Preprint and reproducibility archive; not peer reviewed. What this record contains. The full text of the manuscript "Identical Few-Particle Gauge Hamiltonians, Programmable Collective Charge Response" (M. J. Baek), its Supplementary Information with complete proofs, and the archive that certifies every closed-form value the paper states. The result. Few-particle dynamics are routinely used to infer collective response in paired quantum matter. This work constructs finite-range interacting response twins that defeat that inference exactly. For any prescribed finite p, two positive finite-range parents have complete static gauge Hamiltonians that coincide at every Peierls link phase on every sector N ≤ p and first separate at N = p+1; their zero-twist finite-density ground rays are the same Dicke condensate, yet their collective charge curvatures differ extensively. The obstruction has no universal depth, and the paper locates it exactly: the blind depth equals the spectator body order, the algebraic bound is independent of geometric range, and bounded range still bounds depth only through the number of sites a ball can hold — so no arbitrary-depth family of uniformly fixed range is claimed. Conversely, a structurally closed exact-pair tower closes at depth two. What is hidden is designable: with d(d+1)/2 displacement families the whole symmetric response-tensor difference is locally realizable at interior filling, and a triangular-lattice twin pair differs only in shear while both diagonal responses stay equal. A 4×4 hard-core-qubit discriminator gives an exact factor-of-two contrast with an internal null control and a state-preparation certificate in 17 equal-time settings. What is certified, and what is not. Results proved in closed form are stated as theorems and proved in full in the Supplementary Information; results established by exact rational or symbolic computation are labelled computer-assisted and carry the script that produces them; the 16-qubit discriminator is a blueprint with an end-to-end noise simulation, not a hardware demonstration, which is not claimed anywhere. Scope limits, open proof lines and one refuted conjecture are recorded explicitly rather than omitted. The archive. response_twins_unified_v1.0.zip contains the manuscript and Supplementary sources, the proof notes, the complete fail-closed verification harness (31 executed stages in release mode — 27 claim-bearing certificate stages, 1 non-claim diagnostic stage and three fail-closed gates — and 24 in quick mode) with one frozen log per stage, independent reimplementations of the load-bearing calculations, negative controls that must fail, the exact-arithmetic certificates behind every closed-form value in the paper, the figure-generation scripts, the scientific claim and provenance record (79 rows with status and scope notes, 50 of them naming the in-archive script or certificate that checks them, the rest saying in words why not), and a per-file SHA-256 manifest of all 209 payload files. Verifying it. Extract the archive and run python3 REPRODUCIBILITY/run_all.py --mode quick for the fast subset or --mode release for all 31 stages; bash BUILD.sh rebuilds both PDFs from source. The three gates check journal-format limits, the release payload against its manifest, and every number the documentation states about the artefacts. Running the harness does not modify the archive: all runtime output is written outside the immutable tree, so verification cannot invalidate the manifest. README.md separates reproducing the content (TeX Live 2023 or newer) from reproducing the archived PDF bytes, which needs the exact producing build recorded inside the PDFs. Licences. Manuscript, Supplementary Information and documentation CC BY 4.0; source code MIT; certificate data CC0 1.0. LICENSE.md is the authoritative per-file mapping and is included both in the archive and as a separate file on this record.
