IndisputableMonolith.Verification.Gap45DimensionCert

   1import Mathlib
   2import IndisputableMonolith.Gap45.Derivation
   3import IndisputableMonolith.RecogSpec.Bands
   4import IndisputableMonolith.Patterns
   5
   6/-!
   7# Gap45 + D=3 Dimension Certificate
   8
   9This module certifies the **dimension forcing** result:
  10
  11> The number 45 emerges from the eight-tick structure (T8) combined with Fibonacci,
  12> and lcm(2^D, 45) = 360 forces D = 3 spatial dimensions.
  13
  14## What this certificate does
  15
  161. **Gap = 45 derivation**: 45 = (8 + 1) × 5 = closure_factor × fibonacci_factor
  17   - 8 is the eight-tick period (from T8)
  18   - 9 = 8 + 1 is the closure factor (wrap-around)
  19   - 5 = Fibonacci(4) is the smallest Fibonacci > 1 coprime with 8
  20
  212. **Dimension forcing**: lcm(2^D, 45) = 360 ⟺ D = 3
  22   - Since gcd(2^D, 45) = 1 (45 has no factors of 2), we get lcm = 2^D × 45
  23   - 2^D × 45 = 360 ⟺ 2^D = 8 ⟺ D = 3
  24
  253. **Full period**: lcm(8, 45) = 360 is the complete synchronization period
  26
  27## Why this matters for the certificate chain
  28
  29This proves that D = 3 spatial dimensions is **not an arbitrary choice** but is
  30mathematically forced by:
  31- The eight-tick structure (2^3 = 8)
  32- The 45-gap from Fibonacci + closure
  33- The synchronization requirement lcm = 360
  34-/
  35
  36namespace IndisputableMonolith
  37namespace Verification
  38namespace Gap45Dimension
  39
  40open IndisputableMonolith.Gap45.Derivation
  41open IndisputableMonolith.RecogSpec
  42
  43/-- Certificate structure for Gap45 + D=3 forcing. -/
  44structure Gap45DimensionCert where
  45  deriving Repr
  46
  47/-- Verification predicate: all the dimension forcing results. -/
  48@[simp] def Gap45DimensionCert.verified (_c : Gap45DimensionCert) : Prop :=
  49  -- 1) The gap is 45
  50  gap = 45 ∧
  51  -- 2) 45 = (8 + 1) × 5 = closure × fibonacci
  52  gap = closure_factor * fibonacci_factor ∧
  53  closure_factor = eight_tick_period + 1 ∧
  54  fibonacci_factor = fib 4 ∧
  55  -- 3) 5 is coprime with 8
  56  Nat.gcd fibonacci_factor 8 = 1 ∧
  57  -- 4) lcm(8, 45) = 360
  58  full_period = 360 ∧
  59  -- 5) lcm(2^D, 45) = 360 ⟺ D = 3
  60  (∀ D : ℕ, Nat.lcm (2 ^ D) 45 = 360 ↔ D = 3)
  61
  62/-- The certificate verifies by referencing the proven theorems. -/
  63@[simp] theorem Gap45DimensionCert.verified_any (c : Gap45DimensionCert) :
  64    Gap45DimensionCert.verified c := by
  65  refine ⟨gap_eq_45, ?_, ?_, ?_, ?_, ?_, ?_⟩
  66  · exact gap_forced_from_eight_tick_and_fibonacci.1
  67  · exact gap_forced_from_eight_tick_and_fibonacci.2.1
  68  · exact gap_forced_from_eight_tick_and_fibonacci.2.2
  69  · exact fibonacci_factor_coprime_with_8
  70  · exact full_period_eq_360
  71  · exact lcm_pow2_45_eq_iff
  72
  73end Gap45Dimension
  74end Verification
  75end IndisputableMonolith
  76