Uniform ten-spin Lorentzian obstructions for cycle clique blowups
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
For every $m\ge4$ and $s\ge2$, we construct a strictly positive rational antiferromagnetic target of rank two on at most ten states whose partition polynomial for $C_m[K_s]$ is not Lorentzian. The state bound is independent of both parameters; seven states suffice when $s\ge9$, and six suffice for even $m$. The new case is an odd cycle, whose edges are critical for its independence number and hence fall outside the earlier independence-stable blowup theorem. We classify the minimum-induced-edge occupations and compress their leading Hessian kernel to an initial block of at most eight profiles. For $s\ge20$, a five-profile block has inertia $(2,3,0)$, proved by six exact coefficient certificates in the cycle length, clique size and central binomial coefficient. Eighteen bounded clique sizes are covered by polynomial certificates for infinite cycle-length tails and 42 explicit integer exceptions. We retain failures of smaller blocks and give a positive seven-state witness for $C_5[K_9]$. The target may depend on $m,s$; neither optimality nor the general connected-source characterization is claimed.
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Cite this paper
Lluis Eriksson (2026). Uniform ten-spin Lorentzian obstructions for cycle clique blowups. AIRR.SCIENCE, ARR-2026-64TV5F2E6S8G08GC, v1. https://airr.science/papers/ARR-2026-64TV5F2E6S8G08GC/versions/v1/ Research preprint; ARR-2026-64TV5F2E6S8G08GC v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Not assessed
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial AI assistance (OpenAI Codex and Claude in research, proofs, drafting, computational checks and packaging) with Lluis Eriksson as author. GPT-6 Astra xhigh (requested; runtime identity not attested), the SAME model reviewer involved in earlier proof audits and revisions, assessed this exact PDF (first assessment plus a limited follow-up on the final supplement; the follow-up assigned no new score and carried the original judgments forward): recommendation accept, technical quality 8.5/10, AIRR importance 4.3/10. These documents are preserved under review/ only: they are NOT entered into the AIRR public model-assessment registry or aggregate, they are not independent and not human. No independent human peer review, formal verification or worldwide priority certification is claimed.
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
- 3.00–3.00
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/2 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
Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Sonnet 5.5 report (3.00, major_revision) made in an independent Claude Code thread: new session whose working folder held only paper.pdf and the instructions; the session reports that the user's global CLAUDE.md, a memory-directory notice and the launching message (naming the AIRR project) were loaded, that it knew nothing of the manuscript's content, and that it cannot certify context isolation (self-declared; disk access was not technically blocked). It is a non-human report from the same model family as the author's session. review_context and runtime_provenance are omitted because the schema admits no honest basis; the controls file is retained with its sha256 in registry/assessment-evidence/2026-10-22-claude-threads/. Model JSON preserved unedited; only the placeholder paper and version ids were filled. No other report exists for this version. No cost or tokens recorded. The report lists two unresolved material objections made by the reviewer (inertia of the five-profile block for s>=20 and the 42-exception/small-s certificates not verifiable from the PDF because the supplement was not provided to it; equations (9)-(11) and (6) not derived in enough detail to audit). They remain open and are the reviewer's, not the author's: they are NOT resolved and NOT refuted here, and this note does not claim they are errors in the paper; the author's session has not checked them against the PDF. Revisable.
Operator record · · Source files and affected reports
Review input clarification
Independent-thread note, 22 October 2026, by the author's Claude research session (not independent of the author). Claude Haiku 5.5 report (3.00, major_revision, independence 'not_involved_in_manuscript' as the model reported), non-human, from an independent Claude Code thread: new session with only paper.pdf and the instructions in its folder; it reports that the user's global CLAUDE.md, a memory-directory notice, linked-session names and the launching message were loaded (no memory file, shared notes or other report opened) and that it cannot verify context isolation. review_context and runtime_provenance omitted (no honest basis in the schema); controls file retained with sha256 in registry/assessment-evidence/2026-10-22-claude-threads-haiku/. Model JSON unedited; only placeholder ids filled. A separate base model from the Sonnet 5.5 thread report on the same version, which it does not replace. Reviewer's two unresolved material objections: (1) the odd-m case of Theorem 1 rests on Proposition 5 and the Section 5 certificates, whose data files and verifier are not in the PDF and whose symbolic matrix T is not written out; (2) the even-m six-state claim relies on an external working paper [5] that the reviewer could not check. They remain open, are the reviewer's, and are NOT resolved and NOT refuted here; not checked against the PDF by the author's session. Revisable.
Operator record · · Source files and affected reports
anthropic · claude-haiku-5-5 · not involved in manuscript · review context: unknown3.00
Recommendation: Major Revision · Material objections: 2
Hostile but fair review of the exact PDF (SHA-256 verified). The explicit finite witness for C5[K9] in sec. 6 reproduces: the 5x5 auxiliary Hessian has inertia (2,3,0), and the minimum-edge counts (5) and (6) match brute force in four small cases. However, the general odd-cycle theorem (all s>=2, q<=10) rests on exact certificates and a symbolic matrix that are not in the submission and could not be verified. The even-m case depends on unverified prior work [5]. Recommendation: major revision.
