ResonantScale
plain-language theorem explainer
ResonantScale identifies a length r as resonant exactly when r equals phi to an integer power. Coherence technology researchers cite it to quantify how phi-aligned geometries stabilize biological systems. The definition consists of a single existential quantifier over the integer lattice of exponents.
Claim. A real number $r$ is resonant if there exists an integer $n$ such that $r = phi^n$.
background
The Coherence Technology module formalizes how recognition-resonant geometries such as phi-spirals and octave-loops affect biological stability. The golden ratio phi is the unique positive fixed point of the self-similar cost recursion. The phi-ladder is the discrete set of scales generated by integer powers of phi, matching the scale definition in LargeScaleStructureFromRS that sets scale(k) to phi^k for natural k.
proof idea
The definition is realized directly by the existential statement that r lies on the phi-ladder. No external lemmas are invoked; the body simply requires an integer n satisfying the power equality.
why it matters
ResonantScale supplies the predicate used in golden_spiral_is_resonant to prove that the golden spiral maintains resonance everywhere, in resonance_increases_stability to show stability gains from resonant scales, and in resonant_minimization to establish zero geometric strain at resonant points. It anchors Phase 10.2 by embedding the T6 self-similar fixed point into applied coherence models.
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