Foundation Spin Statistics Boson Rotation Phase Pos One

The Boson Rotation Phase

In quantum mechanics, rotating a particle by 360 degrees does not always return it to its original state. For a spin-1/2 particle like an electron, a full rotation multiplies its quantum state by -1, a minus sign that is physically real and observable. For particles with integer spin, such as photons or certain composite particles, a full rotation returns the state to itself exactly, multiplied by +1. This distinction is a key part of the spin-statistics theorem, which links a particle's spin to whether it must obey Fermi-Dirac or Bose-Einstein statistics.

The Recognition Science framework, which derives physical structure from a discrete record of recognition events called a ledger, models this rotation behavior through its eight-tick cycle. The declaration boson_rotation_phase_pos_one is a theorem in the framework's machine-checked library of formal theorems. It states that for a bosonic state, the phase accumulated under a full 2π rotation is +1. In the framework's terms, a boson's minimal recognition cycle completes in 8 ticks, and the phase at the identity tick (tick 0) is 1, as proven by the theorem phase_0_is_one.

This result is not an isolated fact. It is part of a chain that establishes the complete spin-statistics connection. The companion theorem fermion_rotation_phase_neg_one shows that a fermion, with a 4-tick cycle, acquires a phase of -1 under the same rotation. The framework then proves that the exchange sign for two identical particles is exactly this rotation phase: bosons exchange with a +1 sign, making them symmetric, while fermions exchange with a -1 sign, making them antisymmetric. This symmetry difference is what allows bosons to occupy the same quantum state, as in a laser or a Bose-Einstein condensate, and forbids fermions from doing so, which is the Pauli exclusion principle.

In Recognition Science, this theorem is a derived result, not an assumption. It follows from the framework's foundational structure of an eight-tick recognition cycle. The declaration boson_rotation_phase_pos_one is a precise, machine-checked statement that this particular phase is +1. It does not, by itself, prove the entire spin-statistics theorem, nor does it address the physical mechanism by which a particle's spin is tied to its recognition cycle. Those are separate, larger claims within the framework.

/-- For bosons (8-tick period), the 2π rotation gives phase +1 (via two half-cycles). -/
theorem boson_rotation_phase_pos_one :
    phaseExp ⟨0, by norm_num⟩ = 1 := phase_0_is_one

theorem exchange_sign_boson :
    phaseExp ⟨0, by norm_num⟩ * phaseExp ⟨0, by norm_num⟩ = 1 := by
  rw [phase_0_is_one]; ring

/-- **KEY**: For fermions (4-tick period), the 2π rotation gives phase -1. -/
theorem fermion_rotation_phase_neg_one :
    rotationPhase 4 = -1 := by
  unfold rotationPhase
  exact phase_4_is_minus_one

What this page does not claim

This theorem does not, by itself, prove the entire spin-statistics theorem or the Pauli exclusion principle. The declaration does not describe how a particle's spin is physically connected to its recognition cycle period. The theorem does not claim that all bosons have an 8-tick cycle; spin-0 particles are also classified as bosonic with a 1 or 2-tick cycle.

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

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