arxiv: 2605.11137 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: 2 theorem links

· Lean Theorem

The Alternating Compositions of Weighted Differential Operators Yield The Weights' Wronskian With Which Constant?

Arthemy V. Kiselev, Kian C. Shah

Pith reviewed 2026-05-13 02:13 UTC · model grok-4.3

classification 🧮 math.CO

keywords alternating compositiondifferential operatorsWronskian determinantlate-growing permutationsoperator coefficientscombinatorial constantsweighted differential operators

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The pith

Alternating compositions of 2p weighted p-order differential operators produce a p-order operator with coefficient equal to const(p) times the Wronskian of the weights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the alternating composition of N=2p operators, each consisting of a weight function times the p-th partial derivative with respect to x, yields another operator of exact order p. The leading coefficient of this resulting operator is const(p) multiplied by the Wronskian determinant of the N weight functions, where const(p) depends only on p. This holds with const(1)=1 corresponding to the Lie bracket, and the authors derive a formula for general p using signed sums over late-growing permutations to compute values like const(4)=586656. Readers interested in algebraic structures of differential operators or combinatorial identities would find this reduction to a classical determinant useful for simplifying higher-order calculations.

Core claim

The alternated composition of N=2p differential operators w_j(x) ∂_x^p of strict order p is again a differential operator of strict order p; its coefficient is the constant const(p), depending only on the arity N, times the Wronskian determinant of the originally taken coefficients w_1, …, w_N. The case p=1 fixes const(1)=1. When p=2, const(2)=2 is easy to find; the problem is to determine const(p) for p≥3, which the paper solves by expressing it as a sum with signs over late-growing permutations, yielding exact values up to p=6.

What carries the argument

The alternating composition operator applied to the family of weighted p-th order differential operators w_j ∂_x^p, which isolates the Wronskian in the leading term after cancellation of lower orders.

If this is right

For p=1 the construction recovers the Lie bracket of two vector fields with the constant equal to 1.
The constant const(p) is independent of the choice of weight functions and can be computed combinatorially.
The formula involves a signed sum over the much smaller set of late-growing permutations rather than all permutations.
Explicit computations give const(4) = 586656, const(5) ≈ 1.9 · 10^12, and const(6) ≈ 7.9 · 10^21.
The sequence of these constants for successive p appears to be new.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This identity may allow deriving Wronskian relations in contexts where operator compositions are natural, such as in evolution equations.
The emphasis on late-growing permutations suggests a possible link to sorting algorithms or permutation pattern avoidance in combinatorics.
One could extend the construction to operators with variable order p or to multiple variables to see if analogous determinants appear.
Testing the cancellation of lower-order terms with polynomial weights would directly verify the mechanism for small p.

Load-bearing premise

The composition in alternating order causes all coefficients of derivatives of order less than p to vanish identically.

What would settle it

Compute the full alternating composition explicitly for p=2 with four distinct weight functions and confirm that the result is precisely 2 times the Wronskian times the second derivative, with all first- and zeroth-order terms equal to zero.

read the original abstract

The alternated composition of $N=2p$ differential operators $ w_j(x)\,\partial_x^p$ of strict order $p$ on the line $\mathbb{R}\ni x$ is again a differential operator of strict order $p$; its coefficient is the constant $\mathrm{const}(p)$, depending only on the arity $N$, times the Wronskian determinant of the originally taken coefficients $w_1$, $\ldots$, $w_N$. The case $p=1$ of the Lie bracket for two vector fields fixes $\mathrm{const}(1)=1$. When $p=2$, finding $\mathrm{const}(2)=2$ is easy; we obtain $\mathrm{const}(3)=90$. The problem is to know $\mathrm{const}(p\geqslant 4)$. We express the formula of $\mathrm{const}(p)$ in terms of the sum with signs over the much smaller set of 'late-growing' permutations, thus reaching the exact values $c(p=4)= 586\,656$, $c(p=5)\approx 1.9\cdot 10^{12}$, and $c(p=6)\approx 7.9\cdot 10^{21}$; the positive integer sequence $\mathrm{const}(p)$ seems to be new.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They've derived a signed sum over late-growing permutations for the constant in the alternated operator composition and computed it explicitly up to p=6.

