pith. sign in

arxiv: 2604.28072 · v1 · submitted 2026-04-30 · 🧮 math.NT · math.AG

A torsion-intersection proof of perfect-cuboid nonexistence on 1,072 explicit master-tuple fibers

Ren\'e Peschmann This is my paper

Pith reviewed 2026-05-07 05:52 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords perfect cuboidEuler bricktorsion intersectionelliptic quotientgenus-3 curverank zeromaster-tuple fiberDiophantine equation
0
0 comments X

The pith

A torsion-intersection argument on elliptic quotients establishes the perfect-cuboid conjecture unconditionally on 1,072 explicit master-tuple fibers.

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

The paper proves that no non-degenerate perfect cuboids exist on 1,072 specific master-tuple fibers by excluding all rational (a,b) specializations. It begins with a structural theorem classifying every primitive Euler brick as arising from the standard (a,b,m,n) parametrization up to scaling. On each fiber it then applies a torsion-intersection argument to the elliptic quotients of an associated genus-3 curve: when the rank-zero hypothesis and the required torsion condition hold for one quotient, the group of rational points has order exactly eight and all points are degenerate. These rank-zero conditions are verified for every fiber with max(m,n) at most 100 by combining 2-descent computations with exact evaluation of the L-function at 1, certified via the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound. This rules out an infinite family of candidate solutions for each of the 1,072 fibers.

Core claim

Every primitive Euler-brick arises from the standard (a,b,m,n)-parametrisation up to scaling. On each of the 1,072 master-tuple fibers the torsion-intersection argument applied to the elliptic quotients E_A' and E_A'' forces |H_{m,n}(Q)| = 8 whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of the quotients, with all eight points corresponding to degenerate bricks. The rank-zero hypothesis is certified for each such fiber by PARI's ellrank or by Sage's modular-symbol evaluation of L(E,1)/Omega_E together with Kolyvagin's theorem and Edixhoven's Manin-constant bound.

What carries the argument

The torsion-intersection argument on the elliptic quotients E_A' and E_A'' of the genus-3 curve C_A : w^2 = lambda^8 + A lambda^4 + 1, which forces the rational-point group H_{m,n}(Q) to consist solely of the eight torsion points when rank zero and the torsion condition hold.

If this is right

  • All rational (a,b)-specializations on these 1,072 fibers produce only degenerate bricks.
  • Conjecture B holds unconditionally for every rational point on each of these fibers.
  • The standard (a,b,m,n) parametrization accounts for all primitive Euler bricks.
  • An explicit lift count refines the naive torsion bound when the torsion subgroup is larger than the base case.

Where Pith is reading between the lines

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

  • Extending the same verification procedure to fibers with larger max(m,n) would cover additional infinite families of candidates.
  • The method could be combined with search algorithms that only need to examine the remaining unruled-out fibers.
  • If the full set of fibers can eventually be covered, the perfect-cuboid problem would be reduced to a finite check.
  • The certified rank-zero technique may apply directly to other Diophantine problems whose solution sets are cut out by similar genus-3 curves.

Load-bearing premise

The rank-zero hypothesis and the appropriate torsion condition must hold for at least one of the two elliptic quotients on each of the 1,072 fibers.

What would settle it

A single non-degenerate perfect cuboid whose (a,b,m,n) parameters lie on one of the 1,072 listed fibers would falsify the claim.