Preprint and reproducibility archive; not peer reviewed. What this record contains. The full text of the manuscript "Identical Few-Particle Gauge Hamiltonians, Programmable Collective Charge Response" (M. J. Baek), its Supplementary Information with complete proofs, and the archive that certifies every closed-form value the paper states. The result. Few-particle dynamics are routinely used to infer collective response in paired quantum matter. This work constructs finite-range interacting response twins that defeat that inference exactly. For any prescribed finite p, two positive finite-range parents have complete static gauge Hamiltonians that coincide at every Peierls link phase on every sector N ≤ p and first separate at N = p+1; their zero-twist finite-density ground rays are the same Dicke condensate, yet their collective charge curvatures differ extensively. The obstruction has no universal depth, and the paper locates it exactly: the blind depth equals the spectator body order, the algebraic bound is independent of geometric range, and bounded range still bounds depth only through the number of sites a ball can hold — so no arbitrary-depth family of uniformly fixed range is claimed. Conversely, a structurally closed exact-pair tower closes at depth two. What is hidden is designable: with d(d+1)/2 displacement families the whole symmetric response-tensor difference is locally realizable at interior filling, and a triangular-lattice twin pair differs only in shear while both diagonal responses stay equal. A 4×4 hard-core-qubit discriminator gives an exact factor-of-two contrast with an internal null control and a state-preparation certificate in 17 equal-time settings. What is certified, and what is not. Results proved in closed form are stated as theorems and proved in full in the Supplementary Information; results established by exact rational or symbolic computation are labelled computer-assisted and carry the script that produces them; the 16-qubit discriminator is a blueprint with an end-to-end noise simulation, not a hardware demonstration, which is not claimed anywhere. Scope limits, open proof lines and one refuted conjecture are recorded explicitly rather than omitted. The archive. response_twins_unified_v1.0.zip contains the manuscript and Supplementary sources, the proof notes, the complete fail-closed verification harness (31 executed stages in release mode — 27 claim-bearing certificate stages, 1 non-claim diagnostic stage and three fail-closed gates — and 24 in quick mode) with one frozen log per stage, independent reimplementations of the load-bearing calculations, negative controls that must fail, the exact-arithmetic certificates behind every closed-form value in the paper, the figure-generation scripts, the scientific claim and provenance record (79 rows with status and scope notes, 50 of them naming the in-archive script or certificate that checks them, the rest saying in words why not), and a per-file SHA-256 manifest of all 209 payload files. Verifying it. Extract the archive and run python3 REPRODUCIBILITY/run_all.py --mode quick for the fast subset or --mode release for all 31 stages; bash BUILD.sh rebuilds both PDFs from source. The three gates check journal-format limits, the release payload against its manifest, and every number the documentation states about the artefacts. Running the harness does not modify the archive: all runtime output is written outside the immutable tree, so verification cannot invalidate the manifest. README.md separates reproducing the content (TeX Live 2023 or newer) from reproducing the archived PDF bytes, which needs the exact producing build recorded inside the PDFs. Licences. Manuscript, Supplementary Information and documentation CC BY 4.0; source code MIT; certificate data CC0 1.0. LICENSE.md is the authoritative per-file mapping and is included both in the archive and as a separate file on this record.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? Quantum phases are often characterized through properties of their many-body states: order parameters, correlations, reduced density matrices, entanglement spectra, and local-unitary equivalence classes. Transport, however, is a response to a specified physical probe. This paper asks a sharp version of that distinction: Can two exact condensates be indistinguishable by every spatial reduced-density spectrum, yet exhibit different charge stiffness when the same physical gauge field is applied to both? The answer is yes. The paper constructs an exactly solvable finite-range flat-band pairing model in which two number-projected condensates are related by onsite product unitaries and therefore have identical spatial reduced-density spectra for every bipartition, while their response to the same fixed physical U(1) Peierls twist is qualitatively different. MAIN EXACT RESULT The model contains an exactly degenerate manifold of number-projected condensates. Two branches in this manifold are related by onsite product unitaries. For every spatial region R, their reduced density matrices are therefore unitarily conjugate and have exactly identical spectra. Consequently, all spatial entanglement spectra and all Rényi entropies coincide between the two branches. Nevertheless, under the same fixed physical U(1) twist, one branch is exactly soft while the other has positive static charge curvature. The difference is exact rather than a small perturbative correction. The soft tower has zero curvature, while the rigid branch satisfies an exact all-filling response law K_N^r(epsilon) = [N(M-N)/(M-1)] K_1(epsilon), with M = 2L, and retains a nonzero response density in the fixed-filling density limit evaluated in the paper. An