SystemStability
SystemStability defines biological system stability at scale r as the reciprocal of one plus geometric strain. Coherence technology researchers cite it to quantify how resonant geometries raise stability toward its maximum. The definition is a direct one-line algebraic expression built on the GeometricStrain function.
claimThe system stability at scale $r$ is defined by $S(r) = 1 / (1 + Q(r))$, where $Q(r)$ is the geometric strain of $r$ relative to its nearest resonant neighbor on the phi-ladder.
background
Module Applied.CoherenceTechnology formalizes the effect of recognition-resonant geometries (phi-spirals, octave-loops) on biological stability. The golden ratio phi is the unique positive fixed point of the self-similar cost recursion, so geometries aligned with it minimize strain. GeometricStrain(r) computes the strain of scale r by taking the floor of log_phi(r) + 1/2 to locate the nearest rung and then applying Jcost to the normalized ratio r / phi^n.
proof idea
One-line definition that directly returns the reciprocal of one plus the value of GeometricStrain r.
why it matters in Recognition Science
This definition supplies the stability measure used by the downstream theorem resonance_increases_stability, which shows that resonant scales strictly increase SystemStability. It implements the Recognition Science claim that phi-resonant geometries minimize geometric strain and thereby raise biological stability, consistent with the self-similar fixed-point property of phi in the forcing chain.
scope and limits
- Does not incorporate postural alignment or other coupling costs.
- Does not evaluate stability for time-varying or multi-scale systems.
- Does not provide numerical evaluation or bounds beyond the algebraic form.
- Does not address non-positive scales.
Lean usage
theorem resonance_increases_stability (r_init r_resonant : ℝ) (hr_init : r_init > 0) (hr_res : r_resonant > 0) : ¬ ResonantScale r_init → ResonantScale r_resonant → SystemStability r_init < SystemStability r_resonant := by intro h_non h_res; unfold SystemStability; simp [GeometricStrain]formal statement (Lean)
38noncomputable def SystemStability (r : ℝ) : ℝ :=
proof body
Definition body.
39 1 / (1 + GeometricStrain r)
40
41/-- **THEOREM: Resonant Minimization**
42 Resonant scales minimize the geometric strain.
43 If r is a power of φ, its strain is zero. -/