read the original abstract

Building on the genus-3 reduction $C_A : w^2 = \lambda^8 + A \lambda^4 + 1$ established in our companion paper (arXiv:2604.09328), we give an unconditional proof of the perfect-cuboid conjecture ("Conjecture B") on $1{,}072$ explicit master-tuple fibers, excluding all rational $(a,b)$-specialisations on each such fiber. Our three main contributions are: (i) a structural classification theorem showing that every primitive Euler-brick arises from the standard $(a,b,m,n)$-parametrisation up to scaling; (ii) a torsion-intersection argument applied to the elliptic quotients $E_A'$ and $E_A''$: whenever the rank-zero hypothesis and the appropriate torsion condition hold for one of them, $|H_{m,n}(\mathbb{Q})| = 8$ is forced, with the eight points all corresponding to degenerate bricks; (iii) two complementary techniques to verify the rank-zero hypothesis algorithmically -- PARI's ellrank (2-descent) and, where this is ambiguous, Sage's exact rational evaluation of $L(E,1)/\Omega_E$ via modular symbols, which combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's bound on the Manin constant for semistable curves yields an unconditional rank-zero certificate -- together with an explicit lift count refining the naive torsion-intersection bound when the torsion is larger than the leading case. We exhibit $1{,}072$ such fibers with $\max(m,n) \le 100$ on which Conjecture B is thereby established unconditionally.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript claims an unconditional proof of the perfect-cuboid conjecture (no non-degenerate primitive Euler bricks) on 1,072 explicit master-tuple fibers with max(m,n) ≤ 100. It proceeds by establishing a structural classification that every primitive Euler brick arises (up to scaling) from the standard (a,b,m,n)-parametrization, reducing via the genus-3 curve C_A : w² = λ⁸ + A λ⁴ + 1 (from the companion paper) to a pair of elliptic quotients E_A' and E_A'', and applying a torsion-intersection argument: whenever one quotient has rank zero and satisfies the stated torsion condition, |H_{m,n}(Q)| = 8 with all points degenerate. Rank zero is certified either by 2-descent or by exact computation of L(E,1)/Ω_E via modular symbols, combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's theorem on the Manin constant (for semistable curves).

Significance. If the central claims hold, the work supplies rigorous non-existence results for a concrete, large collection of parametrized families in the perfect-cuboid problem. The explicit enumeration of the 1,072 fibers together with the use of standard, unconditional certificates (2-descent outputs and L-value evaluations grounded in Kolyvagin and Edixhoven) constitutes a reproducible contribution that can serve as a template for further fibers. The algebraic torsion-intersection step, once the rank and torsion data are granted, is clean and falsifiable.

major comments (2)
  1. [Rank-zero certificates section] Rank-zero certificates section: Edixhoven's theorem on the Manin constant is invoked only for semistable curves. The manuscript does not explicitly verify or state that every E_A' or E_A'' arising on the 1,072 fibers is semistable in the cases where the modular-symbol method supplies the rank-zero certificate. This verification is load-bearing for the unconditional character of those certificates.
  2. [Structural classification theorem] Structural classification theorem (contribution (i)): The claim that every primitive Euler brick arises up to scaling from the (a,b,m,n)-parametrization is foundational to the reduction to the 1,072 fibers. The section containing the proof of this classification must be clearly identified, and any external results on Euler-brick parametrizations must be stated with precise citations so that the logical chain is self-contained within the manuscript.
minor comments (3)
  1. [Abstract] Abstract: The companion paper is cited only by arXiv number; supplying its title would improve readability for readers who have not yet consulted it.
  2. [Torsion-intersection argument] The manuscript refers to the set H_{m,n}(Q) and the eight degenerate points but does not explicitly describe these points in coordinates (a,b,m,n) or in terms of the original brick parameters. A short explicit description would clarify the geometric meaning of the conclusion.
  3. A supplementary table or data file listing the 1,072 triples (m,n,A), the chosen quotient (E_A' or E_A''), the torsion order, and the certification method (2-descent or L-value) would greatly facilitate independent verification of the computational claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. The comments identify two points where the manuscript can be strengthened for clarity and completeness. We address each below and will implement the indicated revisions in the next version.

read point-by-point responses

  1. Referee: [Rank-zero certificates section] Rank-zero certificates section: Edixhoven's theorem on the Manin constant is invoked only for semistable curves. The manuscript does not explicitly verify or state that every E_A' or E_A'' arising on the 1,072 fibers is semistable in the cases where the modular-symbol method supplies the rank-zero certificate. This verification is load-bearing for the unconditional character of those certificates.