read the letter

The main point is that the authors reduce the constant in this alternated composition identity to a signed sum over late-growing permutations and compute its values for p up to 6. They start from the definitions of operator composition and the Wronskian, expand algebraically, and isolate the coefficient of the order-p term. For p=1 and 2 they check everything directly, get const(3)=90 by hand, then use the permutation sum to reach 586656 for p=4, about 1.9e12 for p=5, and 7.9e21 for p=6. That sequence does not appear in the literature they cite, so the explicit formula and numbers are new. The work is clean on the combinatorial side. The reduction avoids summing over all (2p)! terms, which is practical, and the values are obtained without any data fitting or post-selection. The potential issue is whether the lower-order terms really cancel for p greater than 3. The manuscript verifies the full operator identity, including vanishing of other coefficients, only up to p=3. For higher p they apply the signed sum directly to extract const(p), assuming the order remains exactly p. If the algebraic expansion justifies the cancellation in general, this is fine, but the write-up does not include an extra check or general argument for that step. This paper suits readers in algebraic combinatorics who care about explicit identities for differential operators. Someone looking for new sequences or permutation-based formulas will find it useful. It has enough concrete content and a clear derivation to warrant sending it out for peer review. I would recommend accepting it for review.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that the alternated composition of N=2p differential operators w_j(x) ∂_x^p (j=1 to 2p) is again a differential operator of strict order p, with leading coefficient equal to const(p) times the Wronskian determinant of the w_j. It verifies the full identity (including vanishing of lower-order terms) directly for p=1,2,3, obtaining const(1)=1, const(2)=2, const(3)=90, and for p=4,5,6 computes explicit values of const(p) (586656, ~1.9e12, ~7.9e21) via signed sums over late-growing permutations, asserting that the lower-order coefficients vanish identically.

Significance. If the central identity holds, the result supplies an explicit combinatorial link between alternated compositions of weighted differential operators and Wronskians, together with a practical formula for the constant that yields concrete integers without data fitting. The direct verifications for small p and the permutation-based expression for larger p are clear strengths that make the constants computable and falsifiable.

major comments (1)

[Abstract; computations for p=4,5,6] The assertion that the alternated composition is always a strict-order-p operator (i.e., that all coefficients of orders k ≠ p vanish) is verified by direct expansion only for p=1,2,3. For p≥4 the manuscript extracts const(p) from the signed permutation sum while assuming the lower-order cancellation persists, but supplies neither an explicit check for p=4 nor a general argument establishing the vanishing. This assumption is load-bearing for the claim that the result is 'again a differential operator of strict order p' for arbitrary p.

minor comments (2)

[Abstract] The abstract notes that the sequence const(p) 'seems to be new'; a brief check against the OEIS or a statement that no match was found would strengthen the novelty claim.
Notation for the alternated composition (the precise meaning of the signed sum over the (2p)! operators) could be made fully explicit in a displayed equation early in the text to aid readers who wish to reproduce the p=1,2,3 cases.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need for stronger evidence regarding the order of the resulting differential operator. We provide a detailed response to the major comment below.

read point-by-point responses

Referee: [Abstract; computations for p=4,5,6] The assertion that the alternated composition is always a strict-order-p operator (i.e., that all coefficients of orders k ≠ p vanish) is verified by direct expansion only for p=1,2,3. For p≥4 the manuscript extracts const(p) from the signed permutation sum while assuming the lower-order cancellation persists, but supplies neither an explicit check for p=4 nor a general argument establishing the vanishing. This assumption is load-bearing for the claim that the result is 'again a differential operator of strict order p' for arbitrary p.

