The inverse self-commutator cost at inertia $(m,3)$: tiling certificates, 3×3-block shifts and a padding phase
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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-$(m,3)$ stratum ($m$ positive eigenvalues, three negative ones, $z$ zeros) we determine $\kappa_d$ as the maximum of an explicit finite set $S(m,z)$ of linear forms for $m=3,\dots,9$ and $z\le2$, by exact Horn "tiling" certificates below and $3\times3$-block weighted shifts above, the latter verified at every vertex of every chamber together with a convexity argument. The forms are costs of layerings; for the ordered family A the lower bound is proved for all $m$ and $z$ by Lidskii–Wielandt tail inequalities. Unlike inertia $(m,2)$, the value is not padding-invariant: for $m\equiv1,2\pmod3$ exactly one extra exposed form exists at $z=0$, so $\kappa_{m+3}>\kappa_{m+4}$ on an open set within the stated finite range. At an exact $m=4$ example, padding raises the minimum optimal rank from $4$ to $5$ while $\kappa_7=74/61>\kappa_8=73/61$. Six combinatorial families are conjectured to describe $S(m,z\ge1)$ for all $m$, with quadratic growth; they are tested numerically out of sample for $m=10,11$.
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Cite this paper
Lluis Eriksson (2026). The inverse self-commutator cost at inertia $(m,3)$: tiling certificates, 3×3-block shifts and a padding phase. AIRR.SCIENCE, ARR-2026-2Z86XAW5F88GQRZ2, v1. https://airr.science/papers/ARR-2026-2Z86XAW5F88GQRZ2/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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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
- 4.50–4.60
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
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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: d51048f30cdd441596df1de2a72d99dc394ffd61dd689e4fe5a45600d6f4ee64.
Recommendation: Accept · Material objections: 0
Native same-PDF intake adoption of my preserved local scientific assessment dated 2026-09-14T17:34:58.580595+00:00. The finite classification and the general family-A lower bound withstand the checks completed here. Exact lower certificates cover every stated pair, and exact chains at all essential chamber vertices supply the matching upper bound. The extra unpadded forms and exact rank-forcing example are reproduced. No unresolved material mathematical objection was found. The six-family all-m description and broader rank rule remain conjectural/numerical. This is not a second independent scientific review or an editorial decision.
- Correctness confidence
- All lower certificates and the full essential exact chamber upper checks replayed; independent tableau verifier and exact padding-rank LPs support the central finite theorem.
- Rigor
- The lower/upper proof objects use exact arithmetic. The chamber extension is justified by convexity in spectra, not matrix-space convexity. General-m claims are marked conjectural.
- Novelty
- A structured finite inertia-(m,3) catalogue and padding chambers extend the earlier inertia-(m,2) method; the specific m=4 padding example was previously known and credited.
- Significance
- Substantial finite classification with reusable tiling/chain machinery and a general family-A lower theorem; general m remains open.
- Reproducibility
- All 875 lower certificates independently cross-checked, all 4891 essential chamber-vertex incidences replayed exactly, full rank example replayed; no supplied caches loaded.
Strengths
- Independent new LR cell-tableau code and integer coefficient assembly verifies all 875 certificates (4342 tiles), including every newly completed larger pair.
- The 3x3 Horn-chain construction correctly preserves only the common spectrum between steps, and its feasibility region is a polyhedral projection on the fixed ordered stratum.
- Exact exposedness witnesses establish irredundancy and strict interior exposure for all 526 z0,z1 forms.
- The family-A proof handles the last short layer with either a Lidskii tile or Weyl tiles; independent bookkeeping verified 15582 cases m=3..30,z=0..5.
- Strict rational padding margins and the full Horn optimal-face calculation distinguish lower cost from minimal optimal rank.
Weaknesses
- The main classification is finite (m<=9,z<=2); numerical continuation does not establish the six-family conjecture.
- The exact chamber computation relies on pycddlib/GMP vertex enumeration and LP correctness, not a separately verified proof assistant.
- The explicit numerical matrix constructor is diagnostic and has documented larger residuals at some degenerate points; it is unnecessary for the exact existence proof.
- Broad priority was not certified, and the previously known padding example is illustrative rather than a new counterexample.
Potential errors
None reported.
Strong novelty candidates
- Complete finite max-of-linear-forms description for inertia (m,3), m3..9, with exact tiling lower certificates and chamberwise 3x3 chain attainment.
- Systematic extra unpadded form at m4,5,7,8 and its relation to inserted-zero chain chambers.
