resonance_increases_stability
Resonant length scales on the phi-ladder produce strictly higher system stability than non-resonant scales. Modelers of coherence effects in biological or postural systems would cite the result when comparing geometric alignment to stability metrics. The proof splits cases: resonant scales are reduced to zero strain via resonant_minimization, while non-resonant scales are shown to carry positive J-cost strain that forces stability below 1.
claimLet $r$ and $s$ be positive reals. If $r$ is not equal to any integer power of the golden ratio $phi$ while $s$ equals some integer power of $phi$, then the system stability at $r$ is strictly less than at $s$, where stability is defined by $S(r) = 1/(1+Q(r))$ and $Q(r)$ denotes geometric strain $Jcost(r/phi^n)$ for the nearest integer rung $n$.
background
ResonantScale(r) holds precisely when $r = phi^n$ for some integer $n$, placing the scale on the phi-ladder. GeometricStrain(r) for $r>0$ equals the J-cost of the ratio of $r$ to its nearest phi-power, using the floor of log_phi(r) plus one-half as the rung index. SystemStability(r) is then the reciprocal of one plus that strain, so lower strain yields higher stability. The module sets this in the context of recognition-resonant geometries (phi-spirals, octave loops) that align with the fundamental scaling law of the ledger. Upstream, resonant_minimization states that any resonant scale has zero geometric strain.
proof idea
The tactic proof opens by applying resonant_minimization to the resonant input, yielding zero strain and therefore stability exactly 1 after unfolding SystemStability. For the non-resonant input it unfolds GeometricStrain, computes the floor-log index n, establishes that the ratio to phi^n differs from 1, invokes Jcost_pos_of_ne_one to obtain positive cost, and concludes the denominator exceeds 1 so that stability is strictly less than 1 by the div_lt_one lemma.
why it matters in Recognition Science
The theorem belongs to the coherence-technology module and demonstrates that phi-ladder alignment raises stability to its maximum value of 1. It supplies a concrete link between resonant geometry and the recognition cost ledger, consistent with the self-similar fixed point phi and the eight-tick octave structure. No downstream theorems are recorded yet; the result therefore functions as a foundational comparison inside applied coherence claims.
scope and limits
- Does not quantify the size of the stability difference.
- Does not treat time evolution or multi-component systems.
- Does not incorporate external loads or environmental coupling.
- Does not assert that resonant scales exist in any given physical specimen.
formal statement (Lean)
68theorem resonance_increases_stability (r_init r_resonant : ℝ) (hr_init : r_init > 0) (hr_res : r_resonant > 0) :
69 ¬ ResonantScale r_init →
70 ResonantScale r_resonant →
71 SystemStability r_init < SystemStability r_resonant := by
proof body
Tactic-mode proof.
72 intro h_non_res h_res
73 -- 1. Resonant stability is 1
74 have h_res_strain := resonant_minimization r_resonant hr_res h_res
75 unfold SystemStability
76 rw [h_res_strain]
77 simp only [add_zero, div_one]
78 -- 2. Non-resonant stability is < 1
79 unfold GeometricStrain
80 simp only [hr_init, ↓reduceDIte]
81 let n := Int.floor (Real.log r_init / Real.log phi + 1 / 2)
82 have h_ratio_ne_one : r_init / (phi ^ (n : ℝ)) ≠ 1 := by
83 intro h_eq
84 have h_r_init : r_init = phi ^ (n : ℝ) := by
85 have h_pow_pos := Real.rpow_pos_of_pos phi_pos (n : ℝ)
86 rw [div_eq_one_iff_eq h_pow_pos.ne'] at h_eq
87 exact h_eq
88 exact h_non_res ⟨n, h_r_init⟩
89 have h_ratio_pos : 0 < r_init / (phi ^ (n : ℝ)) := by
90 apply div_pos hr_init
91 exact Real.rpow_pos_of_pos phi_pos _
92 have h_pos : 0 < Jcost (r_init / (phi ^ (n : ℝ))) := by
93 apply Jcost_pos_of_ne_one
94 · exact h_ratio_pos
95 · exact h_ratio_ne_one
96 -- 1 / (1 + h_pos) < 1
97 have h_denom : 1 < 1 + Jcost (r_init / (phi ^ (n : ℝ))) := by
98 linarith
99 have h_denom_pos : 0 < 1 + Jcost (r_init / (phi ^ (n : ℝ))) := by
100 linarith
101 apply (div_lt_one h_denom_pos).mpr
102 exact h_denom
103
104/-- **DEFINITION: Octave Loop**
105 A closed recognition sequence of exactly 8 steps. -/