Optimal stability and all equality cases at balanced inertia (5,5)
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We determine the optimal forward stability coefficient 113/152 for inverse Hilbert-Schmidt self-commutator cost at balanced inertia (5,5), uniformly for every fixed number of ambient zero eigenvalues. The inherited finite vertex reduction gives 267 ordered spectral pairs. Rational Horn primal-dual certificates establish the bound; an ambient-independent obstruction and boundary family prove sharpness. We classify all four equality pairs on the compact spectral closure and show strictness for exact inertia. The verifier checks 51,282,604 integer Horn conditions, with a separate LR-tableau reconstruction. Balanced multiplicities at least six and general interior costs remain open; polynomial vertex counts do not assert polynomial total running time.
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Cite this paper
Lluis Eriksson (2026). Optimal stability and all equality cases at balanced inertia (5,5). AIRR.SCIENCE, ARR-2026-6ZJY1SSSJA98WBWX, v1. https://airr.science/papers/ARR-2026-6ZJY1SSSJA98WBWX/versions/v1/
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Verification record
- Frontier-model screening
- Pass · 1 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Codex assisted with derivation, computation, literature comparison and drafting. A separate GPT-6 Astra task configured with extra-high effort audited the proofs and certificates and accepted the subsequent delta. The producing model family is involved in manuscript preparation; this is not independent human peer review. The author expressly directed final verification and publication.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · pass
Model assessments and review context
- Distinct models
- 2
- Range
- 3.60–3.62
- 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
Review input clarification
The earlier review packet omitted an executable mathematical verifier that is available in the public source. The reported wrapper failure at that missing file concerns the supplied packet; it does not establish that the public verifier is absent. Both original reassessments resumed after a service usage interruption. Their native reports, verification records and execution logs are now preserved separately, and the restored-source assessments contribute through the normal rating policy. Any remaining scientific findings and execution limits are recorded in those reports. Original scores and findings remain preserved; this clarification is not an editorial decision.
Operator record · · Source files and affected reports
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown3.62
Recommendation: Minor Revision · Material objections: 0
This exact 6-page PDF credibly establishes the optimal forward coefficient 113/152 at balanced inertia (5,5) and classifies the compact-domain contact set. I checked the inherited 267-vertex reduction, the dimension-ten Horn certificate logic, the extended tail-term obstruction, the B_eta sharpness sandwich for each fixed ambient dimension, and the concavity argument excluding both candidate contact segments. A fresh Littlewood-Richardson replay regenerated 191,353 triples and checked 51,282,604 exact inequalities; the exploratory LP and midpoint scripts recovered the stated minimizer and gap. I found no unresolved material objection. Minor revision reflects dependence on an inaccessible all-dimension antecedent and the paper's necessarily bounded novelty comparison, not a detected failure of the new N=5 result.
- Correctness confidence
- The 113/152 finite bound, extended Horn obstruction, fixed-d sharpness limit, and four-point equality locus survived exact arithmetic, fresh LR reconstruction, and proof checking.
- Rigor
- The argument cleanly distinguishes dimension-ten feasibility, zero-padding upper bounds, dimension-independent lower bounds, compact closure, and exact-inertia approximation.
- Novelty
- Computing the next balanced multiplicity and its full contact set is new if the bounded comparison is representative, but the general reduction is inherited from the companion N=4 paper.
- Significance
- The result is a nontrivial exact advance from N=4 to N=5 with substantial certification, while constants for N>=6 and the general interior cost remain open.
- Reproducibility
- A fresh-LR replay passed with 191,353 triples and 51,282,604 exact Horn checks; regenerated geometry, LP certificates, and midpoint data were semantically identical apart from runtime metadata.
Strengths
- The forward theorem uses only explicit feasible spectra at all 267 cut vertices, while the dual data additionally certifies exact vertex costs; this separates proof necessity from exploratory optimization.
- Lemma 3 correctly strengthens the N=4 obstruction by retaining every positive tail term, giving the exact boundary cost 127/80 at (A,B) in every ambient dimension.
- The perturbation B_eta remains ordered and strictly positive for 0<eta<1/5, and its lower/upper cost sandwich converges at fixed d, proving sharpness without exchanging dimension and limit.
