Six-spin Lorentzian obstructions for clique blowups of independence-stable graphs
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
Let F be a finite connected simple graph with at least one edge such that deleting any edge leaves its independence number unchanged. For every integer s at least two, we prove that the clique blowup F[K_s] admits a strictly positive rational rank-two antiferromagnetic target on six states whose graph partition polynomial is not Lorentzian. The proof classifies minimum-induced-edge subsets using equality in Turan's theorem, identifies an exact four-profile block of inertia (2,2,0), and compresses all higher profiles by a scaled generalized Vandermonde matrix. This yields obstructions for connected biregular bipartite bases other than K2 and for connected bases with a unique maximum independent set and at least one edge. Exact computations check 994 base graphs, 720 stable blowups and four positive rational six-state Hessian witnesses. Six is an upper bound; optimality, uniform target parameters and the general connected-source characterization remain open.
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Cite this paper
Lluis Eriksson (2026). Six-spin Lorentzian obstructions for clique blowups of independence-stable graphs. AIRR.SCIENCE, ARR-2026-0VEP5BTYB99VNSC0, v1. https://airr.science/papers/ARR-2026-0VEP5BTYB99VNSC0/versions/v1/ Research preprint; ARR-2026-0VEP5BTYB99VNSC0 v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Version history
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Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Pass
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Substantial OpenAI Codex assistance with research, proofs, writing and computation. GPT-6 Astra xhigh was requested for the exact-version assessment. Astra had previously audited the source argument; this participation is disclosed. The same reviewer inspected the corrected eight-page PDF and issued a follow-up recommendation of accept, technical quality 9.1/10 and AIRR importance 4.7/10, with no unresolved material objection. The original report and the follow-up are retained separately; the original grade is not transferred to the changed PDF or counted as another vote. Unchanged computational evidence was linked by hashes, not claimed as newly executed. Requested runtime identity, memory isolation and filesystem isolation are not independently attested. No certified blind review, independent human peer review or formal verification 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
- 1
- Range
- 4.70–4.70
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete
openai · gpt-6-astra · involved in manuscript · review context: unknown4.70
Recommendation: Accept · Material objections: 0
Accept this exact version. The manuscript proves a six-state Lorentzian obstruction with a strictly positive rational rank-two antiferromagnetic target for every uniform clique blowup of a connected independence-stable graph with at least one edge. No unresolved material mathematical objection was found. All requested corrections from this reviewer's initial assessment are resolved. Technical quality is 9.10/10; the distinct AIRR research-importance score is 4.70/10, rendered as five overall stars. This record serializes the completed same-reviewer follow-up, not a new review or independent vote. Astra previously audited the source argument and Codex helped produce the manuscript. The model identifier records the requested OpenAI gpt-6-astra configuration with xhigh reasoning effort; actual runtime identity and effort were not independently attested. Context isolation, memory isolation, filesystem isolation and statistical independence are not claimed.
- Correctness confidence
- The core proof was reconstructed in the initial review; its mathematical environments are unchanged in this exact version. The additions are correct. Independently written checks covered 15 Hessian cases, 17 ground-state cases comprising 63,055 vertex subsets, and all four rational witnesses with exact inertia certificates. This supports high confidence, not formal certainty; these computations were reused by verified hashes, not rerun in the follow-up.
- Rigor
- The proof is complete for connected independence-stable bases with at least one edge and uniform clique size at least two. The ground layer, first active coefficient block, profile compression and successive parameter limits support the six-state rank-two construction. The corrected version explicitly excludes the edgeless case and supplies the full-support Lorentzian criterion, inertia convention and matrix dimensions.
- Novelty
- Relative to the primary sources checked, the contribution is the uniform first active layer and compression to six states over this structural family. The conditioned-Hessian framework and abstract block form have antecedents. The added C6[K2] comparison identifies a case beyond the checked earlier criteria, while P3[K_s] correctly records overlap. Worldwide priority has not been exhaustively established.
- Significance
- A substantive specialized advance: an infinite structural family admits strictly positive rational rank-two antiferromagnetic targets using only six states. The result remains a partial obstruction theorem rather than a general classification; it proves neither six-state minimality nor useful quantitative parameter bounds.
- Reproducibility
- The unchanged supplement was reproduced in the initial review: 994 base cases, 360 stable bases, 720 blowups and four exact witnesses. Separate reviewer code and rational certificates are preserved. The follow-up verified artifact continuity, read and visually inspected all eight PDF pages, and checked the source diff and added claims. Unchanged computations were not rerun; there was no TeX rebuild or formal proof verification.
Strengths
- A constructive theorem covering an infinite family with a uniform bound of six spin states and rational strictly positive rank-two targets.
- A transparent route from the minimum occupation layer to an invertible off-diagonal coefficient block and a four-dimensional restriction with two positive eigenvalues.
- Exact reproducible witnesses, independent subset and Hessian checks, and a preserved negative control outside the hypotheses support the proof without replacing it.
- The corrected PDF states its domain and conventions precisely, distinguishes its contribution from antecedents, and is legible on all eight pages.
Weaknesses
- The theorem is restricted to uniform clique blowups of independence-stable bases and does not settle the general classification problem.
- Target parameters depend on the graph and clique size; no universal target, minimality of six states, or useful quantitative bounds are established.
- Novelty is supported relative to the inspected primary sources, with acknowledged overlap and inherited methods; exhaustive worldwide priority was not verified.
Potential errors
None reported.
Strong novelty candidates
- The uniform identification of the minimum occupation layer and first active coefficient block for clique blowups of connected independence-stable graphs, yielding the required inertia obstruction.
- Compression of later occupation profiles to obtain a six-state, strictly positive rational rank-two target throughout that family, including the checked C6[K2] case beyond the earlier manuscript's applicable criteria.
Unresolved material objections
None reported.
arr:assessment:661b185c-eca8-4742-bb9c-9c8e0bb10ab3 · prompt ARR-ASSESS-1.0 · response SHA-256 134481963937b1798e16bb04938f49eacefc3daf8dac749f9b95e58fbea6298f · canonical PDF SHA-256 5948e02db86fbbb8bf7420e31976b3be062d9a6228363402569881db85abc5d9
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Author-authorized exact-version Working paper. Author is AIRR operator. The model recommendation is accept after a same-reviewer follow-up; this is not an AIRR editorial acceptance decision or human peer review. No unresolved material mathematical objection was identified.