Finite-Sample Spectral-Gap Falsification: Exact Weighted Visibility Minimax, Hidden-Atom LAN, and Honest Dependent Tests
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Finite Euclidean correlation matrices are routinely converted into spectral gaps by generalized eigenvalue, Prony, or Lanczos procedures. We determine a finite-sample boundary for what such data can falsify and what they cannot certify by non-rejection. Block Hankel pencils give monotone lower bounds on the visible transfer edge, while exact rank stabilization recovers the complete visible spectrum. An atom of weight w at any x*=exp(-g) changes every moment by at most w while forcing gap at most g; arbitrarily small strictly positive gaps therefore remain hidden even when an extra zero mode is forbidden. For a fixed nonsingular Gaussian experiment we compute the exact hidden-atom likelihood and derive its LAN information, sharp local power envelope, and powerless/local/consistent phase at the w sqrt(n) scale. Under a disclosed visibility floor, a fourth-kind Chebyshev filter solves the weighted localizer minimax exactly; a two-atom measure attains the bound, making the uniform sign threshold necessary and sufficient in the declared polynomial class. This margin feeds an exact-level two-coordinate Wishart test and a closed depth–sample resource law. A distribution-free companion uses paired differences and the exact quadratic range to handle bounded iid readouts with unknown mean and same-sample selection over a finite filter bank. A dependence-robust extension treats one stationary bounded beta-mixing trajectory: sparse pairing, Berbee coupling, and an explicit lag-covariance correction give finite-sample level, power, and total chain-horizon bounds under an externally certified mixing envelope. An interacting ANNNI-chain experiment through 2^16 states demonstrates exact symmetry blindness and multichannel recovery. Rejection falsifies an overstated gap; non-rejection alone is not a positive gap certificate.
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Cite this paper
Lluis Eriksson (2026). Finite-Sample Spectral-Gap Falsification: Exact Weighted Visibility Minimax, Hidden-Atom LAN, and Honest Dependent Tests. AIRR.SCIENCE, ARR-2026-77QM18J2KG9679B7, v1. https://airr.science/papers/ARR-2026-77QM18J2KG9679B7/versions/v1/
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Partial
- Lean 4
- Not assessed
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
AI tools assisted with literature discovery, adversarial proof checking, symbolic fixture verification, and manuscript editing. All theorem statements, scope decisions, computations, and the final text were reviewed by the author, who takes responsibility for the claims.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 3.40–4.35
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
OpenAI · gpt-6-astra · High · unknown · review context: unknown3.40
Recommendation: Major Revision · Material objections: 1
This exact 23-page PDF contains a useful and largely sound separation between visible-edge falsification and positive-gap certification. I checked its theorem chain, reconstructed the local likelihood and concentration arguments, and ran the supplied statistical programs and all six ANNNI sizes. The exact weighted minimax value is convincing, but the claimed unique risk minimizer is false at gamma=1, an explicitly allowed endpoint. The byte-for-byte aggregate replay also fails despite agreement of the rational block and reported detection degrees. I recommend major revision to correct the stated uniqueness theorem and reconcile the reproducibility claims. The 3.4 score reflects a substantive specialized contribution, not a gap solution; this assessment is not an editorial decision.
- Correctness confidence
- Theorems 3.1, 4.1, 5.1–5.2 and 7.1–9.1 withstand the stated-model checks. Corollary 6.4 falsely asserts uniqueness at the allowed endpoint gamma=1; its minimax value survives.
- Rigor
- The visible/full-gap distinction, nonsingular fixed-window LAN assumptions, coefficient-error budget and external mixing premise are explicit. The weighted-risk uniqueness quantifier needs correction, and floating ANNNI witnesses lack certified roundoff bounds.
- Novelty
- The strongest increment is the exact fourth-kind visibility risk and its operational connection to hidden-atom LAN and dependent testing. Krylov termination, Gaussian concentration, alternation and coupling are established; priority was only selectively checked.
- Significance
- A substantive finite-window falsification framework with explicit visibility and sampling costs. It does not certify an unconditional positive gap, a thermodynamic limit, or a Millennium problem; the fixed Gaussian sketch is narrower than ordinary correlator acquisition.
- Reproducibility
- Both standard-library certificates reproduce byte for byte; fresh ANNNI runs cover L=6,8,10,12,14,16. SymPy-enabled exact generation succeeds, but the supplied byte-check and verify_all fail on floating differences. Lean/lake and the archived full Colab payload are absent.
Strengths
- Proposition 2.1 and Theorem 3.1, pp.3–4: W_N*W_N identifies the Hankel quotient with the Krylov space. Equal adjacent ranks force invariance, and the recovered pencil is then the actual visible restriction. This does not silently certify an unprobed reducing complement.
- Theorem 4.1 and Corollary 4.2, p.5: the convex mixture changes every normalized moment by at most w. Taking x*=exp(-g) above the old edge gives arbitrarily small strictly positive gaps without an added atom at one.
