GeometricStrain
plain-language theorem explainer
The geometric strain function assigns to each positive length scale r a real value measuring its deviation from the nearest power of φ via the J-cost function. Applied researchers modeling biological coherence would reference this when quantifying how off-resonance scales increase system strain. The definition proxies the nearest rung with floor of log base φ of r plus one half, then evaluates the J-cost on the normalized ratio.
Claim. For $r > 0$, the geometric strain is $J(r / φ^n)$ with $n = ⌊log_φ r + 1/2⌋$; it equals 1 for $r ≤ 0$.
background
The Coherence Technology module examines how φ-spirals and octave-loops affect biological stability. The golden ratio φ is the unique positive fixed point of the self-similar cost recursion, so resonant geometries align with the ledger's scaling law. Geometric strain measures the deviation of a scale r from its nearest resonant neighbor on the phi-ladder. It draws on rung functions imported from constants modules and the J-cost from the Cost module to quantify that deviation.
proof idea
The definition branches on whether r exceeds zero. For positive r it calculates n via floor of (log r over log φ plus 1/2), applies J-cost to r divided by φ to the power n, and returns 1.0 otherwise. No lemmas are invoked; it is a direct computational definition.
why it matters
Geometric strain feeds the SystemStability definition, which inverts it to measure stability, and supports the resonant minimization theorem establishing zero strain at resonant scales. It also enables the resonance increases stability result. This places it in the applied extension of the Recognition framework, linking to the phi fixed point (T6) and the phi-ladder for coherence analysis in biological systems.
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