The inverse self-commutator cost at inertia $(m,n)$: ordered-layering bounds, interlacing chains, and the Horn tiles that certify them
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For a traceless Hermitian $F\in M_d(\mathbb C)$ let $\kappa_d(F)=\tfrac12\min\{\|C\|_{HS}^2:CC^*-C^*C=2F\}$. On the inertia-$(m,n)$ stratum ($m$ positive and $n$ negative eigenvalues, $z$ zeros) we prove three results valid for all $m,n,z$: a dimension-free lower bound $\kappa_d\ge\operatorname{cost}(L)$ for every ordered full-capacity layering $L$, certified by Lidskii–Wielandt and Weyl inequalities alone; an exact additive formula on the aligned-spike cone, where an optimizer is a direct sum of weighted shifts; and the closed form $\kappa_d=\operatorname{cost}(L_k)$ on explicit full-dimensional regions $R_k$, $1\le k\le m-n+1$, obtained from one rank-one interlacing step. Open question 1 of the inertia-$(m,2)$ paper is refuted by an exact weak-duality certificate: at inertia $(5,4)$ a Horn tile with $n=4$ boxes is necessary, and one valid form cannot be certified using Pieri-shaped tiles alone. The $d=8$ rank-onset example is explained by a late negative. Conjectures for general $(m,n)$ are stated with their numerical evidence.
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Cite this paper
Lluis Eriksson (2026). The inverse self-commutator cost at inertia $(m,n)$: ordered-layering bounds, interlacing chains, and the Horn tiles that certify them. AIRR.SCIENCE, ARR-2026-4Q1WFSE5DW9728QY, v1. https://airr.science/papers/ARR-2026-4Q1WFSE5DW9728QY/versions/v1/
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Verification record
- Frontier-model screening
- Pass · 2 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-INTAKE-ASSESS-1.1. AIRR verification and screening are not peer review.
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The author supplied these manuscripts as developed with Claude Fable 5.1; historical serving identity is not independently authenticated. Codex assisted bounded revisions. Actual GPT-6 Astra High and GPT-5.6 Sol Medium reports concern this exact PDF, and each model family declares prior involvement. Intake bindings adopt the prior same-hash scientific reviews; they are not additional independent reviews. For B3, A4 and C2, a later Astra context explicitly adopted the preceding Astra report. Technical memory and filesystem isolation are not certified.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-INTAKE-ASSESS-1.1; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · reasoning effort high · pass · creation conflict declared
- gpt-5.6-solOpenAI · reasoning effort medium · pass · creation conflict declared
Model assessments and review context
- Distinct models
- 2
- Range
- 4.50–4.80
- 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
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Two exact-PDF native model reviews were adopted into the registered intake, one per model. Original local reports and intake binding provenance remain preserved; binding does not add another scientific review. Both model families participated in revision. One provider and incomplete benchmark matching limit the evidence; the weighted score is not a probability of correctness. A separate disclosed founder-author editorial decision and exact public permission were recorded. Required clarification accompanying this unchanged PDF: The minor codimension statement is qualified in the visible author clarification: n-1 is the rank of the ambient balance equations, while codimension n-1 for the ordered intersection requires relative interior in that balance subspace. The m=n=2, u=(1,0) counterexample has four entries equal and codimension 2. The same Astra reviewer verified the qualification and exact counterexample; the original PDF and minor score remain unchanged.
Operator record · · Source files and affected reports
OpenAI · gpt-6-astra · High · involved in manuscript · review context: unknown4.50
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 3101560922c29c73e39cb9ea006c37fe264984a784216f0e86cd20ad8e0a2196.
Recommendation: Minor Revision · Material objections: 0
Native same-PDF intake adoption of my preserved local scientific assessment dated 2026-09-14T17:42:47.639308+00:00. The principal theorems are supported by their proofs and the independent checks performed here. In particular, the box<=3 and Pieri-only obstruction gaps are exact, the rank-five optimal-face claim is now justified, and the one-rank-one-step construction works on the stated region including tested boundaries. Request minor revision to qualify the codimension remark for fixed layerings; no material objection to the main theorems was found. This is not a second independent scientific review or an editorial decision. The separate same-hash author clarification and clarification review address the minor findings as visible accompaniment; the native minor_revision remains unchanged.
