arxiv: 2512.21606 · v3 · submitted 2025-12-25 · ✦ hep-th · math-ph· math.MP

Recognition: 2 theorem links

· Lean Theorem

Shell formulas for instantons and gauge origami

Jiaqun Jiang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:39 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP

keywords shell formulainstantonsgauge origamipartition functionsYoung diagramssuper Yang-MillsDonaldson-Thomas invariants

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The pith

A shell formula unifies closed-form expressions and recursion relations for instanton partition functions whose poles are classified by Young diagrams of arbitrary dimension.

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

The paper introduces the shell formula to handle partition functions organized by multi-dimensional Young diagrams. This single framework produces explicit expressions and recursion relations for 5d pure super Yang-Mills instantons with classical gauge groups and for gauge origami systems such as the magnificent four, tetrahedron instantons, and spiked instantons. It also covers Donaldson-Thomas invariants in C^3 and C^4. A sympathetic reader cares because the approach replaces separate derivations for each theory with one uniform method based on pole classification.

Core claim

We introduce the shell formula-a framework that unifies the description of partition functions whose pole structures are classified by Young diagrams of arbitrary dimension. The formalism yields explicit closed-form expressions and recursion relations for a wide range of physical systems, including instanton partition functions of 5d pure super Yang-Mills theory with classical gauge groups, as well as gauge origami configurations such as the magnificent four, tetrahedron instantons, spiked instantons, and Donaldson-Thomas invariants in C^3 and C^4.

What carries the argument

The shell formula, a unifying expression generator that acts on pole structures classified by Young diagrams of arbitrary dimension.

If this is right

Closed-form expressions appear for instanton partition functions of 5d pure super Yang-Mills with all classical gauge groups.
Recursion relations follow for the magnificent four and tetrahedron instanton configurations.
Similar closed forms and recursions hold for spiked instantons and Donaldson-Thomas invariants in C^3 and C^4.
The same machinery applies whenever a partition function's poles are classified by higher-dimensional Young diagrams.

Where Pith is reading between the lines

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

If the classification extends to other supersymmetric theories, the formula could generate expressions for additional gauge groups or dimensions without new case analysis.
The combinatorial structure might link directly to known identities in representation theory or algebraic geometry for counting invariants.
Numerical checks at low instanton numbers for a classical group would provide an immediate test of the recursions.

Load-bearing premise

The pole structures of the partition functions admit a uniform classification by Young diagrams of arbitrary dimension that permits one formula to produce all the listed expressions and recursions.

What would settle it

An explicit mismatch between the shell formula output and an independently computed instanton partition function for SU(2) 5d SYM at a fixed instanton number, obtained by other methods such as localization or direct integration.

read the original abstract

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 shell formula unifies several instanton and origami partition functions under higher-dimensional Young diagrams, but the single-formula claim needs explicit derivations to confirm it avoids case-by-case tweaks.

read the letter

The paper's main contribution is the shell formula, which treats pole structures of certain partition functions as classified by Young diagrams of arbitrary dimension and claims to deliver closed-form expressions plus recursions for 5d pure SYM instantons (classical groups) and several gauge origami setups including the magnificent four, tetrahedron and spiked instantons, plus DT invariants in C^3 and C^4. That framing is new relative to the usual 2d diagrams for 4d/5d instantons and 3d diagrams for higher origami. The work does a solid job collecting these examples and stating that one construction covers them all with explicit results and recursion relations. Credit for attempting a common language across these calculations. The soft spot is whether the unification is truly single-formula. Standard treatments adjust equivariant weights, dimension shifts, and residue contours separately for each system; if the shell formula absorbs those differences only by redefining the shell per case, the claimed unification is weaker than advertised. The abstract alone does not show the formula or the verification steps, so it is impossible to check whether the closed forms reproduce known results without extra data or fitting. This paper is for people already working on supersymmetric partition functions and enumerative geometry who want to see if the recursions match their existing calculations. A reader with that background can test the claims directly. It deserves peer review so referees can examine the derivations and check reproducibility against known special cases.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the shell formula as a unifying framework for partition functions whose pole structures are classified by Young diagrams of arbitrary dimension. It derives explicit closed-form expressions and recursion relations for the instanton partition functions of 5d pure super Yang-Mills theory with classical gauge groups, as well as for gauge origami configurations including the magnificent four, tetrahedron instantons, spiked instantons, and Donaldson-Thomas invariants in C^3 and C^4.

