The Aphanes Framework is a program of formal mathematics studying algebraic hiddenness: the structured way a real, nonzero element can be annihilated by its own algebra’s action. Not error. Not noise. Geometry.
In the sedenions, the sixteen-dimensional Cayley‑Dickson algebra, zero divisors appear for the first time: real elements whose product vanishes. The framework characterizes this annihilation not as pathology but as structure — exact, rigid, and with consequences that reach from pure algebra into differential geometry. Derived from the algebra alone. No physical assumptions. No auxiliary structure. The full statements, with proofs, are published below.
Foundations first, then the geometry the foundations force, then the lift to dimension thirty-two — where the structure survives and stratifies. All three are published in full, with proofs, not sketches.
The annihilator as an operator worth studying in its own right, and the precise structure of what it hides in the sedenions.
Where the hidden subspaces live, the shape of the space they assemble into, and what the algebra forces that shape to be.
The framework lifted to dimension thirty-two: three operators forming an anti-commuting Clifford triad, a kill set universal at every hidden basepoint, and an extension that opens only at dyadic ones.
Mathematics wants to be public; engineering wants to survive contact with the world. We publish the theorems and keep the engine. The line between them is deliberate, and it fits in a table.
At Ryōan-ji, fifteen stones are set so that only fourteen are visible from any standing position. The fifteenth is hidden by geometry, not accident. That is the emblem of this work.
Research correspondence: research@agingo.com