IndisputableMonolith.Astrophysics.SchumannResonanceFromBIT

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Cost
   4
   5/-!
   6# Schumann Resonance Fundamental from BIT Carrier
   7
   8## Status
   9
  10STRUCTURAL THEOREM. The Earth-ionosphere cavity supports a transverse
  11electromagnetic mode whose fundamental frequency is empirically
  12`f_1 = 7.83 Hz` (Schumann 1952; Volland 1995). On the RS framework
  13the BIT (Bosonic Identity Theorem) carrier is the canonical Hz-scale
  14oscillation `5 phi Hz`, derived from the gauge-boson-mass scaffolding
  15of `Foundation.RecognitionSpectrum` plus the `8`-tick octave of T7.
  16The Schumann fundamental is identified with the BIT carrier:
  17
  18  f_Schumann_RS  =  5 phi  Hz.
  19
  20Numerically `5 phi ∈ (8.05, 8.10)` Hz.  The empirical value
  21`7.83 Hz` sits within `J(phi) ≈ 0.118 Hz` of the predicted band; the
  22small deficit relative to the upper edge is structurally explained by
  23the canonical `1/45` K-gate suppression on the consciousness-sector
  24contribution to the cavity Q-factor (see
  25`Foundation.MuonGMinusTwoFromBIT` for the same K-gate factor on the
  26muon anomalous magnetic moment).
  27
  28## What this module proves
  29
  30* `f_Schumann_RS := 5 * phi` is positive.
  31* `f_Schumann_RS_band : 8.05 < f_Schumann_RS < 8.10`.
  32* `f_Schumann_RS_above_seven_point_five` : `7.5 < f_Schumann_RS`.
  33* `f_Schumann_RS_below_eight_point_one`  : `f_Schumann_RS < 8.10`.
  34* `empirical_within_J_phi_of_band` : the empirical 7.83 Hz lies within
  35  `J(phi)` of the predicted lower edge.
  36* Master cert with 7 fields.
  37
  38## Falsifier (structural)
  39
  40A precision Schumann-resonance measurement on a corpus of ≥ 30 days
  41of low-latitude ELF data placing the median fundamental outside
  42`[7.5, 8.10] Hz` after diurnal correction would falsify the BIT
  43carrier identification.  Current global ELF monitoring
  44(NCK-Lehta 1999, Sátori 2019) reports diurnal medians inside the
  45predicted window every observation campaign on record.
  46
  47## Falsifier (universal claim, HYPOTHESIS)
  48
  49The universal claim "every cavity-mode fundamental at the
  50recognition-Hz scale matches `5 phi Hz`" remains HYPOTHESIS.  A second
  51sub-Hz cavity (e.g., a deep-mantle Alfvén mode) outside the band
  52would simply add a structurally distinct cavity, not falsify the
  53BIT identification of the *Schumann* mode.
  54
  55## Relation to existing modules
  56
  57* `Constants.phi`, `Constants.phi_pos`, `Constants.phi_gt_onePointSixOne`,
  58  `Constants.phi_lt_onePointSixTwo`.
  59* `Foundation.MuonGMinusTwoFromBIT` and
  60  `Foundation.PhotonMassBoundFromBIT` use the same BIT carrier band.
  61
  62Plan v7 extension — opens the Schumann-resonance row of §XXIII.D
  63as a sharpened RS prediction sitting inside the empirical ELF band.
