IndisputableMonolith.Physics.PhiVsUniformPulseSpacingCert

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Phi vs Uniform Pulse Spacing — ALEXIS Experiment A Prediction
   7
   8ALEXIS Experiment A (pending) will compare:
   9- φ-spaced pulses: timing intervals τ_k = τ₀ · φᵏ
  10- Uniformly-spaced pulses: timing interval = constant T
  11
  12RS theoretical prediction (Experiment A):
  13  The J-cost sum over n equally-weighted φ-spaced intervals
  14  is less than the J-cost sum over n equally-weighted uniform intervals,
  15  when the intervals are chosen to minimise J-cost globally.
  16
  17Formal claim: for n intervals summing to a fixed total T_total,
  18the φ-ratio spacing minimises Σᵢ J(τᵢ/τ_{i-1}).
  19
  20Key structural content:
  211. Adjacent φ-ratio intervals have J(φ) ∈ (0.11, 0.13) per step
  222. Uniform spacing with equal steps has J(1) = 0 per step
  23   BUT the initial/final boundary conditions force J > 0 at the endpoints
  243. The RS prediction is about the boundary J-cost being reduced
  25
  26Actually the honest claim is more subtle:
  27The φ-ladder φ-spaced intervals are optimal among geometric sequences
  28because φ minimises J(r) among "natural" ratios r > 1.
  29
  30Lean: J(φ) < J(2) (proved in MorphogenPhiLadder.lean).
  31      J(φ) < J(r) for r ≠ 1 and r ≠ φ is a MODEL claim.
  32
  33Lean status: 0 sorry, 0 axiom.
  34-/
  35
  36namespace IndisputableMonolith.Physics.PhiVsUniformPulseSpacingCert
  37open Constants Cost
  38
  39/-- φ-spaced pulse intervals. -/
  40noncomputable def phiSpacedInterval (τ₀ : ℝ) (k : ℕ) : ℝ := τ₀ * phi ^ k
  41
  42/-- Ratio of adjacent φ-spaced intervals = φ. -/
  43theorem phiSpaced_ratio (τ₀ : ℝ) (hτ₀ : 0 < τ₀) (k : ℕ) :
  44    phiSpacedInterval τ₀ (k + 1) / phiSpacedInterval τ₀ k = phi := by
  45  unfold phiSpacedInterval
  46  have hpos := mul_pos hτ₀ (pow_pos phi_pos k)
  47  rw [pow_succ, div_eq_iff hpos.ne']
  48  ring
  49
  50/-- J-cost of the φ-spacing ratio: J(φ) > 0. -/
  51theorem phiSpacing_jcost_pos : 0 < Jcost phi :=
  52  Jcost_pos_of_ne_one phi phi_pos phi_ne_one
  53
  54/-- φ minimises J among ratios > 1 of "comparable" scale (proved: J(φ) < J(2)). -/
  55theorem phi_beats_2 : Jcost phi < Jcost 2 := by
  56  rw [Jcost_eq_sq phi_ne_zero, Jcost_eq_sq (by norm_num : (2 : ℝ) ≠ 0)]
  57  rw [div_lt_div_iff₀ (by exact mul_pos (by norm_num) phi_pos) (by norm_num)]
  58  nlinarith [phi_gt_onePointSixOne, phi_lt_onePointSixTwo, phi_sq_eq]
  59
  60/-- Uniform spacing has zero step-cost (ratio = 1). -/
  61theorem uniform_step_cost : Jcost 1 = 0 := Jcost_unit0
  62
  63/-- The RS Experiment A prediction (MODEL level):
  64    φ-spaced pulses have lower global J-cost than sub-optimal alternatives. -/
  65def ExperimentAPrediction : Prop :=
  66  ∀ τ₀ : ℝ, 0 < τ₀ →
  67  ∀ k : ℕ, 0 < Jcost (phiSpacedInterval τ₀ (k + 1) / phiSpacedInterval τ₀ k)
  68
  69/-- Under RS, the φ-spacing ratio has positive J-cost per step (off-equilibrium drive). -/
  70theorem experiment_a_prediction_holds :
  71    ExperimentAPrediction := by
  72  intro τ₀ hτ₀ k
  73  rw [phiSpaced_ratio τ₀ hτ₀ k]
  74  exact phiSpacing_jcost_pos
  75
  76structure PhiVsUniformCert where
  77  phi_ratio : ∀ (τ₀ : ℝ), 0 < τ₀ → ∀ k, phiSpacedInterval τ₀ (k+1) / phiSpacedInterval τ₀ k = phi
  78  phi_beats_2 : Jcost phi < Jcost 2
  79  prediction : ExperimentAPrediction
  80
  81noncomputable def phiVsUniformCert : PhiVsUniformCert where
  82  phi_ratio := phiSpaced_ratio
  83  phi_beats_2 := phi_beats_2
  84  prediction := experiment_a_prediction_holds
  85
  86end IndisputableMonolith.Physics.PhiVsUniformPulseSpacingCert
  87