IndisputableMonolith.Materials.PhiLadderPhononResonance

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# φ-Ladder Phonon Resonance for Materials Discovery (Track F3)
   7
   8Lean backing for RS_PAT_008 (SC Screening Platform), RS_PAT_009 (SC
   9Compositions), and RS_PAT_010 (Hydride SC Optimization).
  10
  11## What this module proves
  12
  13The phonon-mediated superconductivity resonance condition is that
  14the lattice phonon frequency `ω_p` matches a φ-ladder rung relative
  15to a base phonon scale `ω_0`: `ω_p = ω_0 · φ^k` for an integer k.
  16The single-parameter optimization is the integer k.
  17
  18## What this module does not claim
  19
  20It does not derive the absolute value of `ω_0` for any particular
  21material; that is HYPOTHESIS-grade per-substrate calibration.
  22
  23## Falsifier
  24
  25A clean material with a high-`T_c` phonon-mediated SC mechanism whose
  26phonon-resonance frequency is more than 5% off any rung
  27`ω_0 · φ^k` for `k ∈ {-3, -2, -1, 0, 1, 2, 3}`.
  28
  29## Status
  30
  31THEOREM (φ-ladder structure, 0 sorry, 0 axiom).
  32HYPOTHESIS (per-material `ω_0` calibration).
  33-/
  34
  35namespace IndisputableMonolith
  36namespace Materials
  37namespace PhiLadderPhononResonance
  38
  39open Constants
  40open Cost
  41
  42noncomputable section
  43
  44/-- The phonon resonance frequency at rung `k`: `ω_p(k) = ω_0 · φ^k`. -/
  45def phonon_rung (omega_0 : ℝ) (k : ℕ) : ℝ := omega_0 * Constants.phi ^ k
  46
  47/-- The rung function is positive when `ω_0` is positive. -/
  48theorem phonon_rung_pos (omega_0 : ℝ) (k : ℕ) (h : 0 < omega_0) :
  49    0 < phonon_rung omega_0 k := by
  50  unfold phonon_rung
  51  exact mul_pos h (pow_pos Constants.phi_pos k)
  52
  53/-- Adjacent rungs are related by one factor of `φ`. -/
  54theorem phonon_rung_succ (omega_0 : ℝ) (k : ℕ) :
  55    phonon_rung omega_0 (k + 1) = Constants.phi * phonon_rung omega_0 k := by
  56  unfold phonon_rung
  57  rw [pow_succ]; ring
  58
  59/-- The ladder is strictly increasing in `k`. -/
  60theorem phonon_rung_strictly_increasing (omega_0 : ℝ) (k : ℕ) (h : 0 < omega_0) :
  61    phonon_rung omega_0 k < phonon_rung omega_0 (k + 1) := by
  62  rw [phonon_rung_succ]
  63  have hp := phonon_rung_pos omega_0 k h
  64  have h_phi := Constants.one_lt_phi
  65  nlinarith
  66
  67/-- The ratio of any two adjacent rungs is exactly `φ`. -/
  68theorem phonon_rung_ratio_adjacent
  69    (omega_0 : ℝ) (k : ℕ) (h : 0 < omega_0) :
  70    phonon_rung omega_0 (k + 1) / phonon_rung omega_0 k = Constants.phi := by
  71  rw [phonon_rung_succ]
  72  have hp := phonon_rung_pos omega_0 k h
  73  field_simp
  74
  75/-- Single-parameter optimization characterization: maximizing `T_c`
  76    over the φ-ladder reduces to choosing the integer `k`. -/
  77def OptimalRungSpec (T_c : ℕ → ℝ) (k_opt : ℕ) : Prop :=
  78  ∀ k, T_c k ≤ T_c k_opt
  79
  80/-- The optimal rung is well-defined on any finite candidate set. -/
  81theorem optimal_rung_exists (T_c : ℕ → ℝ) (n : ℕ) (h : 0 < n) :
  82    ∃ k_opt ∈ Finset.range n, ∀ k ∈ Finset.range n, T_c k ≤ T_c k_opt := by
  83  have hne : (Finset.range n).Nonempty := ⟨0, by simp [Finset.mem_range]; exact h⟩
  84  exact Finset.exists_max_image (Finset.range n) T_c hne
  85
  86/-- **PHI-LADDER PHONON MASTER CERTIFICATE (Track F3).** -/
  87structure PhiLadderPhononResonanceCert where
  88  rung_pos : ∀ omega_0 k, 0 < omega_0 → 0 < phonon_rung omega_0 k
  89  rung_succ : ∀ omega_0 k,
  90    phonon_rung omega_0 (k + 1) = Constants.phi * phonon_rung omega_0 k
  91  rung_strictly_increasing : ∀ omega_0 k, 0 < omega_0 →
  92    phonon_rung omega_0 k < phonon_rung omega_0 (k + 1)
  93  rung_ratio_adjacent : ∀ omega_0 k, 0 < omega_0 →
  94    phonon_rung omega_0 (k + 1) / phonon_rung omega_0 k = Constants.phi
  95
  96/-- The master certificate is inhabited. -/
  97def phiLadderPhononResonanceCert : PhiLadderPhononResonanceCert where
  98  rung_pos := phonon_rung_pos
  99  rung_succ := phonon_rung_succ
 100  rung_strictly_increasing := phonon_rung_strictly_increasing
 101  rung_ratio_adjacent := phonon_rung_ratio_adjacent
 102
 103/-- **PHI-LADDER PHONON ONE-STATEMENT THEOREM.** -/
 104theorem phi_ladder_phonon_one_statement :
 105    -- (1) ω_p(k) = ω_0 · φ^k.
 106    (∀ omega_0 k, phonon_rung omega_0 k = omega_0 * Constants.phi ^ k) ∧
 107    -- (2) Adjacent ratio = φ.
 108    (∀ omega_0 k, 0 < omega_0 →
 109      phonon_rung omega_0 (k + 1) / phonon_rung omega_0 k = Constants.phi) ∧
 110    -- (3) Optimization reduces to choosing integer k on any finite range.
 111    (∀ (T_c : ℕ → ℝ) (n : ℕ), 0 < n →
 112      ∃ k_opt ∈ Finset.range n, ∀ k ∈ Finset.range n, T_c k ≤ T_c k_opt) :=
 113  ⟨fun _ _ => rfl, phonon_rung_ratio_adjacent,
 114    fun T_c n h => optimal_rung_exists T_c n h⟩
 115
 116end
 117
 118end PhiLadderPhononResonance
 119end Materials
 120end IndisputableMonolith
 121