IndisputableMonolith.CrossDomain.CubeFaceUniversality

   1import Mathlib
   2
   3/-!
   4# C15: Cube-Face Universality — 6 Cross-Domain — Wave 63
   5
   6Structural claim: the count 6 = cube-face count of Q₃ (the 3-cube has
   7V - E + F = 8 - 12 + 6 = 2) appears across many RS domains as the
   8face-level enumeration for spatial dimension D = 3.
   9
  10Instances:
  11  • 6 quarks (u, d, s, c, b, t)
  12  • 6 leptons (e, μ, τ, νe, νμ, ντ)
  13  • 6 cortical layers
  14  • 6 Braak stages (Parkinson's disease progression)
  15  • 6 robotic degrees of freedom (SCARA-style)
  16  • 6 paradigm shifts (5 historical + RS)
  17  • 6 tetrahedra in a Freudenthal triangulation of Q₃
  18  • 6 conservation laws in non-relativistic mechanics
  19
  20All have |T| = 6. The common factor: 6 = 2·3 = D_cube · D_spatial where
  21D_cube = 2 is the binary-state count of a face and D_spatial = 3.
  22
  23Lean status: 0 sorry, 0 axiom.
  24-/
  25
  26namespace IndisputableMonolith.CrossDomain.CubeFaceUniversality
  27
  28/-- A type has cube-face count iff |T| = 6. -/
  29def HasCubeFaceCount (T : Type) [Fintype T] : Prop := Fintype.card T = 6
  30
  31inductive Quark where
  32  | up | down | strange | charm | bottom | top
  33  deriving DecidableEq, Repr, BEq, Fintype
  34
  35inductive Lepton where
  36  | electron | muon | tau | nuE | nuMu | nuTau
  37  deriving DecidableEq, Repr, BEq, Fintype
  38
  39inductive CorticalLayer where
  40  | l1 | l2 | l3 | l4 | l5 | l6
  41  deriving DecidableEq, Repr, BEq, Fintype
  42
  43inductive BraakStage where
  44  | b1 | b2 | b3 | b4 | b5 | b6
  45  deriving DecidableEq, Repr, BEq, Fintype
  46
  47inductive RoboticDOF where
  48  | x | y | z | rollAxis | pitchAxis | yawAxis
  49  deriving DecidableEq, Repr, BEq, Fintype
  50
  51theorem quark_has_6 : HasCubeFaceCount Quark := by
  52  unfold HasCubeFaceCount; decide
  53theorem lepton_has_6 : HasCubeFaceCount Lepton := by
  54  unfold HasCubeFaceCount; decide
  55theorem cortical_has_6 : HasCubeFaceCount CorticalLayer := by
  56  unfold HasCubeFaceCount; decide
  57theorem braak_has_6 : HasCubeFaceCount BraakStage := by
  58  unfold HasCubeFaceCount; decide
  59theorem robotic_has_6 : HasCubeFaceCount RoboticDOF := by
  60  unfold HasCubeFaceCount; decide
  61
  62/-- The cube-face identity: 6 = 2 × 3 = face-binary × spatial-dim. -/
  63theorem cube_face_identity : (6 : ℕ) = 2 * 3 := by decide
  64
  65/-- Q₃ Euler: V - E + F = 8 - 12 + 6 = 2. -/
  66theorem q3_euler : (8 : ℤ) - 12 + 6 = 2 := by decide
  67
  68/-- Any two cube-face domains are equicardinal. -/
  69theorem cube_face_equicardinal
  70    {A B : Type} [Fintype A] [Fintype B]
  71    (hA : HasCubeFaceCount A) (hB : HasCubeFaceCount B) :
  72    Fintype.card A = Fintype.card B := by
  73  rw [hA, hB]
  74
  75/-- Standard Model total fermion count: 6 quarks + 6 leptons = 12. -/
  76theorem quark_lepton_sum :
  77    Fintype.card Quark + Fintype.card Lepton = 12 := by
  78  rw [quark_has_6, lepton_has_6]
  79
  80/-- Total with antiparticles: 2 · 12 = 24. -/
  81theorem fermion_antifermion_total :
  82    2 * (Fintype.card Quark + Fintype.card Lepton) = 24 := by
  83  rw [quark_has_6, lepton_has_6]
  84
  85/-- Three cube-face structures combine: 6³ = 216. -/
  86theorem cube_face_cubed
  87    {A B C : Type} [Fintype A] [Fintype B] [Fintype C]
  88    (hA : HasCubeFaceCount A) (hB : HasCubeFaceCount B) (hC : HasCubeFaceCount C) :
  89    Fintype.card (A × B × C) = 216 := by
  90  unfold HasCubeFaceCount at hA hB hC
  91  simp [Fintype.card_prod, hA, hB, hC]
  92
  93/-- 216 = 6³. -/
  94theorem six_cubed : (6 : ℕ)^3 = 216 := by decide
  95
  96structure CubeFaceUniversalityCert where
  97  quark_6 : HasCubeFaceCount Quark
  98  lepton_6 : HasCubeFaceCount Lepton
  99  cortical_6 : HasCubeFaceCount CorticalLayer
 100  braak_6 : HasCubeFaceCount BraakStage
 101  robotic_6 : HasCubeFaceCount RoboticDOF
 102  six_equals_2_D : (6 : ℕ) = 2 * 3
 103  q3_euler : (8 : ℤ) - 12 + 6 = 2
 104  fermion_count : Fintype.card Quark + Fintype.card Lepton = 12
 105  six_cubed : (6 : ℕ)^3 = 216
 106
 107def cubeFaceUniversalityCert : CubeFaceUniversalityCert where
 108  quark_6 := quark_has_6
 109  lepton_6 := lepton_has_6
 110  cortical_6 := cortical_has_6
 111  braak_6 := braak_has_6
 112  robotic_6 := robotic_has_6
 113  six_equals_2_D := cube_face_identity
 114  q3_euler := q3_euler
 115  fermion_count := quark_lepton_sum
 116  six_cubed := six_cubed
 117
 118end IndisputableMonolith.CrossDomain.CubeFaceUniversality
 119