arxiv: 2604.13238 · v1 · submitted 2026-04-14 · 🧮 math.CO

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A quadratic form generalization of rational dinv

Yifeng Huang

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Pith reviewed 2026-05-10 14:21 UTC · model grok-4.3

classification 🧮 math.CO

keywords quadratic formnumerical semigroupdinv statisticgap posetYoung diagramcross-dinvpositive definiteness

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The pith

A quadratic form on the gap poset recovers the dinv statistic for Young subdiagrams and defines nonnegative cross-dinv.

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

The paper defines a quadratic form Q on the space of functions on the gap poset G of a numerical semigroup generated by two coprime integers. It proves that Q applied to the indicator function of an upward closed subset D equals the Gorsky-Mazin dinv statistic of the corresponding Young subdiagram. The associated bilinear form on pairs of subdiagrams equals a new cross-dinv statistic shown to be nonnegative. Using this, the paper establishes that Q satisfies an inequality Q(n) at least one over the size of G times the square of the infinity norm of n, for any decreasing real-valued function n on G, which demonstrates an effective form of positive definiteness on that cone.

Core claim

We introduce the quadratic form Q on functions on G. Its evaluation on the indicator of an upward closed subset recovers dinv exactly. The symmetric bilinear form associated to Q equals the cross-dinv on pairs of subdiagrams and is nonnegative. This implies the inequality showing effective positive definiteness for decreasing functions on G.

What carries the argument

The quadratic form Q on the vector space of real-valued functions on the gap poset G of the numerical semigroup generated by two coprime integers a and b.

Load-bearing premise

The quadratic form Q is defined in a particular way using the poset structure so that it matches the dinv values on the indicator functions of upward closed subsets.

What would settle it

Finding an upward closed subset D in some gap poset G for which the computed value of Q on the indicator of D does not equal the Gorsky-Mazin dinv statistic of D would falsify the main claim.

Figures

Figures reproduced from arXiv: 2604.13238 by Yifeng Huang.

**Figure 1.** Figure 1: For (a, b) = (4, 5), the projection of 1 to other rows are 2 and 3. The projection of 3 to the middle row (row 2) is 6. The projection of 7 to the middle row is 10, which lies outside G. 3. Proof of Theorem 1.1 Though Theorem 1.2 implies the main assertion of Theorem 1.1, we give a separate proof of Theorem 1.1 here, as it motivates the proof method of Theorem 1.2 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: For (a, b) = (4, 7), if D has row lengths (4, 1, 1), then U ∩ G = {1, 6, 2}. The arrow (5, 6) is the only blue arrow and (3, 6) is the only red arrow. Hence Q(D) = |D| − 1 − 1 = 4 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: For (a, b) = (5, 7), if D is the maximum hook, the blue cells are 13, 18 (shaded blue) and the red cells are 9, 16 (shaded red). The red arrow (i, j) = (9, 11) maps to 9 via a parallel shift up to (i ′ , j′ ) = (2, 4). Blue cells: Define Φb(i, j) for (i, j) ∈ Nb to be the cell c at the intersection of the row of i and the column of j. Because j is to the northwest of i, c is due west of i and due south of … view at source ↗

read the original abstract

We introduce a quadratic form $Q$ on the space of functions on the gap poset $G$ of the numerical semigroup $\langle a,b\rangle$. We prove combinatorially that when evaluated on the indicator function of an upward closed subset $D$, this quadratic form precisely recovers the Gorsky--Mazin $\mathtt{dinv}$ statistic of $D$, viewed as a Young subdiagram of $G$. Furthermore, we prove Theorem~1.2 that when evaluated on a pair of subdiagrams of $G$, the symmetric bilinear form associated with $Q$ is equal to a novel cross-$\mathtt{dinv}$ statistic, which is nonnegative. Combining these, we prove the inequality \[ Q(\mathbf{n})\geq \dfrac{1}{|G|}\,\|\mathbf{n}\|_\infty^2\] if $\mathbf{n}$ is a real-valued decreasing function on $G$, showing an effective positive definiteness of $Q$ on the corresponding cone. Theorem~1.2, the main engine of the paper, was autoformalized in Lean/Mathlib by AxiomProver.

