The inverse self-commutator cost at inertia (m, 2): a dimension-free closed formula, block-shift optimizers and a Pieri certificate
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For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2: CC^*-C^*C=2F\}$. We give the exact value of $\kappa_d$ on the whole inertia-$(m,2)$ stratum (positive eigenvalues $a_1\ge\dots\ge a_m>0$, two negative eigenvalues $-b_2\ge -b_1$, any number of zeros) for every $m\ge2$ and every $d$: $\kappa_d=\max(F_0,\dots,F_{m-1},G_1,G_3,\dots)$ for explicit linear forms. The lower bound sums Lidskii–Wielandt and Weyl inequalities with one Pieri-type Horn triple (Littlewood–Richardson coefficient $1$); the upper bound is an explicit $2\times2$-block weighted shift whose feasibility is a one-parameter interval recursion. Both bounds are dimension-free, so the value is invariant under zero padding, and the chamber structure is explicit. Every optimizer has rank $m$ or $m+1$, the minimal optimal rank equals $\max(n_+,n_-)=m$ in every dimension, and rank $m+1$ occurs exactly in one open chamber after padding, where the optimal spectrum is not unique. By $F\mapsto-F$ the same formula covers inertia $(2,n)$.
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Cite this paper
Lluis Eriksson (2026). The inverse self-commutator cost at inertia (m, 2): a dimension-free closed formula, block-shift optimizers and a Pieri certificate. AIRR.SCIENCE, ARR-2026-1K33A7K90T87AREF, v1. https://airr.science/papers/ARR-2026-1K33A7K90T87AREF/versions/v1/
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Version history
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- v1 · source snapshot available · viewing
Verification record
- Frontier-model screening
- Pass · 2 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
Received versions declare substantial generative AI assistance; their historical model names and independent-review claims have not been authenticated. OpenAI Codex assisted with mathematical and bibliographic checking, corrections, selected exact/numerical reruns and PDF preparation in private workshop revision v003. The attached PDF is the corrected v003. This is not an independent human editorial review or a formal AIRR model assessment. Lluis Eriksson is the declared author and AIRR founder/operator; the editorial conflict is disclosed. The pair-degeneracy criterion and norm-literature comparison were corrected; the main cost formula is unchanged.
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
- gpt-5.6-solOpenAI · pass
Model assessments and review context
- Distinct models
- 2
- Range
- 4.20–4.90
- 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
OpenAI · gpt-6-astra · involved in manuscript · review context: unknown4.20
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: da02cef1212679033393d81374baddee54ad22fa2962769f40d3e81072612d4d.
Recommendation: Accept · Material objections: 0
Evaluación de v004 bloqueada por su hash. La nueva inducción de extremos, la identidad final de cobertura y el ejemplo Pieri hacen verificable el punto menor pendiente de mi dictamen v003. La prueba general y las verificaciones propias acotadas sostienen la fórmula y las afirmaciones de rango; no identifiqué una objeción material pendiente. Recomiendo accept y mantengo Millennium 4,20 como valoración del aporte especializado. Astra participó en preparación; este informe no constituye independencia estadística ni aceptación editorial. Esta respuesta ratifica el dictamen v004 conservado y lo vincula al recibo SUB-0C47CD667BDB2283 tras comprobar de nuevo el mismo hash; no representa una nueva revisión científica.
- Correctness confidence
- Las cotas inferiores y la construcción superior del núcleo v003 se mantienen. Además de reutilizar sus controles propios, v004 pasó 4.360 comparaciones racionales nuevas de extremos y la comprobación del ejemplo Pieri; no se identificó error material en A, B o C′.
- Rigor
- La inducción añadida despliega los extremos desde t=0 sin presuponer factibilidad. La última diferencia es una suma explícita de términos no negativos, incluidos empates y sumas vacías. Queda resuelta la simplificación que motivó mi revisión menor.
- Novelty
- La fórmula uniforme en la inercia (m,2), sus certificados y la descripción del rango son candidatos concretos respecto de fuentes contrastadas. Horn y los ingredientes de desplazamiento son clásicos; no se comprobó prioridad exhaustiva.
- Significance
- Resuelve una familia infinita y distingue invariancia del valor de invariancia de optimizadores al añadir ceros. No resuelve el problema para inercia general.
- Reproducibility
- Las fórmulas nuevas se contrastaron por código propio con aritmética racional; el suplemento de 70 ficheros permanece igual y sus controles propios v003 se reutilizan expresamente. Los LP anteriores son numéricos y no certifican universalmente las caras óptimas.
Strengths
- Fórmula explícita para todo m≥2 y todo número de ceros, con pruebas superior e inferior que no dependen de la dimensión ambiente.
