Foundation Eight Tick Phase Eighth Power Is One

The eighth-power identity

In mathematics, the eighth roots of unity are the eight complex numbers whose eighth power is 1. They are found by dividing the circle into eight equal arcs, starting at 1 and moving counterclockwise by 45 degrees each step. The full set is 1, (1+i)/√2, i, (-1+i)/√2, -1, (-1-i)/√2, -i, and (1-i)/√2. Raising any of these to the eighth power returns exactly 1, a property that follows from Euler's formula and the periodicity of the complex exponential.

This classical fact takes on a specific role in the Recognition Science framework, which models reality as a discrete record of events. The framework defines a fundamental clock with eight ticks, or phases, spaced at multiples of π/4 radians. The declaration phase_eighth_power_is_one is a machine-checked theorem in the framework's library of formal theorems. It states that for any of the eight phase steps, the complex exponential of that phase, raised to the eighth power, equals 1. This is not a new mathematical discovery; it is a formal restatement of the classical roots-of-unity property, verified by a proof checker.

Within the framework, this identity is used as a building block. The eight phases are linked to a binary classification: even-numbered ticks (0, 2, 4, 6) are associated with bosons, and odd-numbered ticks (1, 3, 5, 7) with fermions. The theorem spin_statistics_key, also in the library, singles out two special cases. The phase at tick 4 is -1, the sign associated with fermion antisymmetry under particle exchange, and the phase at tick 0 is 1, the sign for boson symmetry. The eighth-power identity guarantees that these signs repeat consistently as the clock cycles.

The theorem also supports a structural claim: the eight phases generate the cyclic group Z/8Z, meaning every phase can be written as a power of the first nonzero step. This group structure is what the framework uses to organize its discrete symmetry arguments. The identity itself, however, does not prove that particles exist, that spin-statistics holds in nature, or that the framework's clock is the correct description of physical time. It is a precise statement about the algebra of the framework's defined phases, not an empirical claim about the universe.

/-- **THEOREM**: The 8th power of each phase gives 1.
    exp(i × k × π/4)^8 = exp(2πik) = 1.
    Uses periodicity: exp(2πin) = 1 for n ∈ ℤ. -/
theorem phase_eighth_power_is_one (k : Fin 8) :
    (phaseExp k)^8 = 1 := by
  unfold phaseExp phase
  rw [← Complex.exp_nat_mul]
  -- 8 * (I * (k * π / 4)) = 2kπI, and exp(2kπI) = 1
  have h : (8 : ℕ) * (Complex.I * ((k.val : ℕ) * Real.pi / 4 : ℝ)) = 2 * Real.pi * Complex.I * k.val := by
    push_cast
    ring
  simp only [] at h
  rw [show (k : ℕ) = k.val from rfl] at h ⊢
  convert Complex.exp_int_mul_two_pi_mul_I k.val using 2
  push_cast
  ring

/-- **SPIN-STATISTICS KEY THEOREM**:
    Phase k=4 (half-cycle) gives -1, which is the fermion antisymmetry sign.
    Phase k=0 (identity) gives 1, which is the boson symmetry sign.
    This connects 8-tick structure to spin-statistics. -/
theorem spin_statistics_key :
    phaseExp ⟨4, by norm_num⟩ = -1 ∧ phaseExp ⟨0, by norm_num⟩ = 1 :=
  ⟨phase_4_is_minus_one, phase_0_is_one⟩

/-- The 8-tick structure generates the group ℤ/8ℤ.
    This is isomorphic to the discrete symmetry group of RS. -/
theorem eight_tick_generates_Z8 :
    ∀ k : Fin 8, ∃ n : ℕ, phaseExp k = (phaseExp ⟨1, by norm_num⟩)^n := by
  intro k
  use k.val
  unfold phaseExp phase
  rw [← Complex.exp_nat_mul]
  congr 1
  push_cast
  ring

What this page does not claim

The theorem does not prove that physical particles obey spin-statistics; it only establishes a formal correspondence within the framework. The identity does not claim that the framework's discrete clock is the correct model of physical time. The theorem does not derive the values of any physical constants.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/EightTick.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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