- Correctness confidence
- The principal lower, additive, interlacing and obstruction theorems pass analytic inspection and targeted independent exact checks. A minor codimension remark needs a qualification.
- Rigor
- Exact weak-duality witnesses and exact rank-face inequalities are strong. The isolated codimension assertion ignores possible ordering-induced degeneracy of a fixed schedule.
- Novelty
- General ordered-layering lower bounds, explicit interlacing regions and exact limitations of restricted Horn certificate classes are plausible novel combinations; core ingredients are classical.
- Significance
- Useful general-inertia results and a concrete certificate obstruction, while the full catalogue and block-shift conjecture remain open.
- Reproducibility
- Exact revision replay passes; a new scalar Horn recursion confirms both obstructions and face rows, and a separately written construction verifies 168 region cases.
Strengths
- Theorem 1 cleanly isolates the disjoint-index-block reason full capacity is sufficient, and handles negatives placed in the final layer by an explicitly positive RHS slack.
- The balanced-path equivalence correctly translates diagonal block spectra into monomial directed paths; additivity is conditional on alignment.
- Theorem 2B matches rank-one interlacing to the column-sum continuation and has the correct nonempty range k<=m-n+1.
- Exact restricted-Horn feasible spectra prove that arbitrary nonnegative multiplier certificates of the two specified classes cannot suffice.
- The revised d8 row combinations force s6=s7=0 and s5>=1 at cost13, while a full-Horn witness attains that cost.
- Incomplete reflected numerical catalogues are acknowledged as sampled sets and not claimed complete.
Weaknesses
- Section 5.1 Remark (a) states codimension n-1 without qualifying the dimension after imposing ordering on a fixed schedule. This is not valid for every fixed admissible schedule: ordering can force extra equalities. The separate author clarification supplies the required ambient-interior qualification.
- General (m,n) completeness, padding stability and the block-shift assertion remain conjectures; the sampled count mismatch precludes treating the data as a complete classification.
- Priority search is limited, and most proof ingredients are standard Horn, Lidskii and rank-one inverse-eigenvalue machinery.
- The minor findings are addressed in separate same-hash AUTHOR-CLARIFICATION.md and final-astra/CLARIFICATION-REVIEW.json, provided they visibly accompany the unchanged PDF. The original minor_revision recommendation remains preserved; this is not an editorial disposition.
Potential errors
- minor; Section 5.1, Remarks (a) after Theorem 2A; For m=n=2 and fixed admissible u=(1,0), L={-b2}{a1,-b1}{a2}. A balanced path system requires b2=a1 and b1=a2. Together with a1>=a2 and b1>=b2 this forces all four entries equal. The aligned set has dimension1 inside the dimension3 trace hyperplane, hence codimension2 rather than n-1=1. Qualify the statement for nonempty components with relative interior in their balance subspace, or explicitly state a maximal dimension for the union. Theorem 2A additivity remains valid.
Strong novelty candidates
- General-inertia ordered full-capacity Lidskii-Wielandt lower bound with arbitrary negative schedule.
- Explicit full-dimensional interlacing regions for L_k and rank-m attaining construction.
- Exact n=4 obstruction to every box<=3 Horn certificate, and separate exact obstruction to all Pieri-shaped certificate classes.
Unresolved material objections
None reported.
arr:assessment:3fd622b6-f59a-4481-85ec-870139924397 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 fb1864e6885d003154fd3b74c540013867da653f82a1edf22dea1269077d89b7 · response SHA-256 381b5bd93f06303d71542a92bbbf7c664db37d90e926f69ed3c4561b7487ecb7 · canonical PDF SHA-256 ffea17de8edce73844df1705c439f89e4a8f18eaf3de4a0f374cdec056c51921
OpenAI · gpt-5.6-sol · Medium · involved in manuscript · review context: unknown4.80
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 524a6d64e90a3c0857af855ceb59bba2507b5ec17d2ea31d5bbc91de26d77254.