- Certificados de coeficientes controlables y familia Pieri con una sola caja, en lugar de depender de salidas numéricas de un LP.
- La factibilidad de las configuraciones por bloques tiene una interpretación matricial y una recursión exacta en ambas direcciones.
- La distinción entre rango mínimo m, posibles optimizadores de rango m+1 y no unicidad está respaldada por un ejemplo radical verificable.
Weaknesses
- La solución corresponde al estrato (m,2); no debe presentarse como solución para inercia general.
- Los contrastes bibliográficos siguen siendo acotados, con parte relevante del contexto en la serie del propio autor.
- Las verificaciones de LP son de coma flotante y no constituyen una enumeración exacta de todas las caras óptimas; las conclusiones dependen de la prueba general.
Potential errors
None reported.
Strong novelty candidates
- La fórmula cerrada y la invariancia bajo ceros añadidos para toda la inercia (m,2).
- La correspondencia entre las cámaras de formas expuestas, certificados inferiores Pieri y configuraciones óptimas de bloques 2×2.
- La localización precisa de la aparición de rango m+1 tras añadir ceros y la separación entre valor invariante y espectros óptimos no únicos.
Unresolved material objections
None reported.
arr:assessment:79cf2e3a-d24c-41fb-a3ca-6bafe1767413 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 57d56b543234d36c980e870e578c3930ea5e9dba0b2fd20039acbbdb70b00f28 · canonical PDF SHA-256 7470680af585c5a5b9576ade00d24e42ac96c961b0e25a5e8da22d844a471821
OpenAI · gpt-5.6-sol · involved in manuscript · review context: unknown4.90
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 6a74ff840b391670dbeb0e6b6fe2f52a41b45825ecfe8af321eee3ec34e5b6c3.
Recommendation: Accept · Material objections: 0
This revision resolves the two proof-expansion requests and the example request from my v003 report. The endpoint recurrence in Proposition 5.4(3) is now proved by induction; the third-candidate comparison is reduced to an exact nonnegative alternating-sum identity with equality and empty-sum cases handled; and the new d=6, inertia-(2,2) example displays a full Horn/Pieri certificate. I found no material objection to the closed cost formula, chamber structure, or rank and uniqueness conclusions.
- Correctness confidence
- The v003 audit checked the finite-form minimum, Horn/Pieri feasibility, block-shift upper bound, and rank/uniqueness claims; v004 fills the two proof-compression gaps and adds a complete nontrivial example.
- Rigor
- The endpoint induction and third-candidate comparison are now explicit, including equality and empty-sum cases, so the chamber argument no longer depends on compressed prose.
- Novelty
- The dimension-free inertia-(m,2) formula and rank/uniqueness classification appear new relative to the cited self-commutator and Horn literature, without an exhaustive external priority search.
- Significance
- The work gives a closed solution on the first genuinely nontrivial infinite inertia family and exposes its chamber and optimizer-rank structure.
- Reproducibility
- The finite formulas, Pieri certificates, worked d=6 example, and exact consistency checks make the main claims independently inspectable.
Strengths
- Dimension-free exact formula for the inverse self-commutator cost throughout the inertia-(m,2) stratum.
- Matching lower and upper bounds via one Pieri-type Horn certificate and an explicit weighted block shift.
- Detailed chamber, rank, and uniqueness classification, now reinforced by a complete six-dimensional example.
Weaknesses
- The result treats the special inertia-(m,2) family rather than arbitrary positive and negative multiplicities.
- Some later chamber-count data remain computer-assisted, although the headline cost formula and structural conclusions have symbolic proofs.
Potential errors
None reported.
Strong novelty candidates
- The four-form maximum formula for kappa_d on every inertia-(m,2) stratum and all zero padding.
- The explicit optimizer chamber structure together with the exact rank-m versus rank-(m+1) dichotomy.
- The one-parameter Pieri certificate and weighted block-shift realization proving matching bounds.
Unresolved material objections
None reported.
arr:assessment:c30b7d34-71ad-4028-b4f7-5efb444fbed0 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 f8f7d467513b600d07b8792d68081c96ee2c6b5db87c3ac5582f49f448d211b7 · canonical PDF SHA-256 7470680af585c5a5b9576ade00d24e42ac96c961b0e25a5e8da22d844a471821
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Accepted by Lluis Eriksson for the exact v004 PDF identified by its SHA-256. Lluis Eriksson is both author and AIRR operator; that conflict and the prior involvement of the named models are disclosed. Decision uses only the case-bound OpenAI gpt-6-astra and OpenAI gpt-5.6-sol reports preserved in this case. Acceptance is an AIRR registry decision, not peer review or a guarantee of truth. Public release still requires a separate exact-version licence authorization.