Kurkov's three-parameter Stirling-minor Vandermonde sum

Mikhail Kurkov's sole FORMULA conjecture in OEIS A396563, approved revision 15, asks whether the unsigned first-kind Stirling determinant $B(n,k,m)=\det[c(n+m+i,n+j)]_{1\le i,j\le k}$ equals

$$ \frac{\displaystyle\sum_{n\ge j_1\ge\cdots\ge j_m\ge0} (-1)^{j_1+\cdots+j_m} \prod_{1\le q<p\le m}(j_q-j_p+p-q)^2 \prod_{t=1}^m\frac{(n+t-1){j_t}(n-j_t+t)^{n+k}}{(j_t+m-t)!}} {\displaystyle\prod{i=0}^{m-1}(n+i)!}. $$

Here $n,k\ge0$, $m\ge1$, $c$ denotes unsigned first-kind Stirling numbers, and $(x)_r$ is the falling factorial. The empty determinant is one. The primary source is https://oeis.org/A396563/internal; the preregistration is https://github.com/the-omega-institute/trureturing/issues/22587.

Statement echo

| Source clause | Lean counterpart | Fidelity | | --- | --- | --- | | The $k\times k$ first-kind minor with one-based indices | B | Fin k values plus one preserve both source indices. | | Nested ranges $n\ge j_1\ge\cdots\ge j_m\ge0$ | Reused KurkovNestedStirlingIdentity.Chain n m | Antitone maps into Fin (n+1) enumerate each source tuple once. | | Sign, squared pair differences, falling factorial, numerator power and denominator factorial | sourceTerm | Rational differences and division; natural factorial indices and exponents. | | Division by the product of $(n+i)!$ | sourceRHS | The entire chain sum retains the full normalization. | | Equality for all natural $n,k$ and $m\ge1$ | claim, result | The source boundaries $n=0$ and $k=0$ are included. |

The four public definitions are the literal source determinant, summand, normalized sum and quantified assertion. There is no extra hypothesis on $n$ or $k$.

The settling declaration is D5/S3/Combinatorics/BinomialBases/KurkovStirlingMinorVandermondeSum.result. Its admission basis is open-problem-resolution (#22587; Proved). The source's A132393 reciprocal-power identity has a different normalization and does not supply this identity with independent $k,m$ and numerator power $n+k$. The module has one public theorem; all mathematical helpers remain private.

The bounded literature screen in #22587 located no identified proof or refutation of the exact three-parameter assertion. It inspected the OEIS record, Stirling determinant literature, arXiv, citation metadata and repository results. This supports the named settlement within the searched scope; literature completeness is ASSUMED-UNVERIFIED.

Write $N=n+m$ and $D_N(X)=\prod_{r=1}^N(1-rX)$. The first-kind coefficient calculation and triangular monic basis change identify the source minor with a corner of the Toeplitz matrix of $D_N$, with sign $(-1)^{km}$. The inverse series has coefficient $[X^r]D_N^{-1}=S(N+r,N)$, where $S$ denotes second-kind Stirling numbers.

The private inverse_corner_minor compares complementary corners of arbitrary rational matrices $A,H$ of size $k+m$ under $AH=I$ and $\det A=1$:

$$ \det[A_{m+i,j}]{0\le i,j<k} =(-1)^{km}\det[H{k+i,j}]_{0\le i,j<m}. $$

For the Toeplitz matrices of $D_N$ and $D_N^{-1}$, the hypotheses follow from series multiplication and the unit diagonal. The map $l_t=n-j_t+t$ identifies nested tuples with increasing $m$-subsets of ${1,\ldots,N}$. Factorial cancellation and Cauchy–Binet turn the literal sum into a signed moment determinant. Stirling inclusion–exclusion evaluates those moments; column reversal identifies it with the inverse Toeplitz corner. The two corner signs cancel, giving result.

A rational inequality between the two literal expressions at any $n,k\ge0$, $m\ge1$ would contradict result in the stated formal system. The symbolic proof includes both zero-boundary parameters and uses no finite checking assumption.

The theorem result : claim is the kernel-checked universal assertion. The private declarations first_minor_coeff, elementary_corner, complete_coeff, source_to_stirling_moment, complete_corner and inverse_corner_minor state the corresponding symbolic proof bridges. The four-slot escape audit for result is unfinished and tracked in https://github.com/the-omega-institute/trureturing/issues/26118. The typed registration decodes as declared_unresolved; it does not establish declared_validated. All divisions are in $\mathbb Q$; natural subtraction is used only for the literal indices, with the chain bounds supplying their intended values.

The published exact experiment is oeis-a396563-stirling-minor. Command: python3 docs/reports/oeis-a396563-stirling-minor/check.py; exit 0; script SHA-256 6041d41d7d2231c3fef7006967f8c14131831aec3a451dd0a74e1e74addb4379. It reproduces all 42 displayed source values, checks 245 triples $0\le n,k\le6$, $1\le m\le5$ and 11,956 chain instances, and reports ALL_CHECKS_OK. This computation is evidence only for its finite tested scope.

A separate local exact rational computation reports GENERIC_CORNER_CHECK_OK 360 seed=396563 0<=k,m<=5 10 trials per pair: 360 seeded unit lower-triangular rational matrix pairs, including zero-size corners. Command: python3 /tmp/op-kurk396563/probe/check_corner.py; exit 0; script SHA-256 9f3075507ebc1293a2e156fbd7405a985298528de37f2e704bd384af1c412b2a. Publication of this additional program in the experiments repository is unverified; the generic identity itself is proved by inverse_corner_minor.

| Target item | Status and evidence | | --- | --- | | Uniform literal identity, including $n=0$ and $k=0$ | Proved in Lean: result. | | First-kind coefficient minor, inverse Toeplitz corner, Cauchy–Binet and inclusion–exclusion bridge | Proved in Lean: first_minor_coeff, elementary_corner, complete_coeff, source_to_stirling_moment, complete_corner, inverse_corner_minor, consumed by result. | | Generic rational inverse-corner comparison under $AH=I$, $\det A=1$ | Proved in Lean: private inverse_corner_minor, for arbitrary $k,m,A,H$. | | 245 triples and 11,956 chain instances | Computed: the pinned exact checker in Evidence, exit 0, ALL_CHECKS_OK. | | 360 generic corner instances, seed 396563, $0\le k,m\le5$, ten trials per pair | Computed: the separate local checker in Evidence, exit 0; its repository publication is unverified. No universal claim follows from this computation. | | $m=0$ extension, Schur-function interpretation, universal positivity and integrality corollaries | Open: these are not separately stated and compiled as exported conclusions. |

What the settlement shows

Proved: the decisive mechanism is complementary-corner duality for determinant-one inverse Toeplitz matrices, together with an exact conversion of the literal chain sum to the inverse corner. The generic rational corner lemma extends beyond the Stirling matrices. The public settlement covers the literal conjecture's entire natural-parameter range.

The source array is the case $m=2$, so the conjectural sum now gives a proved expression for that array. The source programs define the determinant directly; the settlement does not change their definitions. No further source result was identified as conditional on this conjecture. The separate positivity, integrality, $m=0$ and Schur-function questions remain open here.

Literature completeness, historical priority and absence of differently worded settlements outside the screened sources are unverified. The source has no DOI or arXiv identifier. Publication of the additional 360-instance program is unverified. Numerical checks do not replace the universal kernel proof.