arxiv: 2605.09107 · v1 · submitted 2026-05-09 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

Merge-position invariance in quadratically enriched tropical floor diagrams

Yanis Hedjem

Pith reviewed 2026-05-12 01:50 UTC · model grok-4.3

classification 🧮 math.AG

keywords tropical floor diagramsmerge-position invarianceGrothendieck-Witt ringsrational curvesdel Pezzo surfacesquadratic formsPfister elementswall-crossing

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The pith

The Grothendieck-Witt valued floor-diagram count of rational curves is independent of merge positions for adjacent point conditions.

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

The paper proves merge-position invariance for the quadratically enriched tropical floor-diagram formula counting rational curves in smooth toric del Pezzo surfaces. It establishes a wall-crossing factorisation: the difference between counts for any two merge configurations equals a coefficient C times the product over j of (⟨dj⟩ minus ⟨1⟩). Real broccoli invariance then reduces the possible obstruction to a multiple of the virtual Pfister element ⟨⟨2, d1, …, ds⟩⟩. This yields a complete tropical proof of invariance over admissible fields where 2 is a square, and otherwise reduces the question to one explicit mod-2 congruence on the residual coefficient, settled by a single Laurent-series specialisation that uses the existing tropical correspondence and algebraic invariance theorem.

Core claim

For any two merge configurations the difference ΔN equals C times the product from j=1 to s of (⟨dj⟩ − ⟨1⟩), with C admitting a fixed universal lift; the obstruction reduces via real broccoli invariance to a multiple of the virtual Pfister element ⟨⟨2, d1, …, ds⟩⟩. This vanishes over fields in which 2 is a square, and over general admissible fields the same analysis reduces the invariance statement to one mod-2 congruence for the residual coefficient that is verified by a single Laurent-series specialisation.

What carries the argument

The wall-crossing factorisation ΔN = C ∏ (⟨dj⟩ − ⟨1⟩) of the floor formula, combined with reduction of the obstruction to the virtual Pfister form ⟨⟨2, d1, …, ds⟩⟩ via real broccoli invariance.

If this is right

Merge-position invariance holds completely over every admissible field in which 2 is a square.
Over a general admissible field the invariance question reduces to verifying one explicit mod-2 congruence for the residual coefficient.
The congruence is settled by a single Laurent-series specialisation that invokes only the existing tropical correspondence and algebraic invariance theorem.
Abstract invariance from the tropical correspondence and algebraic theorems becomes manifest inside the floor-diagram expression itself.

Where Pith is reading between the lines

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

The same factorisation and reduction may apply to other quadratic enrichments of tropical curve counts.
Invariance statements in enriched enumerative geometry are largely controlled by the arithmetic of quadratic forms and Pfister elements.
Laurent-series specialisations of this type could resolve similar choice-of-position questions in other floor-diagram or tropical formulas.

Load-bearing premise

That the difference after wall-crossing reduces exactly to a multiple of the virtual Pfister element ⟨⟨2, d1, …, ds⟩⟩ using real broccoli invariance, and that the remaining mod-2 congruence can be checked by one Laurent-series specialisation without introducing new choices.

What would settle it

A concrete counterexample in which the mod-2 congruence fails for the residual coefficient after the Laurent-series specialisation, or in which the floor-diagram counts for two merge positions differ by more than the predicted multiple of the Pfister element over a field where 2 is not a square.

Figures

Figures reproduced from arXiv: 2605.09107 by Yanis Hedjem.

**Figure 2.** Figure 2: Examples of twin trees with their multiplicities, schematically adapted from [2, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

read the original abstract

Jaramillo Puentes et al. give a Grothendieck-Witt valued floor-diagram formula for rational curves in smooth toric del Pezzo surfaces with simple and quadratic double point conditions. We study its dependence on the choice of merge positions, namely on which adjacent pairs of point conditions are merged. Although independence of these choices follows abstractly from the tropical correspondence and algebraic invariance, it is not manifest in the floor-diagram expression. We prove a wall-crossing factorisation for the floor formula: for any two merge configurations, the difference is of the form $\Delta N = C \prod_{j=1} ^s (\langle d_j\rangle-\langle 1\rangle)$. The coefficient $C$ admits a fixed universal lift. Using real broccoli invariance, the possible obstruction is reduced to a multiple of the virtual Pfister element $\langle\langle 2,d_1, \ldots,d_s\rangle\rangle$. This gives a complete tropical proof of merge- position invariance over every admissible field in which $2$ is a square. Over a general admissible field, the same tropical analysis reduces the problem to one explicit mod-$2$ congruence for the residual coefficient; this congruence is verified by a single Laurent-series specialisation, using the tropical correspondence of Jaramillo Puentes et al. and the algebraic invariance theorem of Kass-Levine-Solomon-Wickelgren.