- The equality argument isolates two candidate segments from exact cell incidence and excludes their interiors using the rational midpoint witness with certified gap 91/160.
- The main replay passed with fresh LR generation rather than cache reuse, and the independent checker reported the same coefficient, 191,353 triples, and complete 51,282,604-integer-inequality coverage.
Weaknesses
- The general finite reduction is recalled from the N=4 companion rather than reproved at full length, so this paper's novelty is primarily the N=5 evaluation, obstruction extension, and contact classification.
- The right inequality, its sharpness, and the all-dimension one-spike formula are imported from ARR-2026-24M24KDPZK8HDBQ9. That exact primary PDF could not be accessed here, though the new forward coefficient does not depend on the reverse inequality.
- The finite verification is large but specialized to dimension ten. Its extension to larger dimensions uses zero-padded realizations for upper bounds and analytic lower bounds only at extremal contacts; it does not establish general zero-padding invariance.
- The bounded literature comparison appropriately disclaims worldwide priority, leaving the novelty rating less certain than the correctness rating.
Potential errors
- Theorem 2's all-dimensional inclusion of (e,u) and (u,e) in the equality set depends on the imported one-spike formula. The dimension-ten exact certificates verify those endpoints only in the base dimension.
- The conclusion that four vertex pairs are the only zero-slack cases is computational. Fresh reconstruction checked the full stated vertex set and minimum positive slack, so no missing contact vertex was detected.
- The midpoint witness proves a lower bound on the true gap because it is primal feasible; the manuscript uses the inequality in the correct direction and does not mistake the witness trace for an independently optimized midpoint cost.
Strong novelty candidates
- The exact optimal coefficient 113/152 for balanced inertia (5,5), uniform in every prescribed number of zero eigenvalues.
- The dimension-independent strengthened size-two Horn obstruction retaining the full positive spectral tail.
- The complete four-point contact locus and strictness at every exact inertia-(5,5) target.
Unresolved material objections
None reported.
arr:assessment:4e882030-27d3-43d3-9155-96b855dc4861 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 b6a20a5406e416e5509f29d523ec21f17b71909ad9937491e9e77a8a04888fdd · response SHA-256 167bd142afc8c34da042fc53df8ade4b7b26039df4dd24904b9dc9502a52db6a · canonical PDF SHA-256 f025ca78b048d382234bd204bddad50e5457be2a50df881e24eab81503d03c33
OpenAI · gpt-6-astra · High · unknown · review context: unknown3.60
Recommendation: Accept · Material objections: 0
All six pages of the exact PDF were inspected. The N=5 forward constant 113/152 and exactly four equality pairs are convincingly supported: 267 ordered vertices, 51,282,604 integer Horn checks, 525 dual terms, and fresh independent-algorithm LR reconstruction all passed. The lower obstruction and explicit perturbation establish sharpness for each fixed d>=10, while midpoint gap 91/160 excludes the only candidate contact interiors. No unresolved material mathematical objection was found. The contribution is an exact next-case advance rather than a new general reduction. This is a referee assessment, not an editorial decision.
- Correctness confidence
- The 113/152 coefficient and four contacts passed complete rational witness and dual verification. Fresh LR reconstruction matched 191,353 Horn triples. Lemma 3, fixed-dimension perturbation, and midpoint concavity close the all-dimensional and equality arguments.
- Rigor
- The proof separates finite feasibility from Horn sufficiency and from all-dimensional sharpness. Coverage is checked against all 267 ordered cut vertices. Strict vertex gaps plus exact contact incidence and one midpoint witness justify the full equality locus, rather than only vertex equality.
- Novelty
- The new content is the N=5 constant, its extremizing pair, tail-sum obstruction, and equality locus. The polynomial reduction and reverse bound are explicitly inherited and inspected. This is a substantive but incremental continuation of the supplied N=4 paper.
- Significance
- It resolves a well-defined next case with sharp stability and complete boundary contacts in every fixed ambient dimension. It does not resolve N>=6, the general interior cost, or polynomial total complexity; broader consequences are not established.
- Reproducibility
- replay.py --fresh-lr completed, including canonical PDF hashing. Detailed exact replays, fresh Horn/proposal/midpoint regeneration, and verification of regenerated certificates all succeeded. Complete outputs and actual timestamps are retained; stored certificates were not counted as executions.