Authors: We agree that the direct verification of vanishing lower-order terms is limited to p=1,2,3 in the current manuscript. To address this, we will include in the revised version an explicit check for p=4, obtained via symbolic computation of the full alternated composition, which confirms that all coefficients of order different from 4 vanish and that the leading coefficient is indeed const(4) times the Wronskian. This provides concrete evidence for the claim at p=4. However, we do not have a general argument that establishes the vanishing for arbitrary p; the combinatorial expression for const(p) is derived under the assumption that the result is of order exactly p, and proving the cancellation in general would require additional theoretical tools. We therefore revise the manuscript partially to include the p=4 verification and to qualify the general claim accordingly. revision: partial

standing simulated objections not resolved

General argument establishing the vanishing of lower-order terms for arbitrary p

Circularity Check

0 steps flagged

No circularity: derivation proceeds by direct algebraic expansion of operator compositions and combinatorial reduction for the leading coefficient.

full rationale

The paper begins from the explicit definition of the alternating sum of compositions of the operators w_j(x) ∂_x^p and the standard Wronskian determinant. For p=1,2,3 the identity (including vanishing of lower-order terms) is verified by direct expansion, yielding const(p) as a numerical factor. For p≥4 the leading coefficient is extracted via a signed sum over the reduced set of late-growing permutations; this is a computational shortcut that presupposes the general vanishing claim but does not redefine any quantity in terms of itself or rename a fitted parameter as a prediction. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing steps. The entire chain remains self-contained against the input definitions of composition and determinant.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the algebraic properties of differential operator composition (Leibniz rule and associativity) and the definition of the Wronskian determinant. No free parameters are introduced; the only axioms are standard facts from differential algebra and combinatorics.

axioms (2)

standard math The composition of differential operators obeys the Leibniz product rule and is associative.
Invoked implicitly when expanding the alternating composition of the weighted operators.
standard math The Wronskian determinant is the determinant of the matrix formed by the functions and their successive derivatives.
Used as the target object whose multiple appears as the leading coefficient.

pith-pipeline@v0.9.0 · 5545 in / 1499 out tokens · 43441 ms · 2026-05-13T02:13:50.069433+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat.induction / embed_injective unclear
The alternated composition of N=2p differential operators w_j(x) ∂_x^p ... is again a differential operator of strict order p whose coefficient is const(p) times the Wronskian determinant
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We express the formula of const(p) in terms of the sum with signs over the much smaller set of 'late-growing' permutations

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

On associative schlessinger–stasheff algebras and wronskian deter- minants,

A. V. Kiselev, “On associative schlessinger–stasheff algebras and wronskian deter- minants,”Fundamentalnaya i Prikladnaya Matematika, vol. 11, no. 1, pp. 159–180, 2005, In Russian; English translation in Journal of Mathematical Sciences, 2007, 141(1), 1016–1030. arXiv:math/0410185 [math.RA]

work page arXiv 2005
[2]

ROCK: The new Quick-EXAFS beamline at SOLEIL,

A. V. Kiselev, “Wronskians as n-ary brackets in finite-dimensional analogues of sl(2),” Journal of Physics: Conference Series, vol. 3152, no. 1, p. 012 044, Dec. 2025,issn: 1742-6596.doi:10.1088/1742- 6596/3152/1/012044[Online]. Available:http: //dx.doi.org/10.1088/1742-6596/3152/1/012044

work page doi:10.1088/1742- 2025
[3]

Baran and M

H. Baran and M. Marvan,Jets: A software for differential calculus on jet spaces and diffieties,http://jets.math.slu.cz

work page
[4]

Sufficient set of integrability conditions of an orthonomic system,

M. Marvan, “Sufficient set of integrability conditions of an orthonomic system,” Foundations of Computational Mathematics, vol. 9, no. 6, pp. 651–674, Jan. 2009, issn: 1615-3383.doi:10 . 1007 / s10208 - 008 - 9039 - 8[Online]. Available:http : //dx.doi.org/10.1007/s10208-008-9039-8

work page doi:10.1007/s10208-008-9039-8 2009