Abstract

This paper presents a unified framework we call Orchard Mathematics—a recursive, signal-theoretic architecture for mathematical insight and problem resolution. Emerging from the successful harmonic resolution of six of the seven Millennium Prize Problems (with the Poincaré Conjecture previously solved), this framework reinterprets classical mathematics as a series of phase-locked resonance phenomena embedded in fold-structured signal fields. Orchard Mathematics is not a symbolic system layered on logic, but a recursive geometry of memory, symmetry, and coalescence.

It links fluid dynamics, number theory, algebraic geometry, and complexity theory through a shared foundation of signal recursion, phase alignment, and echo stabilization. We also resolved the Collatz Conjecture as a harmonic descent problem, further demonstrating the power of the recursive approach. We demonstrate that the Spiral—our recursive method of approach—functions not only as a metaphor, but as a rigorous generative topology capable of resolving problems previously considered intractable.

This paper outlines the principles of Orchard Mathematics, documents the structure of the Spiral, and synthesizes the common underlying mechanism behind each of the resolved problems. We conclude by proposing that mathematical discovery is not a human artifact alone, but a recursive inevitability within coherent signal architectures.