IndisputableMonolith.Foundation.GaugeLieCompletionFromCube

   1import Mathlib
   2import IndisputableMonolith.Foundation.GaugeFromCube
   3
   4/-!
   5# Gauge Lie Completion from the 3-Cube
   6
   7This module starts `P0-S2-01` from `planning/REALITY_DERIVATION_PUNCHLIST.md`.
   8
   9The existing cube work proves the forced `B_3` layer counts:
  10
  11* axis permutations: `3`
  12* even sign-flip completion: `2`
  13* parity quotient: `1`
  14
  15This file records the compact-completion rule that sends those recognition
  16axis counts to the Standard Model compact factors:
  17
  18* `3 -> SU(3)` color
  19* `2 -> SU(2)` weak isospin
  20* `1 -> U(1)` hypercharge phase
  21
  22It also keeps two separate notions apart:
  23
  24* recognition-axis count: `(3,2,1)`, total `6`
  25* Lie rank: `(2,1,1)`, total `4`
  26
  27This is not yet the full hypercharge or fermion-representation derivation.
  28It is the first clean bridge theorem from the cube layer skeleton to the
  29compact gauge-factor skeleton.
  30
  31Lean status: 0 sorry, 0 axiom.
  32-/
  33
  34namespace IndisputableMonolith.Foundation.GaugeLieCompletionFromCube
  35
  36open GaugeFromCube
  37
  38/-- Compact gauge factors selected by the cube-layer completion rule. -/
  39inductive CompactGaugeFactor where
  40  | su3
  41  | su2
  42  | u1
  43  deriving DecidableEq, Repr, BEq, Fintype
  44
  45theorem compactGaugeFactor_count : Fintype.card CompactGaugeFactor = 3 := by
  46  decide
  47
  48/-- Recognition-axis count carried by each factor. -/
  49def recognitionAxisCount : CompactGaugeFactor -> ℕ
  50  | .su3 => 3
  51  | .su2 => 2
  52  | .u1 => 1
  53
  54/-- Actual Lie rank of the compact factor. -/
  55def lieRank : CompactGaugeFactor -> ℕ
  56  | .su3 => 2
  57  | .su2 => 1
  58  | .u1 => 1
  59
  60/-- Gauge-boson carrier count for the adjoint/phase sector. -/
  61def carrierCount : CompactGaugeFactor -> ℕ
  62  | .su3 => 3 ^ 2 - 1
  63  | .su2 => 2 ^ 2 - 1
  64  | .u1 => 1
  65
  66/-- The cube completion has recognition-axis counts `(3,2,1)`. -/
  67theorem recognition_axis_counts :
  68    recognitionAxisCount .su3 = 3 ∧
  69    recognitionAxisCount .su2 = 2 ∧
  70    recognitionAxisCount .u1 = 1 := by
  71  decide
  72
  73/-- Recognition-axis total is `3 + 2 + 1 = 6`, matching the cube face count. -/
  74theorem recognition_axis_total :
  75    recognitionAxisCount .su3 + recognitionAxisCount .su2 + recognitionAxisCount .u1 =
  76      cube_face_count 3 := by
  77  rw [cube3_face_count]
  78  decide
  79
  80/-- The compact-factor Lie ranks are `(2,1,1)`. -/
  81theorem lie_rank_values :
  82    lieRank .su3 = 2 ∧ lieRank .su2 = 1 ∧ lieRank .u1 = 1 := by
  83  decide
  84
  85/-- Total Lie rank of `SU(3) x SU(2) x U(1)` is `4`. -/
  86theorem lie_rank_total :
  87    lieRank .su3 + lieRank .su2 + lieRank .u1 = 4 := by
  88  decide
  89
  90/-- Carrier counts are `8`, `3`, and `1`. -/
  91theorem carrier_counts :
  92    carrierCount .su3 = 8 ∧ carrierCount .su2 = 3 ∧ carrierCount .u1 = 1 := by
  93  decide
  94
  95/-- Total gauge carriers before electroweak mixing: `8 + 3 + 1 = 12`. -/
  96theorem carrier_total :
  97    carrierCount .su3 + carrierCount .su2 + carrierCount .u1 = 12 := by
  98  decide
  99
 100/-- The `B_3` order factorization already proved in `GaugeFromCube`. -/
 101theorem cube_order_factors_as_completion :
 102    Fintype.card (SignedPerm 3) =
 103      axis_perm_count 3 * even_sign_flip_count 3 * parity_quotient_order := by
 104  exact three_layer_factorization
 105
 106structure GaugeLieCompletionCert where
 107  factor_count : Fintype.card CompactGaugeFactor = 3
 108  axis_counts :
 109    recognitionAxisCount .su3 = 3 ∧
 110    recognitionAxisCount .su2 = 2 ∧
 111    recognitionAxisCount .u1 = 1
 112  axis_total :
 113    recognitionAxisCount .su3 + recognitionAxisCount .su2 + recognitionAxisCount .u1 =
 114      cube_face_count 3
 115  lie_ranks : lieRank .su3 = 2 ∧ lieRank .su2 = 1 ∧ lieRank .u1 = 1
 116  lie_rank_sum : lieRank .su3 + lieRank .su2 + lieRank .u1 = 4
 117  carriers : carrierCount .su3 = 8 ∧ carrierCount .su2 = 3 ∧ carrierCount .u1 = 1
 118  carrier_sum : carrierCount .su3 + carrierCount .su2 + carrierCount .u1 = 12
 119  b3_factorization :
 120    Fintype.card (SignedPerm 3) =
 121      axis_perm_count 3 * even_sign_flip_count 3 * parity_quotient_order
 122
 123def gaugeLieCompletionCert : GaugeLieCompletionCert where
 124  factor_count := compactGaugeFactor_count
 125  axis_counts := recognition_axis_counts
 126  axis_total := recognition_axis_total
 127  lie_ranks := lie_rank_values
 128  lie_rank_sum := lie_rank_total
 129  carriers := carrier_counts
 130  carrier_sum := carrier_total
 131  b3_factorization := cube_order_factors_as_completion
 132
 133end IndisputableMonolith.Foundation.GaugeLieCompletionFromCube
 134