half_cycle_phase
The declaration establishes that four times the phase quantum equals π, encoding a half-cycle shift equivalent to a sign flip under 8-tick discreteness. Researchers modeling chiral anomalies via phase quantization in Recognition Science would cite it when linking discrete time to the π⁰ → γγ process. The proof is a one-line wrapper that unfolds the phaseQuantum definition and normalizes the resulting arithmetic identity with the ring tactic.
claimIn the 8-tick phase model, the accumulated phase after four quanta satisfies $4 q = π$, where $q$ is the fundamental phase quantum arising from the discrete time structure.
background
The module derives quantum anomalies from mismatches between continuous classical phases and the discrete 8-tick structure required by Recognition Science. PhaseQuantum is the quantized phase increment per tick; sibling definitions such as eight_quanta_full_rotation close the full 2π cycle while numDiscretePhases counts the allowed steps. Upstream structures from SpectralEmergence fix the gauge content and 24 chiral fermion flavors consistent with this discreteness, while PhiForcingDerived supplies the convex J-cost minimization that governs local phase updates.
proof idea
The proof unfolds the definition of phaseQuantum to expose its explicit multiple of π and then applies the ring tactic to confirm the arithmetic identity 4 * phaseQuantum = π.
why it matters in Recognition Science
This identity supplies the half-cycle relation required by the subsequent π⁰ lifetime predictions (pi0_lifetime_predicted_units, pi0_prediction_within_2_percent) in the same module. It fills the phase-quantization step in the paper proposition on quantum anomalies from discrete time structure, directly instantiating the eight-tick octave (T7) that produces the chiral anomaly for π⁰ → γγ. No open scaffolding remains at this leaf.
scope and limits
- Does not derive the full chiral anomaly current non-conservation equation.
- Does not compute the numerical π⁰ lifetime or its error bounds.
- Does not address gauge anomaly cancellation or conformal running.
- Does not extend to higher-order phase mismatches beyond the half-cycle case.
formal statement (Lean)
60theorem half_cycle_phase :
61 (4 : ℕ) * phaseQuantum = π := by
proof body
Term-mode proof.
62 unfold phaseQuantum
63 ring
64
65/-! ## The π⁰ → γγ Decay Prediction -/
66
67/-- Predicted π⁰ lifetime from the anomaly (in units of 10⁻¹⁷ seconds). -/