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 gives an explicit tropical wall-crossing factorisation that makes merge-position invariance manifest in the floor-diagram count, reducing the obstruction to a Pfister form or a verifiable mod-2 congruence.

read the letter

The punchline is that this work supplies a concrete tropical mechanism for proving that the Grothendieck-Witt valued count of rational curves is independent of which pairs of point conditions you merge in the floor diagram setup. It does this over admissible fields, with a clean split depending on whether 2 is a square. What is new is the wall-crossing factorisation: the difference between two merge configurations factors as C times the product over j of (⟨d_j⟩ minus ⟨1⟩), where C has a fixed universal lift. They then apply real broccoli invariance to reduce any remaining obstruction to a multiple of the virtual Pfister element ⟨⟨2, d1, ..., ds⟩⟩. When 2 is a square this vanishes in the Grothendieck-Witt ring, giving invariance right away. For general fields the problem reduces to verifying one explicit mod-2 congruence on the residual coefficient, which they do via a single Laurent-series specialisation that re-uses the tropical correspondence from Jaramillo Puentes et al. and the algebraic invariance from Kass-Levine-Solomon-Wickelgren. The paper does well by making the independence manifest in the floor-diagram expression itself, rather than leaving it as an abstract consequence of the correspondence theorem. The argument keeps circularity low by introducing the explicit factorisation and checking the rest externally but in a controlled way. No new choices seem to be introduced in the reduction steps. A minor soft spot is that for fields where 2 is not a square, the proof still depends on that one specialisation argument and the prior results, so the tropical analysis does not stand completely alone for the residual term. But this is proportionate to the claim, and the stress-test shows no hidden circularity or choice issues. This paper is for researchers already familiar with tropical enumerative geometry and quadratic enrichments over del Pezzo surfaces. Someone looking to make computations or formulas more explicit in this area will get direct value from the factorisation. It shows clear thinking and honest use of the literature, so it deserves a serious referee. I would recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The paper claims to prove merge-position invariance for the Grothendieck-Witt valued counts given by quadratically enriched tropical floor diagrams of rational curves in smooth toric del Pezzo surfaces. It establishes a wall-crossing factorisation showing that the difference between counts for different merge configurations is of the form C times the product of (⟨d_j⟩ - ⟨1⟩) for j=1 to s, with C having a universal lift. By invoking real broccoli invariance, the obstruction is reduced to a multiple of the virtual Pfister element ⟨⟨2, d_1, ..., d_s⟩⟩, proving invariance when 2 is a square in the base field. For general admissible fields, the remaining mod-2 congruence is verified using a single Laurent-series specialisation that relies on the tropical correspondence theorem of Jaramillo Puentes et al. and the algebraic invariance theorem of Kass-Levine-Solomon-Wickelgren.

Significance. If the details of the factorisation and reduction hold, the result is significant because it renders merge-position independence manifest within the tropical floor-diagram formula itself, rather than relying solely on abstract correspondence and algebraic invariance theorems. The explicit wall-crossing expression and the reduction to a single verifiable mod-2 congruence (via real broccoli invariance and one Laurent specialisation) provide a concrete, choice-independent mechanism for the enriched counts. The manuscript correctly credits and re-uses prior results without circularity, yielding a complete tropical proof when 2 is a square and a minimal-check argument otherwise. This strengthens the foundations of quadratically enriched tropical enumerative geometry.

minor comments (3)

The notation for Grothendieck-Witt ring elements (e.g., ⟨d_j⟩ and the virtual Pfister form ⟨⟨2,d1,…,ds⟩⟩) is used throughout; a short reminder of the ring operations and the meaning of 'virtual' in the introduction would improve accessibility for readers outside the immediate subfield.
The claim that C 'admits a fixed universal lift' is central to the wall-crossing factorisation; stating explicitly in which ring or module this lift lives (and confirming it is independent of the choice of toric surface) would make the subsequent reduction steps easier to follow.
In the specialisation argument, the single Laurent-series check is said to re-use the tropical correspondence without new choices; adding a one-sentence pointer to the precise statement in Jaramillo Puentes et al. that is being invoked would eliminate any potential ambiguity about parameter dependence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recognizing the significance of establishing merge-position invariance directly within the tropical floor-diagram formula. The report correctly describes our wall-crossing factorisation, the reduction via real broccoli invariance to a multiple of the virtual Pfister element, the complete proof when 2 is a square, and the minimal mod-2 check otherwise. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with external verification

full rationale

The paper derives an explicit wall-crossing factorization ΔN = C ∏(⟨d_j⟩−⟨1⟩) for the floor-diagram count and reduces any obstruction to a multiple of the virtual Pfister form via real broccoli invariance. When 2 is a square this vanishes directly. For the general case the residual mod-2 coefficient is checked by one Laurent-series specialization that invokes only the prior tropical correspondence of Jaramillo Puentes et al. and the algebraic invariance theorem of Kass–Levine–Solomon–Wickelgren. These are independent external results, not self-citations or internal fits. No step equates a claimed prediction to its own input by definition, renames a known pattern, or imports uniqueness from the authors’ prior work. The central tropical proof therefore adds new explicit content rather than reducing to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the tropical correspondence theorem of Jaramillo Puentes et al. and the algebraic invariance theorem of Kass-Levine-Solomon-Wickelgren; these are treated as external inputs. No new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption The tropical correspondence theorem and algebraic invariance results hold for the relevant toric del Pezzo surfaces and point conditions.
Invoked to justify the Laurent-series specialisation that verifies the mod-2 congruence.
domain assumption Real broccoli invariance applies to the floor diagrams under consideration.
Used to reduce the obstruction to a multiple of the virtual Pfister element.

pith-pipeline@v0.9.0 · 5571 in / 1566 out tokens · 57385 ms · 2026-05-12T01:50:05.792308+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We prove a wall-crossing factorisation for the floor formula: for any two merge configurations, the difference is of the form ΔN = C ∏_{j=1}^s (⟨dj⟩−⟨1⟩). ... reduced to a multiple of the virtual Pfister element ⟨⟨2,d1,…,ds⟩⟩.
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear
Using real broccoli invariance, the possible obstruction is reduced to a multiple of the virtual Pfister element ⟨⟨2,d1,…,ds⟩⟩. This gives a complete tropical proof ... over every admissible field in which 2 is a square.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

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work page arXiv 2026
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work page arXiv 2020
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work page 2005