Foundation Eight Tick Phase 0 Is One

The identity phase

The declaration phase_0_is_one is a theorem in the Recognition Science framework's machine-checked library of formal theorems. It establishes that the complex exponential of the phase at tick zero equals 1. The phase at tick zero is 0 radians, and the complex exponential of 0 is 1. This is the identity element of multiplication: multiplying any number by 1 leaves it unchanged.

The framework models reality as operating on a discrete eight-tick cycle, with phases at multiples of π/4: 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4. The theorem phase_0_is_one is the companion to phase_4_is_minus_one, which shows the phase at π equals -1. Together they form the spin-statistics key: the framework connects the even tick at phase 0 to the boson symmetry sign, and the odd tick at phase 4 to the fermion antisymmetry sign. The theorem proves the boson half of that correspondence.

The declaration does not claim that the number 1 is a boson, nor that any physical particle exists at phase zero. It does not prove that the eight-tick structure itself is real; the framework defines the structure, and the library proves facts about that definition. The theorem is a formal identity about a defined function, not an empirical measurement. It also does not claim that phase zero is the only phase returning 1; the full cycle returns to 1 as well, and the theorem phase_periodic records that periodicity separately.

What the declaration changes is the framework's internal consistency: the identity phase behaves as an identity should. That single fact anchors the boson side of the spin-statistics correspondence, giving the framework a starting point for connecting its discrete clock to the symmetry of particles. The reader can now see that the framework's basic vocabulary, the eight-tick phase, has a well-behaved zero point.

/-- Phase at k=0 gives 1 (boson phase).
    This is the identity phase - no change under exchange. -/
theorem phase_0_is_one : phaseExp ⟨0, by norm_num⟩ = 1 := by
  unfold phaseExp phase
  simp only [Nat.cast_zero, zero_mul, zero_div, mul_zero, Complex.ofReal_zero,
             Complex.exp_zero]

/-- The 8-tick phases: kπ/4 for k = 0, 1, ..., 7 -/
noncomputable def phase (k : Fin 8) : ℝ := (k : ℝ) * Real.pi / 4

/-- **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⟩

What this page does not claim

The theorem does not assert that the number 1 is a boson or that any physical particle occupies phase zero. The theorem does not prove that the eight-tick structure is physically real; it is a defined model. The theorem does not claim that phase zero is the only phase returning 1, since the full cycle also returns to 1.

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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