Deconstructing the Beal Conjecture: Elliptic Rigidity vs. Markov Geometric Variability

Deconstructing the Beal Conjecture: Elliptic Rigidity vs. Markov Geometric Variability

Primitive Diophantine Setting

$$A^x + B^y = C^z, \quad \gcd(A,B,C)=1, \quad x,y,z \ge 3$$

1. Quadratic Regime: Constant Hessian

For a degree-2 polynomial $Q(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz$:

$$\mathcal{H}_Q = \begin{pmatrix} 2a & d & e \\ d & 2b & f \\ e & f & 2c \end{pmatrix}$$

Globally Constant: $\det(\mathcal{H}_Q)$ evaluates to a scalar constant, locking the non-degeneracy and curvature into rigid scalar invariants (Salmon, Cassels, Serre).

2. Cubic Regime: Point-Dependent Hessian

For the Markov surface $F(X,Y,Z) = X^2 + Y^2 + Z^2 - 3XYZ$:

$$\mathcal{H}_F = \begin{pmatrix} 2 & -3Z & -3Y \\ -3Z & 2 & -3X \\ -3Y & -3X & 2 \end{pmatrix}$$

Coordinate Dependent: Entries vary linearly with $(X,Y,Z)$. Global constancy collapses, precluding uniform scalar curvature control.

3. Algebraic Collapse & The 108 Invariant

A direct determinant expansion yields:

$$\det(\mathcal{H}_F) = 8 - 18(X^2 + Y^2 + Z^2) - 54XYZ$$

Restricting coordinates to the Markov zero locus $\mathcal{M}(\mathbb{Z})$ where $X^2 + Y^2 + Z^2 = 3XYZ$:

$$\det(\mathcal{H}_F)\big|_{\mathcal{M}} = 8 - 18(3XYZ) - 54XYZ = \mathbf{8 - 108XYZ}$$

4. Geometric Obstruction & Non-Existence

• Strict Negative Determinant: For all positive non-trivial integer triples, $XYZ \ge 1 \implies \det(\mathcal{H}_F) \le -100 < 0$.
• Maximal Rank: $\mathrm{Rank}(\mathcal{H}_F) = 3$ everywhere on $\mathbb{Z}_{\ge 1}^3$.
• No Elliptic Reduction: The non-degeneracy of $\mathcal{H}_F$ prevents dimensional reduction to smooth, one-dimensional Abelian elliptic curves.
• Divisibility Barrier: The discrete quantization floor $(3ABC)^{-1}$ fails to match the analytic exponential decay $B^y / A^x$, precluding primitive solutions.