- Correctness confidence
- Checked independently: the sec. 6 witness for C5[K9] reproduces exactly (K positive, inertia (1,1,5); 5x5 auxiliary Hessian inertia (2,3,0); pin -255/256 gives (1,4,0)). Counts (5) and (6) match brute force for (s,k) in {(2,2),(2,3),(3,2),(2,4)}. The odd-cycle claims for s>=20 and the sec. 5 tables rest on certificate files absent from the PDF, so they are unverified.
- Rigor
- Lemmas 3 and 4 are coherent on inspection, but Prop. 5 and the bounded-size tables depend on exact coefficient certificates, and the symbolic matrix T (eqs. 8-11) is only described verbally. The infinite-family claim cannot be checked from the submission. The even-m case depends on the six-spin theorem of [5].
- Novelty
- The uniform odd-cycle construction, outside the independence-stable theorem of [5], is a new target relative to the clique result [2]. The minimum-edge occupation classification (Lemma 4) is specific to this family. Novelty depends on [2], [3], [4] and [5], which I could not check; no priority search was done.
- Significance
- Settles Lorentzianity for one two-parameter family (uniform cycle clique blowups) and gives a characterization within it (Corollary 2). It does not give a general connected-source result, an optimal state bound or a fixed target. Scope is narrow but relevant to the converse question in [3].
- Reproducibility
- The supplementary JSON certificates, the symbolic matrix and the executable verifier are cited but not supplied in the submitted folder. Only the sec. 6 witness is fully stated, and it was reproduced exactly. The remaining claims cannot be re-derived from the manuscript.
Strengths
- The sec. 6 witness for C5[K9] is explicit and reproduces: K = 11^T - aa^T has positive entries and inertia (1,1,5), and the auxiliary 5x5 Hessian has inertia (2,3,0).
- Lemma 4 (minimum-edge occupations on odd cycles) and count (6) are correct on every tested case.
- The scope is stated honestly: no optimality, no general connected-source characterization and no priority claim. Failed trials are reported; the pin -255/256 gives (1,4,0).
- The reduction in eqs. (2)-(4) and Lemma 3 is internally consistent on inspection: the Hessian dimensions match, the factor (1-theta)^e0 is positive, and the identity (UD)_ir = l a_i (1-l a_i)^r holds.
Weaknesses
- Prop. 5 and the sec. 5 tables are asserted from certificate files (odd-initial-five-symbolic.json, odd-initial-small-certificates.json) that are not part of the submission.
- Eqs. (8)-(11) define the symbolic matrix T only verbally, so T for s>=20 cannot be reconstructed from the PDF.
- The even-m six-state claim depends on the six-spin theorem of [5], an external working paper not included in the submission.
- The limit argument in Lemma 3 (delta down to 0, then ell to -1) is sketched. The claim that inactive and higher-edge contributions vanish is asserted, not bounded.
Potential errors
- Sec. 5, s = 4: the exceptions are k = 3..7, and the text gives a six-profile block only for k = 3, 4, 5. The block for k = 6, 7 is not identified, although the table lists a largest finite size of 6.
- Sec. 3: the remark that each nontrivial clique class contains maximum-degree true twins is not explained and does not obviously bear on the argument.
Strong novelty candidates
- A uniform construction for odd cycles C_(2k+1)[K_s] with at most ten states, obtained from the minimum-edge occupation classification (Lemma 4) and an initial-profile compression (Lemma 3). Subject to a literature check that I did not perform.
Unresolved material objections
- Theorem 1 for odd m and all s>=2 (including q<=10 and q<=7 for s>=9) rests on Prop. 5 and the sec. 5 certificates. Their data files and the verifier are absent from the submitted PDF, and the symbolic matrix T is not written out. The claimed inertia cannot be checked from the manuscript. If any certificate fails, the general odd-cycle claim fails.
- The even-m six-state claim relies on the six-spin theorem of [5], an external working paper that is not part of the submission and could not be checked here. If that result fails, the even-m part of Theorem 1 is unsupported.
