Sra's symmetric alpha-permanent question

Suvrit Sra, A Koteljanskii inequality for permanents, arXiv:2609.39979v1, Section 4(b), asks:

For symmetrizable inverse $M$-matrices and $\alpha\notin{\tfrac12,1}$, decide whether $S\mapsto\operatorname{per}_\alpha(A_S)$ is log-supermodular.

The order is positive as in equation (3.1). The universal affirmative assertion quantifies over $n\ge1$, every real symmetrizable inverse $M$-matrix, every permitted positive order, and every pair of index subsets. The source definitions are quoted in the Library note.

Statement echo

| Source clause | Formal counterpart | Fidelity | | --- | --- | --- | | Every $n\ge1$, real inverse $M$-matrix | claim, imported InverseMMatrix | Exactly the source's nonnegativity, invertibility and inverse off-diagonal sign conditions. | | Positive diagonal similarity to symmetry | Symmetrizable | The diagonal function is strictly positive; its pointwise reciprocal gives the inverse diagonal. | | Equation (3.1), all cycles including fixed points | alphaPermanent | cycleType.card counts nontrivial cycles, and the fixed-point cardinality adds one-cycles. | | Principal restriction, including the empty case | principalAlphaPermanent | Matrix.submatrix on the index subtype; empty product and permutation sum give one. | | $\alpha>0$, $\alpha\ne1/2,1$ and every $S,T$ | claim | All three order conditions and both subset binders are retained. | | Log-supermodular inequality | claim | The $S,T$ product is at most the union-intersection product. | | Universal affirmative assertion is false | result | Literal negation of claim. |

D5/S3/Combinatorics/Permanental/SraSymmetricAlphaPermanentRefutation.result refutes the assertion. The public result is the designated refutation result (basis=refutes) and is exempt from four-slot escape registration (CLAUDE.md §3.9). It is the module's only public theorem.

Preregistration #21689 quotes the source, fixes the definitions and records the Tier 1 literature screen. No settlement was identified within that scope; completeness remains qualified below. Proof shape is bind-only; admission is open-problem-resolution (#21689; Refuted).

The fixed witness is

$$B=\begin{pmatrix}7&2&1&2\2&7&2&1\1&2&7&2\2&1&2&7\end{pmatrix}.$$

Private witness_positive, inverseM and symmetric certify strict entrywise positivity and the source hypotheses. With zero-based $S={0,1,2}$ and $T={0,2,3}$, difference proves that the union-intersection product minus the $S,T$ product is

$$\alpha^2(\alpha+1)(2401\alpha^2+1617\alpha-16).$$

interval_failure proves strict negativity for every $0<\alpha\le1/200$. result specializes to $\alpha=1/200$ and contradicts the universal inequality. The imported InverseMMatrix is the same definition used in the reciprocal-polynomial module; there is no second inverse-$M$ definition.

The displayed matrix with $\alpha=1/200$, $S$ and $T$ satisfies all source hypotheses and violates its affirmative inequality. The argument refutes the universal assertion without classifying all remaining positive orders.

The source is SraSymmetricAlphaPermanentRefutation.lean, with its canonical Blueprint Scribe. Private theorem names below belong to that namespace.

The rational experiment is sra-koteljanskii at fc05f5ee612010d382cf8772f7bf43a1cb830aa1. Its docs/reports/sra-koteljanskii/check.py has SHA-256 bf8bef761479549a67a05ccf46ec6e0ebe90c9adf347d014cec66ea510a92a75. Command: python3 /tmp/op-sra2609/check.py, using the immutable revision's bytes read by git show; exit 0, ALL_CHECKS_OK. This checks the fixed witnesses, the factorization and the finite subset tests below, separately from the Lean proof.

What the settlement shows

1. Proved (kernel-checked): result refutes the universal affirmative extension. witness_positive, inverseM, symmetric, difference and interval_failure certify that this symmetric positive $4\times4$ inverse $M$-matrix fails throughout $0<\alpha\le1/200$, with the factorization displayed above.
2. Computed: all $256$ subset pairs for this fixed $B$ satisfy the inequality at each of $\alpha=1/2$ and $\alpha=1$; the selected difference at $\alpha=1/200$ equals $-63153999/320000000000$. Evidence: the immutable experiment, stated command, exit 0 and SHA-256 above. No other parameter range is asserted to have been enumerated.
3. Open: remaining universal alpha ranges and the source tree case are not newly classified or formalized. Literature reading: the source's theorem and remark retain the distinguished orders and tree hypotheses; no proved source theorem assumes an affirmative answer to either open question. Those source results are not on this module's dependency path.
4. Proved (kernel-checked in the companion module): SraReciprocalPermanentLorentzianRefutation.result refutes Section 4(d). Its mechanism is a persistent positive two-plane after three coordinate derivatives at a strictly positive $5\times5$ inverse $M$-matrix. Computed: its relevant permanents are $546,9996,9996,182784$, product ratio $832/833$, first pivot $1/273$, determinant $1/20783363328$, by the same immutable experiment. Open: symmetric reciprocal-polynomial Lorentzianity and a separately formalized positive diagonal scaling extension. The precise evidence and scope are in the companion dossier.

Literature completeness is ASSUMED-UNVERIFIED. The preregistration's screen includes source v1, Brändén–Huh v8, arXiv, Semantic Scholar, Crossref, MathOverflow and MathDB. OpenAlex yielded no citation list; broad full-text results were not all read, and web search produced no reliable relevant hits. Zero indexed citations does not exclude an undiscovered solution. No journal version was identified or read.