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

1. Problem Statement

The paper addresses the Perfect Cuboid Conjecture (Conjecture B), a long-standing open problem in elementary number theory.

• Definition: An Euler brick is a rectangular parallelepiped with integer edges $(X, Y, Z)$ and integer face diagonals. A perfect cuboid is an Euler brick that additionally possesses an integer space diagonal ($\sqrt{X^2+Y^2+Z^2} \in \mathbb{N}$).
• The Conjecture: No perfect cuboid exists in $\mathbb{N}^3$.
• Current Status: While computational searches have ruled out perfect cuboids with edges up to $5 \cdot 10^{11}$, no general proof exists. Previous algebraic-geometric approaches have been limited to specific sub-families or relied on unproven hypotheses (e.g., Generalized Riemann Hypothesis).

2. Methodology

The author employs a combination of structural number theory, algebraic geometry, and rigorous computational verification. The approach proceeds in three main stages:

A. Structural Reduction to Master Tuples

The paper establishes a structural classification theorem (Theorem 2.4) proving that every primitive Euler brick arises from a unique "master tuple" $(a, b, m, n) \in \mathbb{N}^4$ satisfying specific coprimality and parity conditions.

• The edges of the brick are derived from standard Pythagorean parametrizations: $U_1 = a^2-b^2, V_1=2ab$ and $U_2=m^2-n^2, V_2=2mn$.
• The condition for a perfect cuboid reduces to checking if a specific polynomial $f_1$ (representing the space diagonal squared) is a perfect square.
• This reduces the infinite search space to analyzing specific fibers defined by the parameters $(m, n)$.

B. Genus-3 Curve Decomposition

For a fixed master tuple fiber $(m, n)$, the condition for a perfect cuboid is encoded in a genus-3 hyperelliptic curve $H_{m,n}$.

• The curve is defined by $v^2 = P(t^2)Q(t^2)$, where $t = a/b$.
• The Jacobian of this curve, $\text{Jac}(H_{m,n})$, decomposes (up to isogeny) into a product of three elliptic curves: $E_{PQ}$, $E_{uV}$, and $E_3$.
• The paper focuses on the elliptic quotients $E_{uV}$ and $E_3$.

C. The Torsion-Intersection Argument

The core proof mechanism relies on the Torsion-Intersection Lemma (Lemma 4.2).

• Premise: If an elliptic quotient $E_q$ has rank 0 (i.e., its rational points are finite and consist only of torsion points), then the number of rational points on the genus-3 curve $H_{m,n}$ is bounded by the size of the torsion subgroup of $E_q$.
• Logic:
  1. Verify $\text{rk}(E_q(\mathbb{Q})) = 0$.
  2. Calculate $|E_q(\mathbb{Q})_{\text{tors}}|$.
  3. Map rational points from $H_{m,n}$ to $E_q$. Since the map is degree 2, $|H_{m,n}(\mathbb{Q})| \le 2 \cdot |E_q(\mathbb{Q})_{\text{tors}}|$.
  4. Crucial Step: The curve $H_{m,n}$ always possesses 8 known "trivial" rational points corresponding to degenerate bricks (where one or more edges are zero).
  5. If the upper bound derived from the torsion intersection equals exactly 8, then no non-trivial (perfect cuboid) points exist on that fiber.

D. Unconditional Rank Certification

To ensure the proof is unconditional (not relying on unproven conjectures like GRH), the author uses two complementary techniques to verify the rank-zero hypothesis:

1. 2-Descent: Using PARI/GP's ellrank to get rigorous bounds $[r_{low}, r_{up}]$. If $r_{low} = r_{up} = 0$, the rank is proven zero.
2. Modular Symbols (Kolyvagin's Theorem): For cases where 2-descent is ambiguous (e.g., $[0, 2]$), the author computes $L(E, 1)/\Omega_E$ exactly using Sage's modular symbols.
   • By the Modularity Theorem and Kolyvagin's theorem, if $L(E, 1) \neq 0$, the rank is 0.
   • For semistable curves (which all encountered factors were), the Manin constant is 1, ensuring the modular symbol calculation is exact and unconditional.

3. Key Contributions

1. Structural Classification (Theorem 2.4): Proves the converse direction of the master tuple parametrization, showing that every primitive Euler brick corresponds to a master tuple. This was a missing link in previous literature.
2. Torsion-Intersection Argument: Applies a refined intersection bound to the elliptic quotients $E_{uV}$ and $E_3$. Unlike previous work that focused on $E_A$, this paper exploits the specific geometry of $E_{uV}$ and $E_3$ to force the point count to exactly 8.
3. Refined Lift Counting: When the naive torsion bound is too loose (e.g., torsion size 8 yielding a bound of 12), the author performs an explicit enumeration of the 8 torsion points on the quartic model. By checking which points "lift" rationally to the genus-3 curve, they refine the bound back to 8.
4. Unconditional Certification Pipeline: A robust computational workflow combining PARI/GP and SageMath that avoids reliance on the Generalized Riemann Hypothesis (GRH) or the Birch and Swinnerton-Dyer conjecture, relying instead on Kolyvagin's theorem and Edixhoven's bound on the Manin constant.

4. Results

• Scope: The study covers all coprime pairs $(m, n)$ with $\max(m, n) \le 100$ and $m-n$ odd. There are 2,040 such pairs.
• Success Rate: The method unconditionally proves the nonexistence of perfect cuboids on 1,072 fibers (approx. 52.5% of the search space).
• Breakdown of Proven Fibers:
  • 827 fibers via $E_3$: 359 via sharp 2-descent, 468 via modular symbol certification.
  • 245 fibers via $E_{uV}$: All required the refined lift-counting argument (Case c) because the torsion size was 8.
• Outcome: For these 1,072 fibers, the set of rational points on the associated genus-3 curve is exactly the set of 8 degenerate points. Thus, no perfect cuboid exists for any master tuple derived from these $(m, n)$ pairs.

5. Significance and Limitations

• Significance:
  • This is the first unconditional proof of the nonexistence of perfect cuboids on a large, explicit set of fibers without relying on unproven conjectures.
  • It demonstrates the power of combining explicit descent with modular symbol computations to resolve rank-zero cases in families of curves.
  • It provides a concrete framework (master tuples) for future exhaustive searches.
• Limitations & Future Work:
  • Coverage: The result covers only ~52% of fibers up to $m,n \le 100$. The remaining 968 fibers either have ambiguous ranks that exceed current computational budgets (conductor $> 10^{12}$) or have total Jacobian rank $\ge 3$, rendering standard Chabauty methods inapplicable.
  • Hard Fibers: Fibers where all three elliptic factors have rank $\ge 1$ (e.g., $(5,2)$) remain open.
  • Extensions: The author suggests that Quadratic Chabauty (p-adic methods) and Mordell-Weil sieving are the necessary next steps to tackle the remaining "hard" fibers and potentially extend the bound to higher $m, n$.

In summary, Peschmann provides a rigorous, computer-assisted proof that eliminates the possibility of perfect cuboids for a significant subset of the parameter space, establishing a new benchmark for unconditional results in this classical problem.