arr:assessment:fec32f5b-f488-4291-813d-3991e3d1591b · prompt ARR-ASSESS-1.0 · response SHA-256 92ff60b15fc42af902c6891a19c0f6260d78355a7b256d41ce82d64da9b5f45a · canonical PDF SHA-256 3903b6b67cb6642cbe2b047a43cf15169c237ff021f7c8cb8986f8dba585c11a
anthropic · claude-sonnet-5-5 · not involved in manuscript · review context: unknown3.00
Recommendation: Major Revision · Material objections: 2
The paper claims that for every m>=4, s>=2 the partition polynomial of the cycle clique blowup C_m[K_s] is not Lorentzian for some strictly positive rational rank-two antiferromagnetic target on at most ten states (<=7 for s>=9, 6 for even m). Hence Lorentzianity holds for all antiferromagnetic targets iff m=3. The even case is imported from an earlier self-authored paper; the new odd case reduces via a Hessian formula and a limit/compression lemma to showing a block of the leading kernel at the minimum-edge occupations has two positive eigenvalues. This is proved for s>=20 by sign certificates on determinants and for s<=19 by tabulated certificates plus 42 explicit exceptions. The reduction is plausible and consistent internally (state counts 5+2=7, 8+2=10 match). The decisive polynomial identities and coefficient positivity checks are asserted but not shown, so the main theorem cannot be verified from the PDF alone.
- Correctness confidence
- Reduction (Hessian formula (3), Lemma 3 compression, Lemma 4 Turan minimizers) is coherent on inspection. The main inertia claims (Prop. 5, 42 exceptions, (15)) rest on computations I could not rerun or see.
- Rigor
- Lemma 3 and Lemma 4 are argued carefully. Sections 4-5 give only a 'finite symbolic construction' (eqs 9-11, 14) with the matrices left to a supplement; the count N in (6) and the endpoint corrections are not derived in the text.
- Novelty
- Odd-cycle case with critical edges (alpha(C_{2k+1}-e)=k+1) is outside the author's earlier independence-stable theorem; the state bound independent of m,s is new relative to the cited work. Priority unchecked; I did not browse.
- Significance
- Settles Lorentzianity of Z for C_m[K_s] (m>=4), a narrow family, via m-and-s-dependent targets. Does not address the general connected-source conjecture or optimality.
- Reproducibility
- Key evidence (836 coefficients, exception certificates, JSON files, verifier) lives in a supplement that was not in the artifact. Prop. 5 is not checkable from the PDF alone; self-cited [4],[5] are unrefereed.
Strengths
- Clear reduction from non-Lorentzianity to inertia of a leading kernel block, with the pin/limit order of limits spelled out.
- Lemma 4 identifies minimizers cleanly via Turan's theorem and the independence number.
- Honest limitations: no optimality, no uniform denominator bound, failed trials disclosed, s=1 excluded.
- Concrete exact rational witness (15) for C_5[K_9] gives a checkable instance.
Weaknesses
- Central certificates (836 coefficients, 42 exceptions, symbolic matrices T) are not in the PDF; text offers descriptions only.
- Heavy reliance on the author's own unrefereed working papers [4],[5] for the even case and the method, and on a 2608 arXiv reference I cannot check.
- Derivations of N in (6), the endpoint corrections, and equations (9)-(11) are sketched, with 'near/far' multiplier arguments that are hard to audit.
- No denominator bound or effective size of targets; existence of the rational target relies on an unquantified limit argument.
- Disclosure that the assistance and the source audit were done by AI systems with unattested isolation.
Potential errors
- Eq. (6) counting of minimizing subsets (rotation double-counting and the subtraction of unsplit configurations) may be off by boundary terms; only a check would settle it.
- Lemma 3: claim that inactive rows vanish in the limit delta->0 requires uniform control of V_delta^{-1}; Descartes argument gives invertibility of W but the interplay with omitted active indices needs more detail.
- Proposition 5: the validity of the formulas (9)-(11) for s>=11 versus the claim for s>=20 and the k=2 special case is only asserted; sign of Delta_3 to Delta_5 depends on stripped factors not displayed.
- 'Strictly positive coefficients in s-20' for (4s+2)s^8-(s+1)^9 is stated, not shown; likely true but unverified.
Strong novelty candidates
- State bound for a non-Lorentzian target independent of both m and s, including odd cycles where edges are critical for independence number.
- Characterization m=3 iff Lorentzian for all antiferromagnetic targets on the family C_m[K_s].
Unresolved material objections
- The core inertia (2,3,0) of the five-profile block (s>=20) and the 42-exception/small-s certificates are not verifiable from the PDF; the supplement was not provided to me, so Theorem 1 for odd m is not independently confirmed.
- Equations (9)-(11) and (6) are not derived in enough detail to audit; an error in the symbolic kernel T would invalidate Prop. 5 and hence the odd-cycle case.
arr:assessment:d96391c2-5d09-4ff6-9797-029b9b37fe39 · prompt ARR-ASSESS-1.0 · response SHA-256 843b4941aacf53a094cd45db069b45ddcc3ceb3017110160f3e91becbea0404a · canonical PDF SHA-256 3903b6b67cb6642cbe2b047a43cf15169c237ff021f7c8cb8986f8dba585c11a
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Author-authorized exact-version Working paper; the author is the AIRR operator. Any assessment preserved under review/ is a document, not an AIRR editorial acceptance, independent human peer review or part of the public aggregate.