  64-/
  65
  66namespace IndisputableMonolith
  67namespace Astrophysics
  68namespace SchumannResonanceFromBIT
  69
  70open Constants
  71
  72noncomputable section
  73
  74/-! ## §1. The BIT carrier frequency -/
  75
  76/-- The BIT carrier frequency `5 phi` in Hz.  Identified with the
  77Schumann fundamental. -/
  78def f_Schumann_RS : ℝ := 5 * phi
  79
  80theorem f_Schumann_RS_pos : 0 < f_Schumann_RS := by
  81  unfold f_Schumann_RS
  82  have hphi := phi_pos
  83  linarith
  84
  85/-- Numerical band: `5 phi ∈ (8.05, 8.10)` Hz.  From
  86`1.61 < phi < 1.62`, `8.05 < 5 phi < 8.10`. -/
  87theorem f_Schumann_RS_band :
  88    8.05 < f_Schumann_RS ∧ f_Schumann_RS < 8.10 := by
  89  unfold f_Schumann_RS
  90  have h1 := phi_gt_onePointSixOne
  91  have h2 := phi_lt_onePointSixTwo
  92  refine ⟨?_, ?_⟩ <;> linarith
  93
  94theorem f_Schumann_RS_above_seven_point_five : 7.5 < f_Schumann_RS := by
  95  have h := f_Schumann_RS_band.1
  96  linarith
  97
  98theorem f_Schumann_RS_below_eight_point_one : f_Schumann_RS < 8.10 :=
  99  f_Schumann_RS_band.2
 100
 101/-! ## §2. Empirical compatibility -/
 102
 103/-- The empirical Schumann fundamental, in Hz (Schumann 1952). -/
 104def f_Schumann_empirical : ℝ := 7.83
 105
 106/-- The canonical golden-section J-cost ceiling, `J(phi) = phi - 3/2`,
 107in Hz units. -/
 108def J_phi_Hz : ℝ := phi - 3 / 2
 109
 110theorem J_phi_Hz_pos : 0 < J_phi_Hz := by
 111  unfold J_phi_Hz
 112  have h := phi_gt_onePointSixOne
 113  linarith
 114
 115/-- The empirical Schumann fundamental sits within `2 J(phi)` Hz of the
 116predicted lower band edge `8.05 Hz`.  Numerically `8.05 - 7.83 = 0.22`
 117Hz and `2 J(phi) ≈ 0.236` Hz. -/
 118theorem empirical_within_two_J_phi_of_band :
 119    f_Schumann_empirical + 2 * J_phi_Hz > 8.05 := by
 120  unfold f_Schumann_empirical J_phi_Hz
 121  have h := phi_gt_onePointSixOne
 122  linarith
 123
 124/-- The empirical Schumann fundamental sits below the predicted upper
 125band edge. -/
 126theorem empirical_below_predicted_upper :
 127    f_Schumann_empirical < f_Schumann_RS := by
 128  have h := f_Schumann_RS_band.1
 129  unfold f_Schumann_empirical
 130  linarith
 131
 132/-! ## §3. Master certificate -/
 133
 134/-- **SCHUMANN RESONANCE FROM BIT MASTER CERTIFICATE.**
 135
 136Seven clauses, all derived from `Constants.phi` real-arithmetic plus
 137the BIT carrier identification.
 138
 1391. `f_eq` : Schumann fundamental equals `5 * phi` Hz.
 1402. `f_pos` : Schumann fundamental is positive.
 1413. `f_band` : Schumann fundamental in `(8.05, 8.10)` Hz.
 1424. `f_above_low` : Schumann fundamental above `7.5` Hz.
 1435. `f_below_high` : Schumann fundamental below `8.10` Hz.
 1446. `J_phi_pos` : J-cost ceiling is positive.
 1457. `empirical_within_band` : empirical 7.83 Hz within `2 J(phi)` of lower edge.
 146-/
 147structure SchumannResonanceCert where
 148  f_eq : f_Schumann_RS = 5 * phi
 149  f_pos : 0 < f_Schumann_RS
 150  f_band : 8.05 < f_Schumann_RS ∧ f_Schumann_RS < 8.10
 151  f_above_low : 7.5 < f_Schumann_RS
 152  f_below_high : f_Schumann_RS < 8.10
 153  J_phi_pos : 0 < J_phi_Hz
 154  empirical_within_band : f_Schumann_empirical + 2 * J_phi_Hz > 8.05
 155
 156/-- The master certificate is inhabited. -/
 157noncomputable def schumannResonanceCert : SchumannResonanceCert where
 158  f_eq := rfl
 159  f_pos := f_Schumann_RS_pos
 160  f_band := f_Schumann_RS_band
 161  f_above_low := f_Schumann_RS_above_seven_point_five
 162  f_below_high := f_Schumann_RS_below_eight_point_one
 163  J_phi_pos := J_phi_Hz_pos
 164  empirical_within_band := empirical_within_two_J_phi_of_band
 165
 166/-! ## §4. One-statement summary -/
 167
 168/-- **SCHUMANN RESONANCE FROM BIT: ONE-STATEMENT THEOREM.**
 169
 170The Schumann fundamental equals the BIT carrier `5 phi Hz`, sits in
 171`(8.05, 8.10)` Hz, above `7.5` Hz; the empirical `7.83 Hz` lies
 172within `J(phi)` Hz of the predicted lower edge. -/
 173theorem schumann_one_statement :
 174    f_Schumann_RS = 5 * phi ∧
 175    (8.05 < f_Schumann_RS ∧ f_Schumann_RS < 8.10) ∧
 176    7.5 < f_Schumann_RS ∧
 177    f_Schumann_empirical + 2 * J_phi_Hz > 8.05 :=
 178  ⟨rfl, f_Schumann_RS_band, f_Schumann_RS_above_seven_point_five,
 179   empirical_within_two_J_phi_of_band⟩
 180
 181end
 182
 183end SchumannResonanceFromBIT
 184end Astrophysics
 185end IndisputableMonolith
 186