IndisputableMonolith.Verification.WallpaperEndogenousBridge

   1import Mathlib
   2import IndisputableMonolith.Constants.AlphaDerivation
   3
   4/-!
   5# Wallpaper Endogenous Bridge (Pass 2)
   6
   7This module provides an explicit bridge from cube combinatorics to the
   8crystallographic constant `W = 17`.
   9
  10## Goal
  11
  12The current framework imports `wallpaper_groups = 17` as a classical mathematical
  13fact (Fedorov 1891). This file does **not** re-prove wallpaper classification,
  14but it formalizes an endogenous RS candidate:
  15
  16`W_endogenous(D) := E_passive(D) + F(D)`.
  17
  18For `D = 3`, this gives:
  19- `E_passive = 11`,
  20- `F = 6`,
  21- `W_endogenous = 17`.
  22
  23So the counting-layer identity `11 + 6 = 17` is now explicit and machine-checked,
  24and at `D = 3` it matches the imported `wallpaper_groups`.
  25
  26This is a bridge step toward full endogeneity of `W`.
  27-/
  28
  29namespace IndisputableMonolith
  30namespace Verification
  31namespace WallpaperEndogenousBridge
  32
  33open Constants.AlphaDerivation
  34
  35/-- Endogenous candidate for the `W`-count from RS cube combinatorics. -/
  36def W_endogenous (d : ℕ) : ℕ :=
  37  passive_field_edges d + cube_faces d
  38
  39/-- Expanded closed form:
  40`W_endogenous(d) = d * 2^(d-1) - 1 + 2d`. -/
  41theorem W_endogenous_formula (d : ℕ) :
  42    W_endogenous d = (cube_edges d - active_edges_per_tick) + cube_faces d := by
  43  rfl
  44
  45/-- At `D=3`, the endogenous candidate is exactly 17. -/
  46theorem W_endogenous_at_D3 : W_endogenous D = 17 := by
  47  native_decide
  48
  49/-- At `D=3`, the endogenous candidate matches the imported wallpaper constant. -/
  50theorem W_endogenous_matches_wallpaper_groups :
  51    W_endogenous D = wallpaper_groups := by
  52  native_decide
  53
  54/-- Component decomposition at `D=3`: `11 + 6 = 17`. -/
  55theorem decomposition_at_D3 :
  56    passive_field_edges D = 11 ∧ cube_faces D = 6 ∧ W_endogenous D = 17 := by
  57  native_decide
  58
  59/-- Finite computational scan: up to dimension 64, only `D=3` gives 17. -/
  60def unique17ScanUpTo64 : Bool :=
  61  (List.range 65).all (fun d => decide (W_endogenous d = 17 ↔ d = 3))
  62
  63theorem unique17ScanUpTo64_true : unique17ScanUpTo64 = true := by
  64  native_decide
  65
  66/-- Endogenous candidate at `D=3` as a named constant. -/
  67def W_from_cube : ℕ := W_endogenous D
  68
  69theorem W_from_cube_eq_17 : W_from_cube = 17 := by
  70  simpa [W_from_cube] using W_endogenous_at_D3
  71
  72theorem W_from_cube_eq_wallpaper_groups : W_from_cube = wallpaper_groups := by
  73  simpa [W_from_cube] using W_endogenous_matches_wallpaper_groups
  74
  75/-- Generator-level slot closure at `D=3`:
  76the wallpaper slot is exactly the endogenous cube formula `E_passive + F`. -/
  77theorem wallpaper_slot_iff_endogenous_formula (w : ℕ) :
  78    (w = wallpaper_groups) ↔ (w = passive_field_edges D + cube_faces D) := by
  79  constructor
  80  · intro hw
  81    calc
  82      w = wallpaper_groups := hw
  83      _ = W_from_cube := W_from_cube_eq_wallpaper_groups.symm
  84      _ = passive_field_edges D + cube_faces D := by
  85            simp [W_from_cube, W_endogenous]
  86  · intro hw
  87    calc
  88      w = passive_field_edges D + cube_faces D := hw
  89      _ = W_from_cube := by
  90            simp [W_from_cube, W_endogenous]
  91      _ = wallpaper_groups := W_from_cube_eq_wallpaper_groups
  92
  93/-- Uniqueness form: any `w` satisfying the endogenous wallpaper formula
  94at `D=3` is forced to the imported wallpaper constant. -/
  95theorem wallpaper_slot_unique_from_endogenous_formula (w : ℕ)
  96    (hw : w = passive_field_edges D + cube_faces D) :
  97    w = wallpaper_groups := by
  98  exact (wallpaper_slot_iff_endogenous_formula w).2 hw
  99
 100/-- Endogenous closure package for the counting-layer wallpaper slot:
 101    the cube-derived value is exactly 17 and matches the imported constant. -/
 102theorem endogenous_wallpaper_bridge_complete :
 103    W_from_cube = 17 ∧
 104    W_from_cube = wallpaper_groups ∧
 105    (∀ w : ℕ, (w = wallpaper_groups) ↔ (w = passive_field_edges D + cube_faces D)) := by
 106  refine ⟨W_from_cube_eq_17, W_from_cube_eq_wallpaper_groups, ?_⟩
 107  intro w
 108  exact wallpaper_slot_iff_endogenous_formula w
 109
 110end WallpaperEndogenousBridge
 111end Verification
 112end IndisputableMonolith
 113