arbitrarily weak symmetry-allowed selector epsilon > 0 uniquely chooses the rigid texture while the odd-charge gap remains open. The odd-charge gap is obtained exactly: Delta_odd(epsilon) = 3/4 + epsilon/2. Thus the switch between zero and finite static curvature is not produced by closing the protected odd-charge gap. WHAT INFORMATION IS MISSING? The two condensates are equivalent as static states under onsite local unitary transformations. However, the external gauge coupling is held physically fixed rather than being transformed together with the state. The paper formulates this distinction as an equivariance problem for the twist-defect map. Static onsite-local-unitary orbit invariants determine the charge curvature only if the physical twist-defect map transforms equivariantly along the orbit. In the constructed model, it does not. The conceptual conclusion is therefore more precise than the familiar observation that condensate fraction and superfluid density are different quantities. Here the two states agree on an exceptionally strong collection of static state information: every spatial reduced-density spectrum. They nevertheless respond differently to the same physical probe. The missing datum is the relative orientation between the state orbit and the physical gauge embedding. In other words: state equivalence does not automatically imply response equivalence when the probe is specified independently of the equivalence transformation. The result does not deny that a full microscopic wavefunction written in a fixed physical basis, together with the complete physical coupling to the external field, determines the response. Rather, it identifies exactly which information is lost when one reduces the system to probe-independent state-orbit data. TWIST-INDUCED UNSELECTION AND NONCOMMUTING LIMITS The work also studies competition between the infinitesimal selector and a finite gauge twist. In the joint scaling limit epsilon -> 0+ with A_x^2/epsilon fixed, the leading texture energies cross at lim_(epsilon -> 0+) A_(x,c)^2(epsilon) / epsilon = 24/7. The twist can therefore drive the selected rigid texture back toward the soft texture while the states remain within the same static onsite-local-unitary orbit. This provides an explicit example of noncommuting limits: selecting the state with epsilon > 0 and then taking epsilon -> 0+ does not reproduce every feature of setting epsilon = 0 from the outset. WHY THIS MATTERS For readers outside flat-band superconductivity, the broader issue is: What information about a quantum state is sufficient to determine its response to an external probe? Entanglement spectra, reduced-density spectra, and local-unitary orbit data are powerful state descriptors. This construction shows that even the complete collection of spatial reduced-density spectra does not, by itself, determine static charge response when the physical gauge coupling supplies additional orientation information. The distinction is conceptually relevant to: - quantum many-body response; - quantum geometry; - transport inferred from wavefunction information; - exact paired states; - entanglement-based classification; - local-unitary invariants; - gauge coupling and probe dependence. The value of the construction is that this separation is realized exactly, without disorder, uncontrolled effective theory, or approximate numerical diagonalization. SCOPE AND NONCLAIMS Throughout this work, “stiffness” means finite-volume static charge curvature K_ij^(N) = [d^2 E_(0,N)(A) / dA_i dA_j]_(A=0) and the explicitly evaluated selected density limit. The result is not identified with an unspecified dynamical order of limits, finite-frequency optical conductivity, or a universal dynamical definition of superfluid weight. The thermodynamic sequence used in the manuscript is an L x 1 torus, periodic in both directions. It is not claimed to be a fully two-dimensional thermodynamic family in which both linear dimensions diverge. Neutral collective excitations can soften with increasing size, so finite-twist uniformity and dynamical response limits remain separate problems. VERSION CLARIFICATION The current release should be quoted using the asymptotic scaling statement A_(x,c)^2(epsilon) / epsilon -> 24/7 as epsilon -> 0+. Earlier wording could be read as claiming the stronger finite-epsilon identity A_x^2 = (24/7) epsilon. That stronger interpretation is not supported. The current manuscript explicitly shows finite-epsilon crossing values displaced from 24/7 and states the scaling limit correctly. This clarification does not alter the paper’s main exact result: identical spatial reduced-density spectra can coexist with zero versus finite static charge curvature under the same physical twist. It also does not alter the exact odd-charge-gap formula. REPRODUCIBILITY The deposit contains the Letter, Supplemental Material, exact and numerical certificates, figure generators, manuscript-build scripts, SHA-256 manifests, and a claim ledger. The archive retains refuted or quarantined claims instead of silently removing them and separates exact identities from finite-precision numerical stress tests. The assumptions behind the selected-state, curvature, gap, and scaling-limit statements are explicitly recorded. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; static charge stiffness; Kohn curvature; Peierls twist; exact condensates; eta pairing; quantum geometric nesting; reduced density matrix; entanglement spectrum; Rényi entropy; local-unitary orbit; onsite product unitary; response from wavefunction data; gauge embedding; physical probe orientation; twist-defect map; transport switching; noncommuting limits; quantum many-body response.