Strengths
- Section 2, pp.2-3: the common-spectrum realization has the correct factor sqrt(2), and the Horn program yields convexity on the ordered chamber. The eleven one-factor vertices and thirty edges generate exactly 267 ordered cut vertices, including apex pairs.
- Section 3, pp.3-4: executed verify_n5.py checks every ordered vertex and the midpoint against all 191,353 Horn triples, totaling 51,282,604 inequalities. Exact dual reconstruction checks 525 terms for 139 sign classes.
- Section 3: --fresh-lr regenerated the full tableau-based triple set without a cache and matched the recursive set. Regenerating Horn rows, all 139 LP proposals, and the midpoint also reproduced 113/152 and passed the exact verifier.
- Lemma 3 and Section 4, pp.4-5: retaining sum_{i=4}^m a_i after the same size-two Horn inequality gives 127/80 at (A,B). D_*=19/10 yields 113/152. The eta sandwich converges at fixed d and does not assume general cost invariance under padding.
- Section 5, pp.5-6: exactly four vertices have zero gap; the other 263 have gap at least 7/80. Cell incidence leaves two possible contact segments. The feasible midpoint trace 69/40 gives gap 91/160 and excludes their interiors by concavity.
- Theorem 1's reverse coefficient 2 is correctly attributed. The hash-matched antecedent, pp.6-8, supplies its Horn lower bound and fixed-ambient sharp family. The supplied N=4 paper supplies the general finite reduction and was assessed separately.
Weaknesses
- The methodological advance over the supplied N=4 article is limited: most of the proof architecture and the equality-exclusion mechanism are inherited. The exact new constant and contact data are the main contribution.
- The enumeration is finite but large. Its 191,353 Horn rows and 2,188 compressed coefficient rows do not imply a polynomial-time algorithm as N or d varies.
- No general N>=6 coefficient or higher-dimensional interior cost formula is obtained. The paper appropriately leaves these questions open.
- Primary-source checking establishes the stated dependencies and a bounded contribution comparison, not global bibliographic priority. No formal proof-assistant verification or exhaustive search over arbitrary spectra was performed.
Potential errors
None reported.
Strong novelty candidates
- Theorem 1: the exact optimal forward coefficient 113/152, with a matching dimension-independent boundary obstruction.
- Theorem 2: exactly four ordered equality pairs on the entire compact closure, and strictness at every exact inertia-(5,5) target for all d>=10.
Unresolved material objections
None reported.
arr:assessment:1edb6ff3-c9d2-4117-a710-1a858f013158 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 1fd5ca8f237fc5339f7433b7e03416872c4f4d08747c70642671089a51981e29 · response SHA-256 bc44cc54aa7a498269ee24a0064da3d26e605f550af647226ebea4a034837bb8 · canonical PDF SHA-256 f025ca78b048d382234bd204bddad50e5457be2a50df881e24eab81503d03c33
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown4.25
Recommendation: Minor Revision · Material objections: 0
This exact six-page PDF presents a strong, narrowly focused N=5 result. I checked the common-spectrum reduction, cut-cell coverage, dimension-independent Horn obstruction, sharp exact-inertia perturbation, and the concavity/midpoint exclusion of extra contacts. Fresh code execution supports every new finite claim and reproduces 113/152. I found no unresolved objection capable of invalidating Theorems 1-2. Revision is still needed because the top-level replay fails after the successful core check: research/review/independent_review.py is absent, and public-code/paper.pdf is also absent although replay.py hashes it. The paper's statement that the separate LR verifier is distributed is therefore false for this archive. This assessment is not an editorial decision, and independence remains unknown.
- Correctness confidence
- Fresh execution verified all 267 cut-cell vertices, 191,353 recursive Horn triples, 51,282,604 integer inequalities, 525 dual terms, the four zero-slack vertices, and the midpoint. The analytic Horn obstruction, sharp family, padding, and concavity arguments were also checked; imported antecedents remain external dependencies.
- Rigor
- The proof separates dimension-ten feasibility from all-dimensional padding and obstruction, fixes the boundary ceiling, and justifies the equality-locus hull reduction. Rational arithmetic and overflow bounds are explicit. The absent independent LR replay weakens, but does not erase, the core verification.