Significance. If the derivations hold, the shell formula would provide a valuable unifying tool for computing and relating instanton and DT partition functions across dimensions, potentially simplifying calculations in supersymmetric gauge theories and algebraic geometry by replacing case-by-case residue computations with a single recursive structure.

major comments (2)

[Abstract] Abstract: the central unification claim requires that a single shell formula reproduces known results for both 5d SYM (standard 2d Young diagrams) and higher origami (plane partitions or 3d diagrams) via uniform pole classification; the text must explicitly show how equivariant weights, dimension shifts, and residue contours are encoded without system-specific redefinitions of the shell or normalizations.
[Main derivation] The recursion relations and closed forms for the 5d pure SYM cases with classical groups need direct verification against established literature expressions (e.g., via explicit low-rank examples) to confirm they are parameter-free and not implicitly fitted.

minor comments (1)

[Introduction] Clarify the precise definition of the shell formula with an introductory example computation for a low-dimensional case to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, agreeing where revisions will strengthen the presentation of the unifying shell formula.

read point-by-point responses

Referee: [Abstract] Abstract: the central unification claim requires that a single shell formula reproduces known results for both 5d SYM (standard 2d Young diagrams) and higher origami (plane partitions or 3d diagrams) via uniform pole classification; the text must explicitly show how equivariant weights, dimension shifts, and residue contours are encoded without system-specific redefinitions of the shell or normalizations.

Authors: We agree that explicit demonstration of uniformity strengthens the central claim. The shell formula is defined once for Young diagrams of arbitrary dimension d, with equivariant weights, dimension shifts, and residue contours encoded uniformly through the general pole structure and the single parameter d (d=2 for standard 2d diagrams in 5d SYM; d=3 for plane partitions in origami). In the revised manuscript we will add a dedicated paragraph and explicit mapping table in the introduction, showing the encoding for 5d SYM versus the magnificent four without any system-specific redefinitions of the shell or normalizations. revision: yes
Referee: [Main derivation] The recursion relations and closed forms for the 5d pure SYM cases with classical groups need direct verification against established literature expressions (e.g., via explicit low-rank examples) to confirm they are parameter-free and not implicitly fitted.

Authors: The recursions and closed forms follow directly from specializing the general shell formula to 2d diagrams and classical groups, with no free parameters introduced. To provide the requested direct verification we will add, in the revised version, explicit low-rank computations (SU(2) and SO(3)) together with side-by-side numerical and symbolic comparison to the known expressions in the 5d instanton literature, confirming exact agreement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new shell formula framework introduced as independent unification

full rationale

The paper defines and introduces the shell formula as a new framework for unifying partition functions classified by Young diagrams of arbitrary dimension. It then applies this to derive closed-form expressions and recursions for known systems (5d SYM instantons, gauge origami configurations). No equations or steps are exhibited that reduce the claimed predictions or central results to fitted inputs, self-citations, or prior definitions by construction. The derivation chain is self-contained as a novel formalism rather than a renaming or self-referential fit of existing results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the assumption that pole structures admit a uniform classification by higher-dimensional Young diagrams; no free parameters or invented entities are visible in the abstract, but the shell formula itself functions as the key new construct.

axioms (1)

domain assumption Partition functions of the listed systems have pole structures classifiable by Young diagrams of arbitrary dimension.
Invoked to justify the applicability of the shell formula across all mentioned cases.

invented entities (1)

shell formula no independent evidence
purpose: Unifying framework for closed-form expressions and recursions
New construct introduced to organize the partition functions

pith-pipeline@v0.9.0 · 5369 in / 1257 out tokens · 32369 ms · 2026-05-16T19:39:14.733838+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

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.4 (J-factor). J(x | Y_A) ≡ ∏_{y in S(Y_A)} sh(x - X_A(y))^{Q_{Y_A}(y)} with shell S(Y) = (Y + B_d) ∖ Y and charge Q_Y(x) = sum_{b in B_d, x-b in Y} (-1)^|b|.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The shell formula yields explicit closed-form expressions ... for instanton partition functions of 5d pure super Yang-Mills ... magnificent four, tetrahedron instantons, spiked instantons, and Donaldson-Thomas invariants.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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