Preprint. Third paper of the contextuality-obstructions program, alongside Paper A (the canonical class of the MASA context stack), Paper B (the obstruction spectrum of Weyl context families), Paper D (the observer-context category) and a synthesis note. The slogan "contextual holonomy encodes quantum contextuality" is attractive and, as usually stated, false. This paper gives it a sharp form. On the context stack of maximal abelian subalgebras the canonical degree-2 class has order exactly n, and its cocycle holonomy along closed context loops detects EXACTLY the all-versus-nothing sector of contextuality for closed-triple Weyl families at even local dimension. Both containments are strict, each by an explicit certificate: at d=3 the class is nontrivial of order 3 while the achievable state-independent AvN value is 0, and KCBS exhibits state-dependent contextuality where the class shadow vanishes. The Wigner-friend loop generated by H and S splits by level: its projective (Pancharatnam) holonomy is +1 while its determinant-section holonomy is exactly -i, so any interpretational claim about "the" holonomy of that loop must state which one it means. ATTRIBUTION. The classification of state-independent values for qudit Weyl families, the even/odd dichotomy and the fixed value d/2, on which the state-independent side of the equivalence rests, is DUE TO Abramsky, Cercelescu and Constantin (FSCD 2024). Attainment at general even d is the Raussendorf-Okay-Zurel-Feldmann construction. This paper uses both and rediscovers neither. A RETRACTION IS CARRIED IN THIS PAPER, NOT BURIED. An earlier draft presented the tier-two facets of the d=4 census on a (1/(2*sqrt2))Z grid with a single coefficient magnitude 3/(2*sqrt2), and read the resulting unit-coefficient bounds 4*sqrt2 and 14*sqrt2/3 as placing Tsirelson's sqrt2 inside the tier-two geometry. The whole presentation was a normalization artifact and is withdrawn (Retraction 1): the stored lift has coefficients in {0,+-1} with integer bounds 6 and 7, every facet functional in the census is rational, and Q(sqrt2) is not among the eighteen canonical arithmetic fields. Certificate: d4_arithmetic_profile.py. WHERE THE EVIDENCE IS THIN, STATED IN THE PAPER. Two sweeps quoted here drew no positive instances at all: the d=4 half of the 2,100-family equivalence test, and the 6,500-trial false-certificate sweep, which draws no contextual family at any of d=2,4,8 and therefore establishes the absence of false certificates only vacuously. Both are disclosed where they appear. Theorem 2 is labelled machine-validated rather than proved: the closed form is established end to end for the deletion families treated, while the general closed-triple d=2 statement in compressed coordinates remains open and is recorded as open in the verification ledger. HARDWARE CAVEAT, STATED UP FRONT. The quoted significances, about 56 sigma for the ray-level Wigner-friend holonomy and 35.4 sigma for the Peres-Mermin witness above the deterministic bound 4, are shot-noise statistical only, computed under the implemented witness model and conditional on compatibility and nondisturbance assumptions. They are not systematics-aware and neither measurement is loophole-free; a roughly 3.5 sigma preparation offset on one strand is itself evidence of unmodelled systematics. We also record that the pinned value S = 4.6125 cannot be recomputed from the stored counts (a naive recomputation gives 4.5662, a gap of about 2.7 standard errors); the archived value 4.558 +- 0.018 is the one an independent reader can reproduce from the deposited data. Read these as consistency checks on real hardware, not as precision measurements or certifications. STATUS AND SCOPE. This is a preprint, not a peer-reviewed article. It was held back from the first publication batch and released only after three adversarial review passes; what each pass found, and how each item was closed, is recorded in verification/INDEX.md rather than quietly fixed. One defect found there is NOT closed and belongs to the companion Paper B, which cites a false-certificate artifact that is not in fact in the repository; it is recorded rather than left for a reader to discover. COMPANIONS. Paper A: https://doi.org/10.5281/zenodo.21630431 . Paper B: https://doi.org/10.5281/zenodo.21630630 . Paper D: https://doi.org/10.5281/zenodo.21630697 . Note: https://doi.org/10.5281/zenodo.21630739 . Code archive: https://doi.org/10.5281/zenodo.21498498 AI-ASSISTANCE DISCLOSURE. Large-language-model assistants (Anthropic Claude) were used substantially in drafting, in developing the verification scripts, in literature search and in proposing proofs. Every load-bearing claim is machine-verified independently of that assistance by the scripts named in the text, and the author takes full responsibility for the results, including any errors. VERIFICATION CODE. https://github.com/manuflog/contextuality-obstructions