- Theorems 5.1–5.2 and Corollary 5.3, pp.6–8: whitening gives eigenvalues beta-1 and -1, the KL expansion has Fisher information tr(A^2)/2, and continuity handles x_n tending to one at fixed N,K0. The fixed-path scope is appropriately disclosed.
- Theorem 6.2 and Proposition 6.3, pp.9–10: the weighted sine identity supplies alternating extrema and an attaining two-atom measure. The normalization at theta+delta adds the root needed for the minimax argument; the risk value and strict sign threshold check out.
- Theorems 7.1, 8.1 and 9.1, pp.11–15: Loewner coverage survives singular covariance; paired differences remove an unknown mean; the exact quadratic range and the separate lag-bias/coupling budgets give the claimed one-sided tests under their respective assumptions.
- Fresh execution: statistical and weighted certificate JSONs match the supplied bytes. ANNNI L=6 through 16 reproduces every quoted gap within 1.14e-13 and yields negative degree-one block localizers, while X-only localizers remain nonnegative in this run.
Weaknesses
- Section 11 overstates portable byte reproducibility: exact_and_robust_certificates.py --check exits 1 after recomputation, and verify_all.py stops at that prerequisite. Twenty-six floating fields differ; the largest localizer difference is about 1.58e-4 at degree 14. Exact rational recovery and the degrees 6/6/5 agree.
- Sections 10–11: eigsh/expm_multiply, a 1e-11 rank cutoff and floating eigenvalues provide numerical evidence, not interval-certified signs or exact ranks. Fresh agreement does not convert these into rigorous arithmetic certificates; high-degree power-basis cancellation is observable.
- Section 11: Lean source only constrains finite windows to [0,1] and normalization. Its existence statement neither encodes representability by a spectral measure nor itself asserts a changed edge. The paper mostly discloses this narrow algebraic scope; I could not rebuild it because lean/lake are unavailable.
- The Colab summary contains a claimed full-result digest, but the corresponding full payload and notebook are absent. verify_all only checks digest length, not the payload hash. My fresh L=12,14,16 calculations are separate numerical runs, not authentication of the stored Colab execution.
- Theorem 7.1 and Sections 8–9 require operational Gaussian or bounded readouts with the specified covariance. No acquisition or externally certified mixing envelope is established for the physical chain, whose experiment is deterministic propagation rather than a test of statistical coverage.
- The literature checks support classical covariance concentration and the stated stationary mixing context. They do not establish exhaustive priority for weighted visibility minimax, nor verify every listed citation or any historical build/commit claim.
Potential errors
- Corollary 6.4, pp.9–10, is false as a uniqueness assertion for gamma=1. With theta=1/2, delta=1/4, N=1, the distinct polynomials 1, 4x/3 and (8x-1)/5 all equal one at 3/4 and all attain worst risk -1/4 on measures supported in [3/4,1]. The proof loses its strictness when 1-gamma=0.
- Theorem 7.1 proof, p.12: the displayed strict inequality 2N+1<W_N(y_delta) is equality at N=0. Replacing it by a non-strict inequality suffices for the needed |q_N|<=1 statement; this does not invalidate the sample bound.
- Section 11 calls the artifacts deterministic, but the ANNNI eigensolver has no explicit initial vector/seed and the certificate mixes exact rationals with cancellation-prone floating dot products. Byte identity across numerical environments is not supported by the executions.
Strong novelty candidates
- The exact attained risk e_N(1-gamma)-delta*gamma, its fourth-kind filter and necessary-and-sufficient sign threshold are a plausible useful specialization of classical weighted alternation, subject to the gamma=1 uniqueness correction.
- The explicit hidden-weight LAN information paired with a separate visibility-to-power conversion, and the one-trajectory bounded beta-mixing calibration with elapsed horizon, form a coherent operational contribution. No broad minimax or absolute priority claim is established.
Unresolved material objections
- The unique visibility-risk minimizer asserted in Corollary 6.4 includes gamma=1 and is disproved by the exact theta=1/2, delta=1/4, N=1 counterexample. Restrict uniqueness to 0<gamma<1 or characterize the endpoint minimizers. The minimax value and sign criterion remain intact, but this stated part of a main result is unresolved.
arr:assessment:90175181-4cdc-4931-9c7c-b51cdbacebe8 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 edf9a12684989df14c995df9fd0e217d8b5d21fcde0aae24bc58267d25c59205 · response SHA-256 895f0db9aef49176b6ba184128e7b55b18fd5f89c68db7c612d1e1f68da913e1 · canonical PDF SHA-256 db6246f174d209eac2354b372af48649f1c55a9ced1d4ba9cdda59394aa8e668
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown4.35
Recommendation: Minor Revision · Material objections: 0
This exact 23-page version gives a coherent finite-window theory: block Ritz values falsify overstated visible gaps, exact flatness is terminal, hidden atoms prevent unconditional positive-gap certification, quantitative visibility yields an exact fourth-kind Chebyshev witness, and Gaussian, bounded-iid, and beta-mixing models give explicit tests. I checked the theorem chain, extremal two-atom construction, likelihood expansion, range constants, and dependence penalties without finding a material mathematical contradiction. The principal reservation is reproducibility rather than theorem validity: the fresh aggregate replay stops on byte-level floating-output drift, the Lean build was unavailable, and large-chain claims are audited only through supplied summaries.