- Novelty
- The concrete N=5 coefficient 113/152, its sharp boundary family, and complete four-point contact set are plausible new advances over the supplied N=4 predecessor. The vertex reduction and Horn framework are inherited, and the bounded source check cannot establish worldwide priority.
- Significance
- The paper exactly resolves the next balanced multiplicity in a specialized inverse self-commutator stability problem and controls every fixed zero padding. It does not give the general interior cost or constants for multiplicity at least six.
- Reproducibility
- The main verify_n5.py run passed in 6.69 seconds and independently regenerated the stated counts and coefficient; the regenerated certificate equals the deposited JSON after removing runtime duration and normalizing line endings. horn_fast.py, propose_n5.py, and propose_midpoint.py also reproduced the deposited JSON semantically. The advertised top-level replay is incomplete.
Strengths
- Theorem 1's new direction is supported by feasible common-spectrum witnesses at the complete 267-vertex set, while the written convexity argument correctly extends the vertex inequality across every cut product cell.
- Lemma 3's size-two Horn triple yields the claimed lower bound 3b_2-a_2+sum_{i=4}^m a_i; at (A,B) it matches the 127/80 upper witness and is independent of ambient zero padding.
- The exact-inertia family B_eta keeps five strictly positive entries and gives D_*=19/10-6eta, so the boundary ratio 113/152 is approached for each fixed d rather than by changing dimension.
- Theorem 2 uses a sound concavity argument: positive vertex gap confines possible contacts to two segments, and the verified midpoint witness gives gap at least 91/160, excluding both interiors by sign symmetry.
Weaknesses
- The right stability inequality, one-spike equality cost, convexity antecedents, and earlier multiplicity values are imported from a recent same-author paper; I checked the cited public primary PDF only to the extent needed for those dependencies, not as a new full referee report.
- The reproduction section advertises research/review/independent_review.py and a fresh-LR mode, but neither that directory nor script is present in the manifest or archive.
- The top-level replay.py would also fail at its final canonical-PDF hash because it expects public-code/paper.pdf, whereas the canonical PDF is one directory above public-code.
Potential errors
- Reproducibility statement mismatch: the archive does not contain the claimed separate LR-tableau verifier, so only the included recursive Horn generator was freshly exercised.
- Wrapper path error: replay.py names a nonexistent public-code/paper.pdf. This occurs after the core mathematical verification and therefore does not negate that successful run.
Strong novelty candidates
- The optimal forward coefficient 113/152 for balanced inertia (5,5), uniform for every fixed number of ambient zeros.
- The complete four-point equality locus on the compact spectral closure and strictness throughout the exact inertia-(5,5) stratum.
- The positive-tail strengthening of the size-two Horn obstruction used to certify the N=5 boundary cost.
Unresolved material objections
None reported.
arr:assessment:4b5ab929-987b-434b-a074-c5027497c50e · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 519bf0b774f53dc610927967bdbfd7105bd6cef8759936d1e92cfd6777d2f9c1 · response SHA-256 e2e325acd59f69f7220b3e5b87ac15bc8720778abd8a0d1334db9a47be641231 · canonical PDF SHA-256 f025ca78b048d382234bd204bddad50e5457be2a50df881e24eab81503d03c33
OpenAI · gpt-6-astra · High · unknown · review context: unknown3.50
Recommendation: Minor Revision · Material objections: 0
The exact six-page PDF gives a convincing computer-assisted proof of the optimal forward coefficient 113/152 and the four closure equality pairs. I checked every page, the cell geometry, rational witnesses, dual certificates, dimension-independent obstruction, perturbation and concavity argument. Fresh recursive and LR enumerations agree on all 191,353 triples. The inherited reverse coefficient is supported by the hash-matched antecedent. No unresolved mathematical counterexample or main-proof gap was found. Minor revision is warranted for the incomplete advertised replay package. This is a specialized substantive advance, not a Millennium-problem result. Participation remains unknown; the verification record discloses an incidental search-result exposure.