This source-only preprint gives a four-qubit counterexample to the arbitrary-phase extension of the first-excited parity conjecture discussed in Wong, Bravyi, Gosset and Liu, Lee-Yang tensors and Hamiltonian complexity, arXiv:2602.03605v1, Section 5.6. For an unweighted four-cycle with phase pi on one edge and a common parameter 0 <= s < 1, it derives the complete 16-dimensional spectrum and an exact parity threshold. The rational value s = 2/3 is an explicit witness; a specified fixed-witness robustness interval is also proved. The original zero-phase conjecture and the general all-graph gap problem remain open. Ground-state uniqueness and the non-strict gap bound are known inputs, explicitly credited to Wei, arXiv:2609.26802v2, Corollary 5.8 and Section 5.4, and Bravyi, Gosset, Liu and Wong, arXiv:2607.14401v1, Theorem 1. The concrete phase counterexample, complete spectrum and threshold are distinguished from these known results. The primary-source comparison is dated and bounded; worldwide priority is not certified. Read the unchanged PHASE_EPR_FINAL_MANUSCRIPT_v0.1.3_R2_20261010.tex or .md first, together with the current public source/verification supplement and PUBLIC_RELEASE_NOTE. Older versions and administrative private-draft wording, if present as provenance, describe historical preparation stages and do not define this public version or its reviewed scope. Auxiliary six-qubit exploration is outside this paper's completed claim. Shunpei Shimizu initiated and directed this personal independent project, arranged verification, and coordinated the materials. No university or institutional affiliation is declared. AI systems performed research exploration, proof construction, computational checks, analytical review and manuscript preparation. The reviews are internal AI analyses; independent human verification of every proof, specialist human peer review and complete proof-assistant verification are not claimed. No PDF of editorial revision R2 has been generated or visually verified: the available compiler failed during environment setup. Editable TeX, a readable Markdown manuscript and verification materials are provided. This is a limited, unrefereed research preprint, released under Creative Commons Attribution 4.0.
Preprint and reproducibility archive; not peer reviewed. WHAT IS THE QUESTION? In an ordinary dispersive band, charge stiffness is closely related to kinetic energy. In a perfectly flat band, ordinary band velocity vanishes, so superconducting or superfluid stiffness can instead depend on quantum geometry and interactions. In the quantum-geometric-nesting (QGN) class of frustration-free flat-band models, previous work established a rigorous bound relating the many-body superfluid stiffness to the inverse mass of a Cooper pair and reported numerical saturation at approximately the 0.1% level. This raises a natural question: Is perfect quantum geometric nesting by itself sufficient to make the pair-mass bound an exact equality? This work does not assume that the answer is yes and does not attempt to prove a universal saturation theorem. Instead, it calculates the finite-size static charge-twist response exactly and identifies the structural obstruction that decides whether saturation occurs. MAIN RESULT: TWO EXACT RESPONSE CHANNELS For the isolated QGN condensates studied here, the finite-size second derivative of the ground-state energy under a charge twist has the exact structure E_0^(N)''(0) = alpha C_3(N) + gamma C_2(N), where C_3(N) and C_2(N) are fixed hypergeometric filling factors. The amplitudes alpha and gamma are independent of pair number N and can be determined entirely within the one-pair sector. The second amplitude satisfies gamma >= 0, so its contribution has a fixed sign. This converts the question of saturation from a numerical observation into a structural condition: The familiar one-channel pair-mass form is saturated exactly when the second channel vanishes. The paper then determines how this second channel can arise. The spin trace first decides whether a generator contributes at all. Traceless spin structures make the relevant contribution vanish identically. For the contributing charge-type sector of the on-site construction, the remaining finite-size term is an aliasing contribution associated with nonzero lattice Fourier or Poisson modes rather than the continuum mode. EXPLICIT FINITE-SIZE COUNTEREXAMPLE The paper constructs an explicit admissible QGN model with a numerically certified unique gapped ground state for which gamma / alpha = 0.846 at N_k = 4. Relative to the one-channel saturating form, the finite-size departure is approximately 42% at half filling and rises to approximately 169% at N = 3, where the reference one-channel contribution is itself small. These percentages are finite-size departures from the one-channel form. They are not claimed to be thermodynamic violations of the rigorous pair-mass bound. THERMODYNAMIC RECOVERY The same analysis also explains why near-perfect saturation may appear in larger regular systems. Under the stated joint analyticity and regularity assumptions, uniform normalization