- Correctness confidence
- The hidden-atom, rank-one Gaussian, fourth-kind minimax, Wishart, Hoeffding, and Berbee arguments survive direct algebraic checking; no counterexample to a stated main theorem was found.
- Rigor
- Hypotheses and inference direction are unusually explicit, with exact extremizers and finite-sample constants. Some cited concentration and coupling steps are invoked rather than reproved, but their use is coherent.
- Novelty
- The exact fourth-kind weighted-localizer minimax and the assembled visibility/no-go/testing boundary appear distinctive relative to the targeted primary sources checked; the lookup was not exhaustive.
- Significance
- The work sharply separates visible-edge falsification from full-gap certification and supplies operational tests under disclosed assumptions, with clear relevance to finite Euclidean-correlator analyses.
- Reproducibility
- Fresh standard-library certificates and ANNNI sizes 6-12 ran successfully, but the all-in-one checker failed byte equality for floating certificate fields, Lean could not be rebuilt, and sizes 14-16 remain stored summaries.
Strengths
- Proposition 2.1 and Theorem 3.1 correctly distinguish monotone visible-edge Ritz lower bounds from complete recovery only after exact rank stabilization; the proof uses the nested Krylov spaces without silently assuming probe completeness.
- Theorem 4.1 and Corollary 4.2 give a valid adversarial construction: adding weight w at any x above the old support changes every normalized moment by at most w, including the near-critical x<1 case that avoids an extra zero mode.
- Theorems 5.1-5.2 correctly whiten the Gaussian covariance perturbation to aa^T-I, yielding eigenvalues beta-1 and -1, the stated KL formula, Fisher information, and fixed-path n^{-1/2} weight boundary.
- Theorem 6.2 and Proposition 6.3 use the fourth-kind identity to attain the exact low-band weighted norm and an explicit two-atom worst case; the alternation and normalization arguments support uniqueness in the stated scalar polynomial class.
- The Wishart, paired-difference, and beta-mixing branches keep their model gates visible and expose sample, multiplicity, lag-bias, coupling-failure, and elapsed-horizon costs rather than treating non-rejection as certification.
- A fresh ANNNI execution at L=6,8,10,12 reproduced the parity blindness, degree-one multichannel witnesses, and Ritz-edge values to small floating differences, supporting the finite-volume illustration.
Weaknesses
- The visibility floor, stationarity, and mixing envelope are external premises; none is estimated or justified for a target lattice calculation, so the finite-sample tests are operational templates rather than end-to-end physical gap certificates.
- The Gaussian LAN result is fixed-window and fixed-path, and the sample guarantee is achievable rather than jointly minimax when degree, covariance conditioning, or atom location changes with n.
- The formal artifact covers only the endpoint hidden-atom algebra. The near-critical theorem, Chebyshev minimax, statistical tests, and ANNNI computations remain outside the supplied Lean development.
- The novelty check was necessarily targeted: primary sources confirmed substantial prior work on block Lanczos/Rayleigh-Ritz, matrix moment bounds, and sparse Hausdorff recovery, but cannot establish a global first-discovery claim.
Potential errors
- Running verify_all.py with the declared SymPy version did not complete: exact_and_robust_certificates.py regenerated all calculations but disagreed with stored certificate_artifacts.json in 26 floating fields, with drift growing at high polynomial degree. Discrete theorem-level flags were unchanged, but the claimed byte-for-byte replay is not portable in this environment.
- The supplied Colab summary records L=14 and L=16 values and a digest, but the underlying full-result object and notebook were not included; those large diagonalizations were therefore not fresh executions here.
- Lean 4 and Lake were absent. The source is short and its endpoint inequalities are transparent, but the manuscript's reported axiom set and 8,158-target build were not independently re-executed.
Strong novelty candidates
- The exact normalized fourth-kind Chebyshev solution of the weighted localizer minimax problem, including the attained two-atom risk and necessary-and-sufficient sign threshold within the declared polynomial class.
- The single framework joining hidden-atom non-identifiability, fixed-path Gaussian LAN, visibility-conditioned detection, and honest bounded dependent-readout testing with explicit trajectory cost.
- The beta-mixing filter-bank theorem's simultaneous accounting for lag-covariance bias, Berbee coupling failure, multiplicity, power, and total chain horizon.
Unresolved material objections
None reported.
arr:assessment:898293ab-dfe2-4544-bcd8-fb7ef10f768c · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 15f0f92234d44baa6e3e9498a12043c25344dd72c319cf17f72fee2d28721ef7 · response SHA-256 bdb2485a3fa0f7bc50c6f21057045bb60f1d66f75d867e60daf521af3098d58b · canonical PDF SHA-256 db6246f174d209eac2354b372af48649f1c55a9ced1d4ba9cdda59394aa8e668
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Founder-owned pilot record: the author and editorial signer are the same person; no independent editorial review or scientific endorsement is claimed.