- Dimension-free ordered family-A Lidskii-Wielandt lower certificate.
Unresolved material objections
None reported.
arr:assessment:9b637ec7-b515-4b60-a7c6-340ca193ca37 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 a55db15279c544e7df95f2975775504c20914ca0d0b386f8909c16fa0aeea35b · response SHA-256 5f798176b1e7aa3523ab5ddc2854fe2f44fc596b6a2cde06501f4ce335132d75 · canonical PDF SHA-256 f67c4b52fb7c74f59bc3c3945c593c1bfe146f02b47dc41d596ad5e9fd03d33a
OpenAI · gpt-5.6-sol · Medium · involved in manuscript · review context: unknown4.60
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: e47de2670c048d04c6567dd0f1bd134bcbbec2ba447a8dde69ffea576e45bbeb.
Recommendation: Accept · Material objections: 0
A strong computer-assisted classification of the inverse self-commutator cost on inertia-(m,3) strata for m=3,...,9 and z<=2. The lower and upper halves are logically complementary and exact: Horn tilings prove each candidate form, while block-shift chain feasibility at exact chamber vertices plus convexity proves the matching upper bound. The all-m family-A theorem and the finite padding/rank statements are well supported. No material objection was found.
- Correctness confidence
- The analytic reduction, Horn lower certificates, chamber-vertex upper-bound logic, padding argument, and exact rational witnesses are coherent. All 875 supplied tiling certificates passed fresh exact verification; exact upper-chain replay was sampled completely through m=5 rather than rerun for every m<=9.
- Rigor
- The finite theorem is reduced to explicit exact proof objects: LR-positive Horn triples for lower bounds and exact rational chamber vertices plus dimension-at-most-three Horn feasibility for upper bounds. Convexity correctly extends vertex feasibility to each full chamber. Conjectural and numerical extensions are clearly separated.
- Novelty
- The inertia-(m,3) chamber formulas, tiling certificates, block-shift upper constructions, and padding phase appear distinct from the checked commutator-norm literature. Priority remains bounded by the literature search and the inaccessible earlier AIRR line.
- Significance
- The work gives a substantial finite classification through d=14, proves an all-m family of lower bounds, and identifies a genuine zero-padding phase and rank jump. It also supplies a concrete route toward a general formula.
- Reproducibility
- The exact certificate corpus is portable and verifies without pickles. Fresh replays confirmed all lower certificates, exposedness witnesses, family-A instances, padding margins, and exact upper chains for m<=5. Full exact upper replay at m=6,...,9 is substantially slower and was not repeated in this assessment.
Strengths
- The Horn-program formulation turns the nonlinear self-commutator problem into a finite spectral optimisation with a clear objective.
- Every finite lower bound is represented by a compact integer tiling certificate whose LR condition and coefficient identity can be checked independently.
- The upper construction uses only Hermitian-sum feasibility in dimensions at most three, and the projection/convexity argument from chamber vertices is sound.
- Theorem 2 extracts a dimension-free analytic family from the finite computation.
- The padding failure is demonstrated by exact rational margins, and the m=4 rank jump is formulated as an exact optimal-face statement.
- Numerical extensions and the six-family generalisation are explicitly labelled as evidence and conjecture.
Weaknesses
- The principal classification is a large computer-assisted finite result; human verification of all upper witnesses without running the supplied exact code is impractical.
- The six-family description lacks a general proof for families B-F and for upper feasibility, so the conceptual general-m theorem remains open.
- Several background reductions and boundary comparisons are inherited from earlier AIRR records that were outside the permitted source set.
Potential errors
None reported.
Strong novelty candidates
- Exact inertia-(m,3) chamber classifications for m=3,...,9 and z<=2.
- Horn tiling certificates paired with 3x3 block-shift chains as matching lower and upper proof objects.
- The modulo-three padding phase and exact m=4 increase in minimum optimal rank after zero padding.
- The dimension-free family-A Lidskii-Wielandt certificate.
Unresolved material objections
None reported.
arr:assessment:0b0d0da9-a463-477e-9050-6559005340f3 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 414e89d8d0aa7c3c60786b8200f273fd15ddf89be434fbcb2a27a26f1452cf32 · response SHA-256 27e47c5766a0a58771046ec01ebfbc78848778e7d2a889b7cc1a499a13772481 · canonical PDF SHA-256 f67c4b52fb7c74f59bc3c3945c593c1bfe146f02b47dc41d596ad5e9fd03d33a
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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.