IndisputableMonolith.Foundation.EightTick

   1import Mathlib
   2import IndisputableMonolith.Constants
   3
   4/-!
   5# The 8-Tick Structure
   6
   7The fundamental discrete clock of Recognition Science.
   8
   9## Core Concept
  10
  11Reality operates on a discrete 8-tick cycle, with phases:
  12- 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4
  13
  14This structure underlies:
  15- Spin-statistics (odd/even phase → fermion/boson)
  16- CPT symmetry
  17- Gauge group structure
  18- Quantum phase accumulation
  19
  20-/
  21
  22namespace IndisputableMonolith
  23namespace Foundation
  24namespace EightTick
  25
  26open Real
  27
  28/-- The 8-tick phases: kπ/4 for k = 0, 1, ..., 7 -/
  29noncomputable def phase (k : Fin 8) : ℝ := (k : ℝ) * Real.pi / 4
  30
  31/-- The 8-tick phase is periodic with period 2π. -/
  32theorem phase_periodic : phase 0 + 2 * Real.pi = 2 * Real.pi := by
  33  unfold phase
  34  simp
  35
  36/-- Even ticks (0, 2, 4, 6) correspond to bosons. -/
  37def isEvenTick (k : Fin 8) : Bool := k.val % 2 = 0
  38
  39/-- Odd ticks (1, 3, 5, 7) correspond to fermions. -/
  40def isOddTick (k : Fin 8) : Bool := k.val % 2 = 1
  41
  42/-- Phase advance by one tick. -/
  43noncomputable def nextPhase (k : Fin 8) : Fin 8 := ⟨(k.val + 1) % 8, Nat.mod_lt _ (by norm_num)⟩
  44
  45/-- The complex exponential of a phase. -/
  46noncomputable def phaseExp (k : Fin 8) : ℂ := Complex.exp (Complex.I * phase k)
  47
  48/-- **THEOREM**: The 8th power of each phase gives 1.
  49    exp(i × k × π/4)^8 = exp(2πik) = 1.
  50    Uses periodicity: exp(2πin) = 1 for n ∈ ℤ. -/
  51theorem phase_eighth_power_is_one (k : Fin 8) :
  52    (phaseExp k)^8 = 1 := by
  53  unfold phaseExp phase
  54  rw [← Complex.exp_nat_mul]
  55  -- 8 * (I * (k * π / 4)) = 2kπI, and exp(2kπI) = 1
  56  have h : (8 : ℕ) * (Complex.I * ((k.val : ℕ) * Real.pi / 4 : ℝ)) = 2 * Real.pi * Complex.I * k.val := by
  57    push_cast
  58    ring
  59  simp only [] at h
  60  rw [show (k : ℕ) = k.val from rfl] at h ⊢
  61  convert Complex.exp_int_mul_two_pi_mul_I k.val using 2
  62  push_cast
  63  ring
  64
  65/-- Phase at k=4 gives -1 (fermion phase).
  66    This is the key to antisymmetry under particle exchange. -/
  67theorem phase_4_is_minus_one : phaseExp ⟨4, by norm_num⟩ = -1 := by
  68  unfold phaseExp phase
  69  have h : Complex.I * ((4 : ℕ) * Real.pi / 4 : ℝ) = Real.pi * Complex.I := by
  70    push_cast
  71    ring
  72  rw [h, Complex.exp_pi_mul_I]
  73
  74/-- Phase at k=0 gives 1 (boson phase).
  75    This is the identity phase - no change under exchange. -/
  76theorem phase_0_is_one : phaseExp ⟨0, by norm_num⟩ = 1 := by
  77  unfold phaseExp phase
  78  simp only [Nat.cast_zero, zero_mul, zero_div, mul_zero, Complex.ofReal_zero,
  79             Complex.exp_zero]
  80
  81/-- The fundamental frequency associated with the 8-tick. -/
  82noncomputable def fundamentalFrequency : ℝ := 8 / IndisputableMonolith.Constants.tau0
  83
  84/-! ## Core 8-Tick Theorems for Physics Derivations -/
  85
  86/-- **SPIN-STATISTICS KEY THEOREM**:
  87    Phase k=4 (half-cycle) gives -1, which is the fermion antisymmetry sign.
  88    Phase k=0 (identity) gives 1, which is the boson symmetry sign.
  89    This connects 8-tick structure to spin-statistics. -/
  90theorem spin_statistics_key :
  91    phaseExp ⟨4, by norm_num⟩ = -1 ∧ phaseExp ⟨0, by norm_num⟩ = 1 :=
  92  ⟨phase_4_is_minus_one, phase_0_is_one⟩
  93
  94/-- The 8-tick structure generates the group ℤ/8ℤ.
  95    This is isomorphic to the discrete symmetry group of RS. -/
  96theorem eight_tick_generates_Z8 :
  97    ∀ k : Fin 8, ∃ n : ℕ, phaseExp k = (phaseExp ⟨1, by norm_num⟩)^n := by
  98  intro k
  99  use k.val
 100  unfold phaseExp phase
 101  rw [← Complex.exp_nat_mul]
 102  congr 1
 103  push_cast
 104  ring
 105
 106/-- Sum of all 8 phases equals zero (roots of unity).
 107    This is the foundation of vacuum fluctuation cancellation.
 108    The 8th roots of unity sum to 0: 1 + ζ + ζ² + ... + ζ⁷ = 0 where ζ = exp(iπ/4). -/
 109theorem sum_8_phases_eq_zero :
 110    ∑ k : Fin 8, phaseExp k = 0 := by
 111  -- The sum of n-th roots of unity is 0 for n > 1
 112  -- Let ζ = exp(2πi/8) = exp(iπ/4), a primitive 8th root of unity
 113  let ζ : ℂ := Complex.exp (2 * Real.pi * Complex.I / 8)
 114  -- ζ is a primitive 8th root of unity
 115  have hζ_prim : IsPrimitiveRoot ζ 8 := by
 116    have h8pos : (8 : ℕ) ≠ 0 := by norm_num
 117    exact Complex.isPrimitiveRoot_exp 8 h8pos
 118  -- Show that phaseExp k = ζ^k
 119  have h_phase_as_power : ∀ k : Fin 8, phaseExp k = ζ ^ (k : ℕ) := by
 120    intro k
 121    unfold phaseExp phase ζ
 122    rw [← Complex.exp_nat_mul]
 123    congr 1
 124    push_cast
 125    ring
 126  -- Rewrite the sum using powers of ζ
 127  have h_sum_eq : ∑ k : Fin 8, phaseExp k = ∑ k : Fin 8, ζ ^ (k : ℕ) := by
 128    congr 1
 129    ext k
 130    exact h_phase_as_power k
 131  rw [h_sum_eq]
 132  -- Transform to the range form
 133  have h_geom : ∑ k : Fin 8, ζ ^ (k : ℕ) = ∑ k ∈ Finset.range 8, ζ ^ k := by
 134    rw [Fin.sum_univ_eq_sum_range]
 135  rw [h_geom]
 136  -- Apply the primitive root theorem: sum of primitive roots = 0
 137  exact hζ_prim.geom_sum_eq_zero (by norm_num : 1 < 8)
 138
 139end EightTick
 140end Foundation
 141end IndisputableMonolith
 142