svinin 2015 discrete polynomial identity

A. K. Svinin, On integrals for some class of ordinary difference equations admitting a Lax pair representation, arXiv:1505.06394v2, Conjecture 1, Eq. (48), asks whether

$$ \langle S,T\rangle_s^r(i)=\sum_{j=0}^{r-2}(-1)^j u(i+r-j-2)u(i+s+j)S_{s+j+2}^{r-j-2}(i)T_{s-r+j}^{j}(i+r-j) $$

is an identity for every $r\ge1$, $s\ge r-1$ and $i\in\mathbb Z$ in the specialization $(h,n)=(1,1)$. The source says: “Unfortunately, by now we can prove this conjecture only for r = 1, 2, 3.” The settlement is Proved, under preregistration #19287. The coefficient field is an arbitrary function $u:\mathbb Z\to R$ into any commutative ring $R$.

Statement echo

| Source clause | Lean counterpart | Correspondence | | --- | --- | --- | | Eq. (4): strictly increasing tuples $0\le\lambda_1<\cdots<\lambda_r\le s-1$ | T | Maps Fin r → Fin s.toNat, filtered by StrictMono; factor $a$ has offset $a$. | | Section 2.3: weakly increasing tuples with descending offsets | S | The Monotone filter and integer offset $r-1-a$ reproduce the factor pattern. The printed first-offset $k$ is read as $r$. | | Section 3: $(h,n)=(1,1)$ | T, S | The lattice increments in the two literal sums are one. | | Section 3.4: alternating binary operation | bracket | The sum has indices $j=0,\ldots,r$ and exactly the source's degree, subscript and lattice shifts. | | Conjecture 1, Eq. (48), all degrees | claim, result | All rings, functions, degrees $r\ge1$, subscripts $s\ge r-1$ and lattice points are quantified. | | Degree-zero and nonpositive-subscript conventions | T, S | Empty products are one; positive-degree maps into Fin 0 do not exist. Natural subtraction is truncated; lattice arithmetic uses integers. |

No nonvanishing, field-inverse, characteristic or integral-domain hypothesis is added. For $r=1$ the right-hand side is empty. The Library note Library/RecurrenceComposition/svinin2015discrete.md gives the verbatim source expressions and locator.

The frozen settling declaration is D5/S1/Recurrence/Algebraic/SvininDiscretePolynomialIdentity.result. It establishes the prerequisite that the source uses for subsequent relations between difference-equation families and first integrals in the same specialization. It proves the entire quantified identity rather than a finite list of degrees.

The source proves only $r\le3$. The literature screen recorded on #19287 inspected arXiv:2106.00539v2, 2307.05866, 2502.06908v1 (Remark 5.12), 2604.13115v1, 2606.24925 and 2605.29722, together with Hone–Kouloukas–Quispel (2018), Hone–Roberts–Vanhaecke (2024) and Flajolet (1980). Usmani (1994) was unread. No general proof was identified in that searched scope; completeness and priority are ASSUMED-UNVERIFIED.

Partition weighted monotone and strictly monotone tuples according to whether their first coordinate is zero. Explicit inverse correspondences give the left Pascal recurrences. Reversing both tuple coordinates and the input lattice gives the right recurrences. Let $Q^p_s(i)=S^p_s(i)$ for $p\le1$, and for $p\ge2$ put

$$Q^p_s(i)=S^p_s(i)-u(i+p-2)u(i+s)S^{p-2}_{s+2}(i).$$

The recurrences telescope to $\langle Q,T\rangle^r_s(i)=\langle Q,T\rangle^r_{s-1}(i+1)$ for $s\ge r$. At $s=r-1$, every term with positive $T$-degree vanishes. The remaining $Q^r_{r-1}(i)$ is a finite-window function invariant under translation for every input function. Cut off that input to zero beyond its observation window; a disjoint translated window then sees only zero. Locality and shift invariance make the boundary zero. Induction on $s-r+1$ and expansion of the quadratic correction yield Eq. (48). No division occurs.

A commutative ring, function $u$, degree $r\ge1$ and integers $s\ge r-1,i$ violating the displayed identity would contradict result. Noncommutative coefficients or other coprime $(h,n)$ lie outside its statement. The finite symbolic experiment below checks only its stated range and supplies no unbounded inference.

D5.S1.Recurrence.Algebraic.SvininDiscretePolynomialIdentity.result is the kernel-checked theorem of claim. The axiom closure of every public declaration is contained in {propext, Classical.choice, Quot.sound}. The module has no D5 imports; its prerequisites are pinned Mathlib.

Experiment: symbolic polynomial check. The command git -C /Users/auric/trureturing-experiments show dafb15a7b6763e3fe0f10914f8e2fb2b135b3c8d:docs/reports/svinin-2015-discrete-polynomial-identity-48/check.py > /tmp/op-svinin48-experiment.py && python3 /tmp/op-svinin48-experiment.py exits 0. The script SHA-256 is 4518365271cf5f20710ae33baffdb21dda8c9458caa449d3a4a47f5492fb1838. It verifies polynomial equality in independent $u_k$ for $r=1,\ldots,5$ and $s=r-1,\ldots,r+3$, at $i=0$: 25 cases, all true, RESULT PASS.

The escape audit for the sole public theorem result is unfinished: #19360. The dependent infinite-family registration lacks a complete typed source reconstruction, source-to-law bridge, whole-family variation and role sensitivity, and actual observational dependence. No Reg registration or declared_validated status is claimed.

What the settlement shows

1. Proved (kernel-checked). Telescoping makes the corrected convolution invariant under $(s,i)\mapsto(s-1,i+1)$. Evidence: private bracket_shift, instantiated by bracket_Q_zero, with the Pascal recurrences derived from mono_pascal and strict_pascal.
2. Proved (kernel-checked). At $s=r-1$, all positive-degree $T$ factors in the convolution vanish (T_vanish); the residual boundary is zero by finite-window locality and shift invariance (Q_boundary). These are consumed in bracket_Q_zero.
3. Proved (kernel-checked). The identity holds in every commutative ring, including zero divisors and zero input values, without division. Evidence: the complete telescope of public result.
4. Proved (kernel-checked). The locality component requires only Zero-valued input and output types, without ring structure. Evidence: private local_translation_invariant_zero, used by Q_boundary.
5. Source-level consequence; not separately formalized. The source's downstream difference-equation equivalences and first-integral statements are conditional on Conjecture 1. This theorem supplies that prerequisite for $(h,n)=(1,1)$. Those separate implications and their other hypotheses are not kernel-checked here; the evidence for their dependency is the source's argument.
6. Open. General coprime $(h,n)$, noncommutative coefficients and wider recurrence systems are outside this claim. Their literal source-to- recurrence correspondence and remaining proof obligations are unresolved.

The continuant/inverse-entry interpretation is a source-level explanatory route; no continuant, walk-bijection or top-matching-coefficient theorem is asserted as kernel-checked in this module. The symbolic range in Evidence is computed; it is distinct from the unbounded theorem in item 3.

Literature completeness and publication priority are ASSUMED-UNVERIFIED. The finite search does not establish the absence of a differently represented owner or an uninspected proof. The source's downstream equivalences and first integrals are not separately formalized. The escape audit remains unfinished under #19360. No claim is made for the extensions listed as open in Triage.