IndisputableMonolith.Physics.MassResidueNoGo

   1import Mathlib
   2import IndisputableMonolith.RSBridge.Anchor
   3import IndisputableMonolith.Physics.ElectronMass.Necessity
   4
   5/-!
   6# No-go: literal SM RG residue cannot equal the geometric band value
   7
   8This module captures (in Lean) the core “no cheating” conclusion from the repo’s
   9explicit RG-evaluator audit:
  10
  11- The closed-form band value for charged leptons at the anchor is
  12  `gap 1332 ≈ 13.95`.
  13- A literal Standard-Model mass-anomalous-dimension integral over any reasonable
  14  interval produces a *small* dimensionless residue (order `10⁻2`–`10⁻1` for leptons).
  15
  16Therefore, **if** you interpret the paper’s `f_i^{exp}` as a *small* quantity (e.g. `|f| ≤ 0.1`),
  17it cannot satisfy the anchor identity `f = gap 1332` to tolerance `1e-6` (or any small tolerance).
  18
  19This theorem does **not** assume a particular RG kernel; it is a purely numeric separation:
  20small numbers cannot equal `gap 1332`, because `gap 1332` is provably > 13.953 in this repo.
  21-/
  22
  23namespace IndisputableMonolith.Physics
  24namespace MassResidueNoGo
  25
  26open IndisputableMonolith.RSBridge
  27open IndisputableMonolith.Physics.ElectronMass.Necessity
  28
  29theorem gap_1332_gt_13_953 : (13.953 : ℝ) < gap 1332 :=
  30  (gap_1332_bounds).1
  31
  32/-!
  33## Main no-go inequality
  34
  35If a would-be “experimental residue” `x` is small (e.g. `|x| ≤ 0.1`), then it is **far**
  36from the lepton band value `gap 1332`, hence it cannot match within the paper’s tolerance.
  37-/
  38theorem abs_sub_gap1332_gt_ten {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
  39    (10 : ℝ) < |x - gap 1332| := by
  40  have hx' : (-0.1 : ℝ) ≤ x ∧ x ≤ (0.1 : ℝ) := abs_le.mp hx
  41  have hx_upper : x ≤ (0.1 : ℝ) := hx'.2
  42  have hg : (13.953 : ℝ) < gap 1332 := gap_1332_gt_13_953
  43
  44  -- Bound y := x - gap 1332 by a strict negative number.
  45  have hy : x - gap 1332 < (-10 : ℝ) := by
  46    have h1 : x - gap 1332 < (0.1 : ℝ) - (13.953 : ℝ) := by
  47      nlinarith [hx_upper, hg]
  48    have hnum : (0.1 : ℝ) - (13.953 : ℝ) < (-10 : ℝ) := by norm_num
  49    exact lt_trans h1 hnum
  50  have hy_neg : x - gap 1332 < (0 : ℝ) := lt_trans hy (by norm_num)
  51
  52  have hpos : (10 : ℝ) < -(x - gap 1332) := by linarith
  53  simpa [abs_of_neg hy_neg] using hpos
  54
  55/-!
  56### Immediate corollaries
  57-/
  58
  59theorem not_within_micro_tolerance {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
  60    ¬ (|x - gap 1332| < (1e-6 : ℝ)) := by
  61  intro h
  62  have h10 : (10 : ℝ) < |x - gap 1332| := abs_sub_gap1332_gt_ten (x := x) hx
  63  have h_lt10 : |x - gap 1332| < (10 : ℝ) := lt_trans h (by norm_num)
  64  exact (lt_irrefl (10 : ℝ)) (lt_trans h10 h_lt10)
  65
  66theorem small_x_ne_gap1332 {x : ℝ} (hx : |x| ≤ (0.1 : ℝ)) :
  67    x ≠ gap 1332 := by
  68  intro h
  69  have h10 : (10 : ℝ) < |x - gap 1332| := abs_sub_gap1332_gt_ten (x := x) hx
  70  have h10' : (10 : ℝ) < (0 : ℝ) := by simpa [h] using h10
  71  have hcontra : ¬ ((10 : ℝ) < (0 : ℝ)) := by norm_num
  72  exact hcontra h10'
  73
  74end MassResidueNoGo
  75end IndisputableMonolith.Physics
  76