IndisputableMonolith.Verification.AlphaCorrectionAnalysis

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Constants.Alpha
   4import IndisputableMonolith.Constants.AlphaDerivation
   5import IndisputableMonolith.Constants.ExternalAnchors
   6import IndisputableMonolith.Numerics.Interval.AlphaBounds
   7
   8/-!
   9# Alpha Correction Term: First-Principles Analysis
  10
  11This module characterizes the ~0.001 correction needed to close the 8 ppm
  12gap between α⁻¹_RS and α⁻¹_CODATA, and evaluates candidate correction
  13terms from the cube geometry.
  14
  15## The Gap
  16
  17α⁻¹_RS = 4π·11 − w₈·ln φ + 103/(102π⁵) ≈ 137.0349
  18α⁻¹_CODATA = 137.035999206(21)
  19
  20Required correction: δ₂ ≈ +0.00110 (to be added to α⁻¹_RS)
  21
  22## Structural Constraints on δ₂
  23
  24Any admissible correction must:
  25(A1) Be expressible in terms of counting-layer integers and transcendentals (π, φ).
  26(A2) Be small relative to the existing terms (~10⁻³ vs ~10² for seed).
  27(A3) Not introduce new free parameters.
  28(A4) Have a combinatorial interpretation within the cube geometry.
  29
  30## Candidate Evaluation
  31
  32We evaluate several candidate expressions and their numerical proximity
  33to the required correction.
  34-/
  35
  36namespace IndisputableMonolith
  37namespace Verification
  38namespace AlphaCorrectionAnalysis
  39
  40open Constants
  41open Constants.AlphaDerivation
  42open Constants.ExternalAnchors
  43
  44noncomputable section
  45
  46/-! ## The Required Correction -/
  47
  48/-- The exact required correction to match CODATA. -/
  49def required_correction : ℝ := alpha_inv_CODATA - alphaInv
  50
  51/-- The required correction is positive once `alphaInv < alpha_inv_CODATA` is
  52established for the chosen α closure model. -/
  53theorem correction_positive
  54    (hα : alphaInv < alpha_inv_CODATA) : 0 < required_correction := by
  55  unfold required_correction
  56  linarith
  57
  58/-- Current interval bounds imply a narrow correction window around zero. -/
  59theorem correction_window_from_current_bounds :
  60    (-0.004 : ℝ) < required_correction ∧ required_correction < (0.006 : ℝ) := by
  61  simp only [required_correction, alpha_inv_CODATA]
  62  constructor
  63  · have hα := Numerics.alphaInv_lt
  64    linarith
  65  · have hα := Numerics.alphaInv_gt
  66    linarith
  67
  68/-- Sign lemma: once `alphaInv < alpha_inv_CODATA` is established, positivity follows immediately. -/
  69theorem correction_positive_of_alphaInv_lt
  70    (h : alphaInv < alpha_inv_CODATA) :
  71    0 < required_correction := by
  72  unfold required_correction
  73  linarith
  74
  75/-! ## Candidate Correction Terms -/
  76
  77/-- Candidate 1: 1/(F × W × π²) = 1/(102π²).
  78    Uses the same seam_denominator as the existing curvature term,
  79    but with π² instead of π⁵.
  80    Interpretation: "lower-order curvature correction from face × wallpaper channels." -/
  81def candidate_1 : ℝ := 1 / (102 * Real.pi ^ 2)
  82
  83/-- Candidate 2: 1/(V × seam_numerator) = 1/(8 × 103) = 1/824.
  84    Uses vertices × seam numerator.
  85    Interpretation: "vertex-level correction to the curvature seam." -/
  86def candidate_2 : ℝ := 1 / (8 * 103 : ℝ)
  87
  88/-- Candidate 3: A/(seam_denominator × (π² − 1)).
  89    Uses the "reduced π²" factor.
  90    Interpretation: "active-edge coupling through the curvature channels,
  91    with the (π² − 1) factor accounting for the non-spherical correction." -/
  92def candidate_3 : ℝ := 1 / (102 * (Real.pi ^ 2 - 1))
  93
  94/-- Candidate 4: (ln φ)² / (2 × seam_denominator).
  95    A second-order gap correction.
  96    Interpretation: "second-order self-similar coupling through curvature channels." -/
  97def candidate_4 : ℝ := (Real.log phi) ^ 2 / (2 * 102)
  98
  99/-! ## Numerical Evaluation of Candidates -/
 100
 101/-- Candidate 1: 1/(102π²) ≈ 0.000994.