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

The paper defines a quadratic form Q on the gap poset that recovers dinv on indicators and pairs with a Lean-verified nonnegative cross-dinv bilinear form.

read the letter

The main thing here is a new quadratic form Q on functions over the gap poset G of a numerical semigroup generated by coprime a and b. It matches the Gorsky-Mazin dinv statistic exactly on indicators of upward-closed subsets, and the associated symmetric bilinear form equals a new cross-dinv statistic that is nonnegative. Theorem 1.2, which encodes that nonnegativity, was autoformalized in Lean/Mathlib. From those two pieces the paper derives an inequality showing Q is effectively positive definite on real decreasing functions on G. That is the core output. The combinatorial recovery uses the standard poset structure on the gaps and does not appear to loop back on itself. The Lean formalization supplies concrete, machine-checked support for the nonnegativity engine, which is a genuine plus in this area. The definition of Q is chosen to fit the dinv case, but that is explicit and the cross-dinv extension is new. No obvious circularity or missing cases in the central claims. One softer point is the step from indicators to general real-valued decreasing functions; the derivation is claimed to follow directly, but the details would need a close look in the full text to confirm there are no hidden approximation steps. The work stays tightly inside rational Catalan combinatorics and does not claim broader reach. Readers already working on dinv statistics, positivity questions for Young diagrams, or combinatorial interpretations of quadratic forms will find the cross-dinv and the formalization useful. Others outside that niche will not. I would bring this to a reading group focused on algebraic combinatorics or formal methods in combinatorics. I would not cite it in the next year unless I were actively using dinv or gap posets. It deserves peer review: the new statistic plus the Lean check give it enough substance for referees to evaluate the combinatorial steps and the inequality extension.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a quadratic form Q on the space of functions on the gap poset G of the numerical semigroup ⟨a,b⟩ with a,b coprime. It claims a combinatorial proof that Q evaluated on the indicator function of an upward-closed subset D recovers the Gorsky-Mazin dinv statistic of D viewed as a Young subdiagram of G. Theorem 1.2 asserts that the associated symmetric bilinear form equals a novel cross-dinv statistic which is nonnegative (autoformalized in Lean/Mathlib), and these facts are combined to prove the inequality Q(n) ≥ (1/|G|) ‖n‖_∞² for real-valued decreasing functions n on G, establishing effective positive definiteness on the cone of decreasing functions.

Significance. If the combinatorial identifications hold, the work supplies a quadratic-form generalization of rational dinv that recovers the statistic on indicators and extends via nonnegative cross-dinv to an inequality on real decreasing functions. The Lean autoformalization of Theorem 1.2 is a clear strength, furnishing machine-checked support for the nonnegativity engine that underpins the main inequality. This supplies a rigorous, parameter-free bridge between poset combinatorics, quadratic forms, and dinv statistics.

minor comments (2)

[§2] §2: the explicit coordinate formula for the quadratic form Q on the gap poset would benefit from an accompanying small example (e.g., a=3, b=5) showing recovery of dinv on a concrete upward-closed set.
[Introduction] Introduction and §3: a short diagram of the gap poset G together with the partial order would improve readability for readers outside numerical-semigroup combinatorics.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, which accurately summarizes the main contributions of the manuscript, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper explicitly defines the quadratic form Q on functions over the gap poset G of the numerical semigroup generated by coprime a and b. It then supplies a combinatorial proof that Q applied to the indicator of an upward-closed subset D recovers the Gorsky-Mazin dinv statistic, relying on the standard poset structure rather than defining Q in terms of dinv. Theorem 1.2 identifies the associated symmetric bilinear form with a novel combinatorially nonnegative cross-dinv statistic on pairs of subdiagrams; this identification is machine-checked via Lean/Mathlib formalization by AxiomProver, providing independent verification. The inequality Q(n) ≥ (1/|G|) ||n||_∞² for real decreasing n is derived directly from the two preceding facts. No step reduces by construction to its own inputs, no parameters are fitted to data, and no load-bearing claim rests on self-citation chains. The derivation is self-contained against external combinatorial and formal benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

No numerical parameters are fitted to data. The construction relies on the standard poset structure of the gap set of a two-generated numerical semigroup and the correspondence between upward-closed subsets and Young diagrams.

axioms (1)

domain assumption The gap poset G of the numerical semigroup generated by two coprime integers admits a natural partial order whose upward-closed subsets correspond to Young subdiagrams.
Invoked to interpret indicator functions of subsets as Young diagrams for the dinv recovery statement.

invented entities (1)

Quadratic form Q no independent evidence
purpose: To generalize the dinv statistic to all real-valued functions on G and to produce a cross-dinv bilinear form
Newly defined object whose evaluation on indicators recovers dinv; no independent existence proof outside the paper is given.

pith-pipeline@v0.9.0 · 5481 in / 1453 out tokens · 25185 ms · 2026-05-10T14:21:40.806108+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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