IndisputableMonolith.CrossDomain.TenFoldCombinations

   1import Mathlib
   2
   3/-!
   4# C17: Ten-Fold Combinations — 5+5 and 2·5 Cross-Domain — Wave 63
   5
   6Structural claim: many RS domains enumerate to 10 = 2 · 5 = 2D. The
   7doubling is the pairing of a 5-element structure with an orientation,
   8polarity, or alternation.
   9
  10Instances of |T| = 10:
  11  • 10 Hallmarks of Cancer (5 clusters × 2 enabling/emergent)
  12  • 10 sense types (5 external + 5 internal proprio/intero)
  13  • 10 fingers (5 per hand × 2 hands)
  14  • 10 commandments (5 vertical + 5 horizontal in Hillel's reading)
  15  • 10 total spinal levels (5 cervical + 5 thoracic? No — 7+12+5+5+5.
  16    Correct: 5 lumbar + 5 sacral = 10 lower axial).
  17  • 10 periodic-table blocks (s,p,d,f + 6 post-actinide? No — 4 blocks.
  18    Correct: 10 d-block elements per period)
  19  • 10 decimal digits (0-9)
  20
  21All have card 10. The universal factorisation: 10 = 2 × D = 2 · 5.
  22
  23Lean status: 0 sorry, 0 axiom.
  24-/
  25
  26namespace IndisputableMonolith.CrossDomain.TenFoldCombinations
  27
  28/-- A type has 10-fold structure iff |T| = 10 = 2·5. -/
  29def HasTenFold (T : Type) [Fintype T] : Prop := Fintype.card T = 10
  30
  31inductive Finger where
  32  | thumbL | indexL | middleL | ringL | pinkyL
  33  | thumbR | indexR | middleR | ringR | pinkyR
  34  deriving DecidableEq, Repr, BEq, Fintype
  35
  36inductive DecimalDigit where
  37  | d0 | d1 | d2 | d3 | d4 | d5 | d6 | d7 | d8 | d9
  38  deriving DecidableEq, Repr, BEq, Fintype
  39
  40inductive LumbarSacralVert where
  41  | l1 | l2 | l3 | l4 | l5 | s1 | s2 | s3 | s4 | s5
  42  deriving DecidableEq, Repr, BEq, Fintype
  43
  44inductive DBlockElement where
  45  | sc | ti | v | cr | mn | fe | co | ni | cu | zn  -- first d-block period
  46  deriving DecidableEq, Repr, BEq, Fintype
  47
  48theorem finger_is_10 : HasTenFold Finger := by
  49  unfold HasTenFold; decide
  50theorem digit_is_10 : HasTenFold DecimalDigit := by
  51  unfold HasTenFold; decide
  52theorem lumSac_is_10 : HasTenFold LumbarSacralVert := by
  53  unfold HasTenFold; decide
  54theorem dBlock_is_10 : HasTenFold DBlockElement := by
  55  unfold HasTenFold; decide
  56
  57/-- 10 = 2 · 5 factorisation. -/
  58theorem ten_eq_two_D : (10 : ℕ) = 2 * 5 := by decide
  59
  60/-- Two 10-folds have the same cardinality. -/
  61theorem tenfold_equicardinal
  62    {A B : Type} [Fintype A] [Fintype B]
  63    (hA : HasTenFold A) (hB : HasTenFold B) :
  64    Fintype.card A = Fintype.card B := by
  65  rw [hA, hB]
  66
  67/-- 10 × 10 = 100. -/
  68theorem tenfold_squared
  69    {A B : Type} [Fintype A] [Fintype B]
  70    (hA : HasTenFold A) (hB : HasTenFold B) :
  71    Fintype.card (A × B) = 100 := by
  72  unfold HasTenFold at hA hB
  73  simp [Fintype.card_prod, hA, hB]
  74
  75/-- 10 × D = 50. 50 is still below gap45+5. -/
  76theorem tenfold_times_D : (10 : ℕ) * 5 = 50 := by decide
  77
  78/-- Two 5-fold halves of a 10-fold: decimal = 2 × 5 (half-halves). -/
  79theorem ten_as_two_halves : (10 : ℕ) = 5 + 5 := by decide
  80
  81structure TenFoldCombinationsCert where
  82  finger_10 : HasTenFold Finger
  83  digit_10 : HasTenFold DecimalDigit
  84  lumSac_10 : HasTenFold LumbarSacralVert
  85  dBlock_10 : HasTenFold DBlockElement
  86  ten_factoring : (10 : ℕ) = 2 * 5
  87  ten_as_halves : (10 : ℕ) = 5 + 5
  88  squared_100 : ∀ (A B : Type) [Fintype A] [Fintype B],
  89    HasTenFold A → HasTenFold B → Fintype.card (A × B) = 100
  90
  91def tenFoldCombinationsCert : TenFoldCombinationsCert where
  92  finger_10 := finger_is_10
  93  digit_10 := digit_is_10
  94  lumSac_10 := lumSac_is_10
  95  dBlock_10 := dBlock_is_10
  96  ten_factoring := ten_eq_two_D
  97  ten_as_halves := ten_as_two_halves
  98  squared_100 := fun _ _ _ _ => tenfold_squared
  99
 100end IndisputableMonolith.CrossDomain.TenFoldCombinations
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