assumptions, and a uniform charge-gap condition, the finite-size aliasing contribution decays and the conjectured one-channel relation is recovered in the thermodynamic limit. The work therefore separates two logically different statements: 1. perfect QGN alone does not force exact finite-size saturation; 2. with additional regularity and gap assumptions, the finite-size obstruction disappears in the thermodynamic limit. This distinction is important because an exact identity at finite system size and a thermodynamic definition of stiffness are not interchangeable statements. ADDITIONAL STRUCTURAL RESULTS The paper derives further constraints on the pairing form factor. When the generators that actually enter the Hamiltonian act irreducibly on each flat-band fibre and the momentum graph generated by them is connected, the pairing form factor has a degenerate, momentum-independent singular-value spectrum and the condensate is unique up to an overall scale. Counterexamples to converses are given, so these properties are not claimed to follow from QGN alone. The admissible band-space bilinears are also organized by a Z2-graded real structure. In an appropriate real basis, the even sector is real symmetric and the odd sector is i times a real antisymmetric matrix, with dimensions m(m+1)/2 and m(m-1)/2, respectively. This grading clarifies which algebraic sectors can carry the finite-size obstruction. The work additionally checks that, in the translation-invariant setting considered, optimization over the relevant all-spin matrix-valued antisymmetrized-geminal-power variational family reproduces the exact finite-size curvature to second order. The additional response channel is therefore not merely an artifact of evaluating a single fixed trial vector. WHY THIS MATTERS The scientific point is not simply that one numerical conjecture has a finite-size exception. The stronger result is that the apparent equality between a collective many-body charge response and a two-particle pair-mass quantity has an identifiable structural condition. The exact two-channel decomposition explains both: - why many QGN examples may appear almost perfectly saturated, and - how a valid QGN system can depart strongly from the one-channel form at finite size. For a broader reader, the work illustrates an important problem in many-body physics: When can information obtained from a low-particle sector determine a collective many-body response? In this QGN class, the one-pair sector carries unexpectedly strong information: it determines both response amplitudes. However, an additional algebraic and finite-size aliasing condition decides whether the response collapses to the commonly assumed pair-mass form. The lesson is therefore not that few-particle information is generally insufficient, but that its sufficiency must be established structurally rather than inferred from numerical agreement. SCOPE AND NONCLAIMS The observable treated here is the static charge-twist curvature in the conventions stated in the manuscript. The work does not claim that the 42% or 169% finite-size departures survive as thermodynamic violations. The thermodynamic recovery theorem requires the regularity, normalization, and gap hypotheses stated in the paper. Finite-range or infinite-volume situations outside those hypotheses are separate questions. The surjectivity theorem in release v1.1 explicitly assumes a momentum-independent, invertible singlet nesting matrix. That hypothesis is essential and should not be omitted when quoting the theorem. The relation between the graded algebraic sectors and the complete QGN interaction class also requires the stated orbital-coverage conditions; the manuscript gives cases in which those conditions fail. VERSION CLARIFICATION Release v1.1 tightens the formal scope of the v1.0 statements. Most importantly, it explicitly adds the momentum-independence hypothesis required by the surjectivity argument and separates analytically proved statements from numerically certified properties of the explicit counterexample. The finite-size two-channel response result and the thermodynamic scope should be quoted in the v1.1 form given above. REPRODUCIBILITY The deposit includes the manuscript and a fail-closed reproducibility package containing exact or symbolic checks, numerical certificates, figure generators, deterministic reference outputs, and a frozen manifest. The explicit finite-size counterexample, response formulas, convention factors, gap diagnostics, and auxiliary algebraic claims are mapped to executable verification routines. SEARCH AND INDEXING CONTEXT Relevant concepts include: flat-band superconductivity; quantum geometric nesting; QGN; superfluid stiffness; static charge curvature; Kohn twist; Cooper-pair mass; pair-mass bound; quantum metric; quantum geometry; exact pairing; antisymmetrized geminal power; AGP; finite-size aliasing; Poisson summation; thermodynamic limit; frustration-free Hamiltonian; Z2 grading.
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.