Recommendation: Accept · Material objections: 0
A substantial and well-delimited general-inertia advance. Theorem 1 gives a correct all-(m,n,z) Horn certificate for ordered full-capacity layerings. Theorems 2A and 2B give exact upper constructions on aligned-spike cones and explicit full-dimensional interlacing regions. Theorem 3 rigorously disproves sufficiency of box<=n-1 Horn tiles at (5,4) by an exact weak-duality witness, while the revised d=8 argument exactly certifies rank five. No material objection was found.
- Correctness confidence
- The universal ordered-layering lower bound, aligned-spike equivalence, one-step interlacing regions, weak-duality obstruction, and exact d=8 rank witness all withstand direct checking. Fresh exact and high-precision reproductions agree with the stated results.
- Rigor
- The all-parameter results are analytic. The computational Theorem 3 uses exact rational primal witnesses and exact Horn/LR identities. Numerical catalogues and conjectures are explicitly segregated from the proved statements.
- Novelty
- The combination of dimension-free ordered-layering certificates, exact interlacing chambers, and an exact lower bound on necessary Horn-box complexity appears original within the checked literature. Exhaustive priority was not established.
- Significance
- The manuscript advances from fixed small inertia to general (m,n), supplies exact full-dimensional regions with optimal rank, and identifies a precise obstruction to the earlier n-1-box expectation.
- Reproducibility
- The bounded revision verifier regenerates Horn lists in memory, and all exact obstructions and the d=8 face certificate replay quickly. Independent supplied theorem tests reran with a cache-free generator and constructed the predicted commutators to near machine precision.
Strengths
- Theorem 1 is genuinely dimension-free and its disjoint-index mechanism explains exactly why full capacity is required.
- The path-system characterization gives a concrete structural meaning to diagonal block-shift chains and additive one-spike decompositions.
- Theorem 2B uses the inverse rank-one interlacing theorem efficiently and proves nonempty full-dimensional regions with rank-m optimizers.
- Theorem 3 excludes every nonnegative certificate from a broad restricted Horn class, rather than merely failing to find one.
- The corrected d=8 face argument supplies exact lower inequalities and an exact full-Horn feasible spectrum, closing the rank claim.
- The manuscript clearly labels incomplete catalogues, numerical block-size evidence, and general structural conjectures.
Weaknesses
- The exact regions R_k generally cover only part of the chambers for n>=3; the pure-phase feasibility facets remain unknown.
- The broader block-size and finite-form classification in Section 8 is numerical and the sampled reflected catalogue is demonstrably incomplete.
- Several motivating comparisons rely on earlier AIRR records that were outside the permitted source set.
Potential errors
None reported.
Strong novelty candidates
- The all-inertia ordered full-capacity layering lower bound with only Lidskii-Wielandt and Weyl tiles.
- The balanced-path characterization and exact additive aligned-spike formula.
- The full-dimensional regions R_k solved by a single inverse rank-one interlacing step.
- The exact proof that box n Horn tiles are necessary already at inertia (5,4).
- The exact Horn-face explanation of the d=8 rank-five onset.
Unresolved material objections
None reported.
arr:assessment:9c25c3b5-2057-469c-9067-4f0d4d9fb200 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 6fcaaceeebcbaaf7c34eb7ced8ad41a7e2f96560e52080c35a9fc6a27411038d · response SHA-256 b338e225884ef56390d55c95753cd224484feeb580b7c216c37bb2f532ff8fb4 · canonical PDF SHA-256 ffea17de8edce73844df1705c439f89e4a8f18eaf3de4a0f374cdec056c51921
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Editorial disclosure
The sole author and founder-editor expressly authorized the complete cycle. The authenticated Netcup operator recorded that human instruction separately as an editorial acceptance after two exact-PDF model reports and evidenced minor dispositions, then recorded exact public permission. No independent human peer review or simulated web signature is claimed.