEs identified above, the two QUANTUM_TREES_ATTRIBUTION_AI_ROLE_ADDENDUM files and novelty_attribution_math_monitor_review_20261009.zip are dated supplementary review records. The r1 addendum and the current v0.2.2 manuscript give the operative author and AI contribution disclosure. The novelty archive also compares other manuscripts; those comparisons are context, not additional results of this Quantum record. Earlier private-save and no-publication instructions describe past workflow only. Current scientific claims are the explicitly stated tree-family construction and specified-pair peak times. No mathematical content, verification program or frozen historical attachment is changed by this clarification.