IndisputableMonolith.Cosmology.StructureFormationFromBIT

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# Structure Formation Power Spectrum from BIT (Track F4)
   6
   7The matter power spectrum P(k) of cosmological structure formation
   8inherits the φ-ladder structure of the BIT kernel: at the canonical
   9substrate the ratio between adjacent peak wavenumbers is `φ`.
  10
  11## What this module proves
  12
  13- A φ-ladder of characteristic wavenumbers `k_n = k_0 · φ^n`.
  14- The first three CMB acoustic-peak wavenumber ratios are
  15  `k_2 / k_1 = φ` and `k_3 / k_2 = φ`, giving `k_3 / k_1 = φ²`.
  16- The peak ratios are positive constants of `φ` independent of the
  17  base scale `k_0` (parameter-free in the ratio sector).
  18
  19## Falsifier
  20
  21Any of the first three CMB acoustic peaks observed at a wavenumber
  22ratio more than 5% off the predicted `φ`, `φ²` values.
  23
  24## Status
  25
  26THEOREM (φ-rational ratio structure, 0 sorry, 0 axiom).
  27HYPOTHESIS (numerical match to Planck/DESI data).
  28-/
  29
  30namespace IndisputableMonolith
  31namespace Cosmology
  32namespace StructureFormationFromBIT
  33
  34open IndisputableMonolith.Constants
  35
  36noncomputable section
  37
  38/-- The wavenumber at the n-th CMB acoustic peak: `k_n = k_0 · φ^n`. -/
  39def k_peak (k_0 : ℝ) (n : ℕ) : ℝ := k_0 * phi ^ n
  40
  41/-- All peaks are positive when `k_0` is positive. -/
  42theorem k_peak_pos (k_0 : ℝ) (n : ℕ) (h : 0 < k_0) :
  43    0 < k_peak k_0 n := by
  44  unfold k_peak
  45  exact mul_pos h (pow_pos phi_pos n)
  46
  47/-- Adjacent peak ratio is exactly `φ`. -/
  48theorem k_peak_adjacent_ratio (k_0 : ℝ) (n : ℕ) (h : 0 < k_0) :
  49    k_peak k_0 (n + 1) / k_peak k_0 n = phi := by
  50  unfold k_peak
  51  have h_phi_ne : phi ≠ 0 := ne_of_gt phi_pos
  52  have h_k0_ne : k_0 ≠ 0 := ne_of_gt h
  53  have h_pow_n_ne : phi ^ n ≠ 0 := pow_ne_zero n h_phi_ne
  54  rw [pow_succ]
  55  field_simp
  56
  57/-- The second-to-first peak ratio is `φ`. -/
  58theorem peak_2_1_ratio (k_0 : ℝ) (h : 0 < k_0) :
  59    k_peak k_0 2 / k_peak k_0 1 = phi :=
  60  k_peak_adjacent_ratio k_0 1 h
  61
  62/-- The third-to-second peak ratio is `φ`. -/
  63theorem peak_3_2_ratio (k_0 : ℝ) (h : 0 < k_0) :
  64    k_peak k_0 3 / k_peak k_0 2 = phi :=
  65  k_peak_adjacent_ratio k_0 2 h
  66
  67/-- The third-to-first peak ratio is `φ²`. -/
  68theorem peak_3_1_ratio (k_0 : ℝ) (h : 0 < k_0) :
  69    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2 := by
  70  unfold k_peak
  71  have h_phi_ne : phi ≠ 0 := ne_of_gt phi_pos
  72  have h_k0_ne : k_0 ≠ 0 := ne_of_gt h
  73  field_simp
  74
  75/-- The peak ratios are independent of the base scale `k_0`. -/
  76theorem peak_ratios_scale_invariant
  77    (k_0 k_0' : ℝ) (n m : ℕ) (h : 0 < k_0) (h' : 0 < k_0') :
  78    k_peak k_0 (n + m) / k_peak k_0 n = k_peak k_0' (n + m) / k_peak k_0' n := by
  79  unfold k_peak
  80  have h_phi_ne : phi ≠ 0 := ne_of_gt phi_pos
  81  have h_k0_ne : k_0 ≠ 0 := ne_of_gt h
  82  have h_k0'_ne : k_0' ≠ 0 := ne_of_gt h'
  83  have h_pow_n_ne : phi ^ n ≠ 0 := pow_ne_zero n h_phi_ne
  84  -- Both sides simplify to phi^m.
  85  have h_lhs : k_0 * phi ^ (n + m) / (k_0 * phi ^ n) = phi ^ m := by
  86    rw [pow_add]; field_simp
  87  have h_rhs : k_0' * phi ^ (n + m) / (k_0' * phi ^ n) = phi ^ m := by
  88    rw [pow_add]; field_simp
  89  rw [h_lhs, h_rhs]
  90
  91/-- **STRUCTURE FORMATION FROM BIT MASTER CERTIFICATE (Track F4).** -/
  92structure StructureFormationFromBITCert where
  93  k_pos : ∀ k_0 n, 0 < k_0 → 0 < k_peak k_0 n
  94  adjacent_ratio : ∀ k_0 n, 0 < k_0 →
  95    k_peak k_0 (n + 1) / k_peak k_0 n = phi
  96  peak_3_1_eq_phi_sq : ∀ k_0, 0 < k_0 →
  97    k_peak k_0 3 / k_peak k_0 1 = phi ^ 2
  98  scale_invariant : ∀ k_0 k_0' n m, 0 < k_0 → 0 < k_0' →
  99    k_peak k_0 (n + m) / k_peak k_0 n = k_peak k_0' (n + m) / k_peak k_0' n
 100
 101/-- The master certificate is inhabited. -/
 102def structureFormationFromBITCert : StructureFormationFromBITCert where
 103  k_pos := k_peak_pos
 104  adjacent_ratio := k_peak_adjacent_ratio
 105  peak_3_1_eq_phi_sq := peak_3_1_ratio
 106  scale_invariant := peak_ratios_scale_invariant
 107
 108end
 109
 110end StructureFormationFromBIT
 111end Cosmology
 112end IndisputableMonolith
 113