The Exponent Chamber Lift: Beal, Reciprocal Shells, and the First Hyperbolic Shadow
Authors/Creators
Description
This note builds a reciprocal-exponent atlas for Beal's conjecture. It does not prove Beal's conjecture. Its purpose is structural: to describe the exponent chamber in coordinates
\[
u=1/x,\qquad v=1/y,\qquad w=1/z,
\]
where the Fermat-Catalan boundary is the plane \(u+v+w=1\). Inside the Beal chamber \(x,y,z>2\), the only boundary point is the cubic corner \((3,3,3)\). The first hyperbolic shell beyond this corner is the signature \((3,3,4)\), for which
\[
\frac13+\frac13+\frac14=\frac{11}{12},\qquad
\chi=-\frac{1}{12},\qquad
\delta=\frac{1}{12}.
\]
The corresponding fixed equation \(A^3+B^3=C^4\) is known to have no primitive positive solutions by Bruin's Chabauty and covering-method resolution of \(x^3+y^3=z^4\). The atlas therefore correctly singles out a real solved lock: the first hyperbolic shadow is empty. A global proof of Beal would require a uniform obstruction for all hyperbolic exponent chambers, not just this first one. Version 1.3 sharpens the shell-envelope proof with a finite exchange argument and adds a shifted-coordinate decomposition: after subtracting the cubic corner, the exponent chamber is the nonnegative integer cone \(\Z_{\ge0}^3\), with triangular shells, tetrahedral cumulative counts, and limiting strata. Small-denominator coincidences with zeta or the 45-degree cone remain explicitly excluded unless a common identity can be written down.