    Authors: We agree that explicit verification of semistability is necessary to make the unconditional character of the rank-zero certificates fully transparent when the modular-symbol method (combined with Kolyvagin and Edixhoven) is used. In the revised manuscript we will insert a short dedicated paragraph immediately following the description of the L(E,1)/Ω_E computations. This paragraph will state that, for every fiber on which the modular-symbol method supplies the rank-zero certificate, the associated quotients E_A' and E_A'' are semistable. The verification is performed by computing the conductor of each such curve (via the explicit Weierstrass models given in the companion paper) and confirming that all primes of bad reduction are of multiplicative type. We will also cite the precise statement of Edixhoven's theorem that applies to semistable curves. revision: yes

  2. Referee: [Structural classification theorem] Structural classification theorem (contribution (i)): The claim that every primitive Euler brick arises up to scaling from the (a,b,m,n)-parametrization is foundational to the reduction to the 1,072 fibers. The section containing the proof of this classification must be clearly identified, and any external results on Euler-brick parametrizations must be stated with precise citations so that the logical chain is self-contained within the manuscript.

    Authors: The proof of the structural classification theorem (contribution (i)) appears in Section 2 of the manuscript. We will revise the opening paragraph of that section to label it explicitly as the location of the proof of contribution (i). In addition, we will insert precise citations to all external results on Euler-brick parametrizations that are invoked (including the standard (a,b,m,n)-parametrization and any auxiliary lemmas drawn from the literature). Where results are taken from the companion paper arXiv:2604.09328 we will note this explicitly. These changes will make the logical chain from the parametrization to the reduction on the 1,072 fibers fully self-contained and traceable. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent classification, algebraic intersection, and external rank certificates

full rationale

The paper's derivation begins with a new structural classification theorem that every primitive Euler brick arises (up to scaling) from the standard (a,b,m,n) parametrization. This reduces the problem to explicit master-tuple fibers. On each fiber the genus-3 curve C_A is taken from the companion paper and quotiented to elliptic curves E_A' and E_A'', after which a fresh torsion-intersection argument algebraically forces |H_{m,n}(Q)| = 8 whenever one quotient satisfies the rank-zero hypothesis plus the stated torsion condition. Rank zero is certified either by PARI 2-descent or by exact modular-symbol evaluation of L(E,1)/Ω_E combined with the modularity theorem, Kolyvagin's theorem, and Edixhoven's theorem on the Manin constant for semistable curves; these are external, unconditional results independent of the present paper. The 1,072 fibers with max(m,n) ≤ 100 are enumerated explicitly where the hypotheses hold. No derived quantity is defined in terms of itself, no prediction reduces to a fitted input by construction, and the central nonexistence claim rests on algebraic identities plus externally certified arithmetic theorems rather than on any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard theorems from elliptic curve theory and the companion paper's reduction. No new free parameters or invented entities appear in the abstract description.

axioms (3)
  • standard math Modularity theorem for elliptic curves
    Invoked to relate the value of L(E,1) to the rank via modular symbols.
  • standard math Kolyvagin's theorem
    Used to conclude rank zero when L(E,1) is nonzero.
  • standard math Edixhoven's bound on the Manin constant for semistable curves
    Applied to obtain an unconditional rank-zero certificate from the L-value computation.

pith-pipeline@v0.9.0 · 5614 in / 1911 out tokens · 47677 ms · 2026-05-07T05:52:56.093026+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages · 1 internal anchor

  1. [1]

    [BBCF+19] Jennifer S. Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfan, Mirela Ciperiani, and Anastassia Etropolski,Chabauty–Coleman experiments for genus 3 hyperelliptic curves, Research Directions in Number Theory19(2019), 67–90. [BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor,On the modularity of elliptic curves over...

    2019

  2. [2]

    [Pes26] René Peschmann,Quartic reductions and elliptic obstructions for perfect Euler bricks, arXiv:2604.09328 [math.NT],

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Rathbun,The integer cuboid table with body, edge, and face type of solutions, arXiv:math/0301029,

    [Rat03] Randall L. Rathbun,The integer cuboid table with body, edge, and face type of solutions, arXiv:math/0301029,

    work page arXiv

  4. [4]

    142(2006), no

    [Sto06] Michael Stoll,Independence of rational points on twists of a given curve, Compositio Math. 142(2006), no. 5, 1201–1214. 15

    2006