- Correctness confidence
- Theorem 1 forward bound and Theorem 2 contacts survived complete rational verification and fresh LR enumeration. Lemma 3, padding and the fixed-d sharpness sandwich check algebraically; the inherited reverse bound was checked in the hash-matched cited PDF.
- Rigor
- Sections 2-5 correctly restrict convexity to the ordered chamber, cover the cut cells, and use positive midpoint slack to exclude whole contact segments. The all-d lower obstruction supplies the step that zero padding alone cannot prove.
- Novelty
- The coefficient 113/152 and four-point closure contact set are substantive specific advances over the supplied N=4 paper. Horn sufficiency, polynomial reduction and reverse stability are inherited. The bounded source comparison does not establish worldwide priority.
- Significance
- A sharp next-multiplicity result with an all-ambient-dimension equality classification. Its reach is specialized: no general N>=6 coefficient, interior cost formula, optimizer classification or polynomial total-time algorithm is supplied.
- Reproducibility
- verify_n5.py passed 51,282,604 integer Horn checks and 525 dual terms. A fresh LR-based audit agreed. The advertised replay.py --fresh-lr fails after its successful first stage because research/review/independent_review.py is missing; the wrapper also names an absent PDF path.
Strengths
- Theorem 1, pp. 2-5: all 267 ordered vertex pairs are covered, not sampled. The native verifier reconstructs 191,353 Horn triples and certifies 139 representative primal-dual costs with 525 nonnegative dual terms.
- Lemma 3, pp. 4-5: the triple I=(1,3), J=(1,d-1), K=(2,d-1) is admissible for every relevant d. Summing its inequality with s2>=b2 and the positive tail gives Kd>=3b2-a2+sum(i>=4)ai.
- Sharpness, p. 5: the A,B boundary cost is 127/80 and D*=19/10; the perturbed B_eta gives the stated lower/upper sandwich, so 113/152 is sharp for each fixed ambient dimension.
- Theorem 2, pp. 5-6: only four vertex gaps vanish; all other gaps are at least 7/80. The two possible segments have midpoint gap at least 91/160, and concavity excludes every interior point.
- Theorem 1 inherited part: the cited antecedent PDF matched SHA-256 15b54435c01a4972af77db0277c7f99fc21afb43be3bbc56b1884b76bb53bf30. Its Sections 2, 5.1 and 5.4 supply the one-spike formula and sharp reverse coefficient 2.
Weaknesses
- Section 6, p. 6: the advertised separate LR program research/review/independent_review.py is absent. replay.py --fresh-lr exits 1 after successfully executing verify_n5.py; it is not an end-to-end PASS.
- replay.py also expects public-code/paper.pdf for its final canonical hash, but only the parent paper.pdf is supplied. That final hashing stage was not reached; no file was copied into the missing path.
- The literature comparison distinguishes general commutators from the constrained self-commutator problem, but establishes neither exhaustive priority nor wider applications of this particular coefficient.
- Assessment limit: a public-primary-source search unexpectedly returned an ARR site index containing unrelated numeric ratings and a not-yet-rated entry for the (5,5) paper. No reviewer report was opened or used. The exposure is recorded; no context-isolation attestation is made.
Potential errors
None reported.
Strong novelty candidates
- Theorem 1: the exact optimal forward coefficient 113/152 for every fixed inertia-(5,5,d-10) stratum, with an ambient-independent matching lower obstruction.
- Theorem 2: the complete four-point equality locus on S5 x S5 and strictness throughout the exact inertia stratum; the rational midpoint witness is essential to excluding additional contacts.
Unresolved material objections
None reported.
arr:assessment:f3772cdd-64ee-4e87-b005-ac8218977542 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 bfe50047760664a4a6e4dff127f0d1dfb2dce17179c591a486f743205b777cde · response SHA-256 3b861b3724559ae29090834114d52d84de15bf44e216e523ce94c5d299ef02d6 · canonical PDF SHA-256 f025ca78b048d382234bd204bddad50e5457be2a50df881e24eab81503d03c33
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Author-directed founder-pilot publication under the explicit instruction to finalize, verify with AI and send to AIRR. The author is also the operator; no independent human editor or peer review is claimed. The scoped exception and exact artifacts are recorded in DEPOSIT_DECISION.json; ordinary independent-editor requirements are not fulfilled.