 102    Deviation from target: ~10% low.
 103    102 × π² ≈ 1006.08, so 1/1006.08 ≈ 0.000994. -/
 104theorem candidate_1_bounds :
 105    0.000993 < candidate_1 ∧ candidate_1 < 0.000996 := by
 106  constructor
 107  · unfold candidate_1
 108    have hden_pos : 0 < (102 : ℝ) * Real.pi ^ 2 := by positivity
 109    have hpi_hi : Real.pi < (3.141593 : ℝ) := Real.pi_lt_d6
 110    have hpi2_hi : Real.pi ^ 2 < (3.141593 : ℝ) ^ 2 := by
 111      nlinarith [Real.pi_pos, hpi_hi]
 112    have hden_hi : (102 : ℝ) * Real.pi ^ 2 < (1007.049 : ℝ) := by
 113      have hscale : (102 : ℝ) * Real.pi ^ 2 < (102 : ℝ) * (3.141593 : ℝ) ^ 2 := by
 114        nlinarith [hpi2_hi]
 115      have hnum : (102 : ℝ) * (3.141593 : ℝ) ^ 2 < (1007.049 : ℝ) := by
 116        norm_num
 117      exact lt_trans hscale hnum
 118    have hmul : (0.000993 : ℝ) * ((102 : ℝ) * Real.pi ^ 2) < 1 := by
 119      have hscale : (0.000993 : ℝ) * ((102 : ℝ) * Real.pi ^ 2) <
 120          (0.000993 : ℝ) * (1007.049 : ℝ) := by
 121        exact mul_lt_mul_of_pos_left hden_hi (by norm_num)
 122      have hnum : (0.000993 : ℝ) * (1007.049 : ℝ) < 1 := by
 123        norm_num
 124      exact lt_trans hscale hnum
 125    exact (lt_div_iff₀ hden_pos).2 hmul
 126  · unfold candidate_1
 127    have hden_pos : 0 < (102 : ℝ) * Real.pi ^ 2 := by positivity
 128    have hpi_lo : (3.141592 : ℝ) < Real.pi := Real.pi_gt_d6
 129    have hpi2_lo : (3.141592 : ℝ) ^ 2 < Real.pi ^ 2 := by
 130      nlinarith [Real.pi_pos, hpi_lo]
 131    have hden_lo : (1004.017 : ℝ) < (102 : ℝ) * Real.pi ^ 2 := by
 132      have hnum : (1004.017 : ℝ) < (102 : ℝ) * (3.141592 : ℝ) ^ 2 := by
 133        norm_num
 134      have hscale : (102 : ℝ) * (3.141592 : ℝ) ^ 2 < (102 : ℝ) * Real.pi ^ 2 := by
 135        nlinarith [hpi2_lo]
 136      exact lt_trans hnum hscale
 137    have hmul : (1 : ℝ) < (0.000996 : ℝ) * ((102 : ℝ) * Real.pi ^ 2) := by
 138      have hscale : (0.000996 : ℝ) * (1004.017 : ℝ) <
 139          (0.000996 : ℝ) * ((102 : ℝ) * Real.pi ^ 2) := by
 140        exact mul_lt_mul_of_pos_left hden_lo (by norm_num)
 141      have hnum : (1 : ℝ) < (0.000996 : ℝ) * (1004.017 : ℝ) := by
 142        norm_num
 143      exact lt_trans hnum hscale
 144    exact (div_lt_iff₀ hden_pos).2 hmul
 145
 146/-- Candidate 2: 1/824 ≈ 0.001214.
 147    Deviation from target: ~10% high. -/
 148theorem candidate_2_value : candidate_2 = 1 / 824 := by
 149  simp [candidate_2]
 150  norm_num
 151
 152/-! ## Structural Observation
 153
 154The required correction δ₂ ≈ 0.00110 lies BETWEEN:
 155  - Candidate 1: 1/(102π²) ≈ 0.000994  (10% low)
 156  - Candidate 2: 1/824 ≈ 0.001214      (10% high)
 157
 158A weighted combination could close the gap exactly, but that would
 159introduce a free parameter. The honest status is:
 160
 1611. The correction has magnitude ~1/(F×W×π²), which is natural as a
 162   "next-order curvature term" in the same series as 103/(102π⁵).
 1632. No single counting-layer expression hits the target exactly.
 1643. The gap may involve a term not yet identified in the cube geometry,
 165   or may require the full QED vacuum polarization computation.
 166-/
 167
 168/-- Summary structure for the correction analysis. -/
 169structure CorrectionAnalysis where
 170  /-- The correction is positive (RS underpredicts) -/
 171  sign_positive : String := "RS underpredicts CODATA by ~0.001 (8 ppm)"
 172  /-- Order of magnitude is ~1/(F×W×π²) -/
 173  magnitude_natural : String := "~10⁻³, consistent with next-order curvature term"
 174  /-- No single counting-layer expression is exact -/
 175  no_exact_match : String := "Candidates bracket the target (0.000994 to 0.001214)"
 176  /-- Three resolution paths remain -/
 177  resolution_paths : List String :=
 178    [ "Path A: Identify exact higher-order geometric term from cube topology"
 179    , "Path B: Show RS computes α at a specific recognition scale, not Q²=0"
 180    , "Path C: Compute QED VP correction between RS scale and CODATA extraction" ]
 181
 182/-- The current correction analysis. -/
 183def analysis : CorrectionAnalysis := {}
 184
 185end
 186
 187end AlphaCorrectionAnalysis
 188end Verification
 189end IndisputableMonolith
 190