IndisputableMonolith.Quantum.RecognitionFirst.EightTickWeyl

   1import Mathlib.Analysis.SpecialFunctions.Complex.Circle
   2import Mathlib.Analysis.SpecialFunctions.Complex.Log
   3import Mathlib.Data.ZMod.Basic
   4
   5/-!
   6# The eight-tick Weyl relation: the recognition root of canonical non-commutativity
   7
   8Recognition-first physics (`plans/RS_Recognition_First_Physics_Program_20260623.html`).
   9Conventional QM POSTULATES the canonical commutator `[x,p] = iℏ`. RS DERIVES it: on the
  108-tick recognition cycle `ZMod 8`, "occupation" and "cost-rate" are the shift and clock
  11operators of the finite Heisenberg–Weyl group. They satisfy the Weyl relation
  12`clock ∘ shift = ω • (shift ∘ clock)` with `ω` a primitive 8th root of unity, so they
  13do NOT commute. Canonical non-commutativity is the cyclic recognition structure, not an
  14axiom. The continuum limit gives `[x,p] = iℏ` (node D3, OPEN derive-tick work), and the
  15magnitude is tied to `ℏ = φ⁻⁵` through the J-cost quantum.
  16
  17Closed 2026-06-26 (axiom-clean: `[propext, Classical.choice, Quot.sound]`). The braiding is
  18ring-generic: its only ring-specific content is `ω^8 = 1` and `ω ≠ 1`. The exponent
  19reconciliation `ω^(k.val) = ω^((k-1).val + 1)` is a finite `ZMod 8` fact discharged by
  20`decide`, with the single wraparound case `k = 0` (`ω^0 = ω^8`) closed by `omega_pow_eight`.
  21The narrow imports (no full `import Mathlib`) keep the file light. The continuum limit
  22`[x,p]=iℏ` and the magnitude `ℏ=φ⁻⁵` remain OPEN (node D6), not asserted here.
  23-/
  24
  25namespace IndisputableMonolith
  26namespace Quantum
  27namespace RecognitionFirst
  28
  29open scoped Real
  30open Complex
  31
  32/-- The 8-tick phase: a primitive 8th root of unity. The cost-rate advances by this
  33phase per recognition tick. -/
  34noncomputable def omega : ℂ := Complex.exp (2 * Real.pi * Complex.I / 8)
  35
  36/-- Recognition occupation shift on the 8-tick cycle (advance occupation by one tick). -/
  37def shift (ψ : ZMod 8 → ℂ) : ZMod 8 → ℂ := fun k => ψ (k - 1)
  38
  39/-- Recognition cost-rate clock on the 8-tick cycle (phase by `ω^k`). -/
  40noncomputable def clock (ψ : ZMod 8 → ℂ) : ZMod 8 → ℂ := fun k => omega ^ (k.val) * ψ k
  41
  42/-- The 8-tick phase closes the cycle: `ω^8 = 1`. (`ω^8 = exp(2πi) = 1`.) -/
  43theorem omega_pow_eight : omega ^ 8 = 1 := by
  44  have h : omega ^ 8 = Complex.exp (2 * Real.pi * Complex.I) := by
  45    rw [omega, ← Complex.exp_nat_mul]
  46    congr 1
  47    push_cast
  48    ring
  49  rw [h, Complex.exp_two_pi_mul_I]
  50
  51/-- The cycle is nontrivial: `ω ≠ 1`. This is what makes the non-commutativity real.
  52(If `ω = 1` then `ω^4 = 1`, but `ω^4 = exp(πi) = -1 ≠ 1`.) -/
  53theorem omega_ne_one : omega ≠ 1 := by
  54  intro h
  55  have hsq : omega ^ 4 = 1 := by rw [h]; ring
  56  have hpi : omega ^ 4 = -1 := by
  57    have : omega ^ 4 = Complex.exp (Real.pi * Complex.I) := by
  58      rw [omega, ← Complex.exp_nat_mul]
  59      congr 1
  60      push_cast
  61      ring
  62    rw [this, Complex.exp_pi_mul_I]
  63  rw [hpi] at hsq
  64  norm_num at hsq
  65
  66/-- **The eight-tick Weyl relation.** `clock (shift ψ) = ω • (shift (clock ψ))`,
  67pointwise. This is the recognition root of the canonical commutator: occupation and
  68cost-rate do not commute, their failure to commute is exactly the 8-tick phase `ω`. -/
  69theorem eightTick_weyl (ψ : ZMod 8 → ℂ) (k : ZMod 8) :
  70    clock (shift ψ) k = omega * shift (clock ψ) k := by
  71  have hval : ∀ j : ZMod 8,
  72      j.val = (j - 1).val + 1 ∨ (j.val = 0 ∧ (j - 1).val = 7) := by decide
  73  have e : omega ^ (k.val) = omega ^ ((k - 1).val + 1) := by
  74    rcases hval k with h | ⟨h0, h7⟩
  75    · rw [h]
  76    · rw [h0, h7, pow_zero]; exact omega_pow_eight.symm
  77  simp only [clock, shift]
  78  rw [e]
  79  simp [pow_succ, mul_comm, mul_assoc, mul_left_comm]
  80
  81/-- **Canonical non-commutativity emerges.** The clock and shift operators do not
  82commute. This is the finite, exact RS root of `[x,p] ≠ 0`; the continuum limit (node D3)
  83turns it into `[x,p] = iℏ`. -/
  84theorem canonical_noncommutativity :
  85    ∃ ψ : ZMod 8 → ℂ, clock (shift ψ) ≠ shift (clock ψ) := by
  86  refine ⟨fun _ => 1, fun h => omega_ne_one ?_⟩
  87  have h1 := congrFun h 1
  88  simp only [clock, shift, mul_one] at h1
  89  rw [show ZMod.val (1 : ZMod 8) = 1 by decide, pow_one,
  90      show ((1 : ZMod 8) - 1).val = 0 by decide, pow_zero] at h1
  91  exact h1
  92
  93end RecognitionFirst
  94end Quantum
  95end IndisputableMonolith
  96