GeometricStrain
plain-language theorem explainer
GeometricStrain quantifies how far a positive scale r deviates from its nearest resonant neighbor on the phi-ladder by normalizing to the closest power of phi and applying the J-cost function. Researchers modeling coherence effects in biological or engineered systems cite this definition when comparing resonant versus non-resonant geometries. The definition estimates the nearest rung via floor of (log r over log phi plus one-half) then evaluates Jcost on the residual ratio, defaulting to 1 for non-positive inputs.
Claim. For positive real scale $r$, the geometric strain $Q(r)$ equals $J(r / phi^n)$ where $n = floor(log_phi r + 1/2)$ and $J$ is the J-cost function; the value is 1 for non-positive $r$.
background
The Coherence Technology module formalizes how recognition-resonant geometries such as phi-spirals and octave-loops affect biological stability. The golden ratio phi serves as the unique positive fixed point of the self-similar cost recursion, so scales aligned with its powers are resonant. Geometric strain measures deviation from the nearest such power using the J-cost function imported from the Cost module.
proof idea
The definition is a direct conditional computation. For r greater than zero it forms n as the integer floor of (log r divided by log phi plus one-half), normalizes the scale by phi to the power n, and applies Jcost to the result; non-positive inputs return the constant 1.0. No lemmas are invoked.
why it matters
GeometricStrain supplies the deviation measure required by resonant_minimization, which proves that resonant scales achieve zero strain, and by SystemStability, which inverts strain to obtain a stability score. It also feeds resonance_increases_stability. The definition closes a concrete step in the Phase 10.2 coherence framework by linking scale alignment to the phi fixed point and Recognition Composition Law.
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