PhaseEvolution
PhaseEvolution pairs a natural-number tick count with a continuous real phase and derives the discrete phase as ticks times pi over 4. Researchers modeling discrete-time QFT anomalies would cite it when tracing chiral or conformal symmetry breaking to 8-tick quantization. The declaration is a direct structure definition that imports the periodic phase map from the EightTick module.
claimA structure consisting of a natural number $n$ (tick count), a real number $p_c$ (continuous phase), and the derived discrete phase $p_d := n π / 4$.
background
In Recognition Science, anomalies arise when continuous classical phase evolution meets the discrete 8-tick quantum evolution. The upstream EightTick.phase supplies the quantized values $k π / 4$ for $k = 0,…,7$, periodic with period $2π$. The module frames this mismatch as the source of chiral anomalies (axial current non-conservation) and conformal anomalies (scale breaking), consistent with the forcing chain’s T7 eight-tick octave.
proof idea
One-line structure definition that directly encodes the discrete-phase formula from the tick count. It draws the phase quantization convention from the EightTick.phase definition and the mismatch interpretation from the module doc-comment.
why it matters in Recognition Science
The structure supplies the basic object for the QFT-014 derivation of anomalies from 8-tick phase mismatches. It sits inside the T7 eight-tick octave of the forcing chain and supports the paper proposition on discrete-time structure. No downstream uses are recorded, leaving open its integration into explicit anomaly coefficients or π⁰ lifetime predictions.
scope and limits
- Does not compute anomaly coefficients or decay rates.
- Does not enforce periodicity or alignment at multiples of 8.
- Does not incorporate J-cost minimization or phi-ladder scaling.
- Does not reference gauge groups or fermion generations.
formal statement (Lean)
177structure PhaseEvolution where
178 /-- Number of discrete ticks. -/
179 ticks : ℕ
180 /-- Corresponding continuous phase. -/
181 continuous_phase : ℝ
182 /-- The discrete phase (ticks × π/4). -/
183 discrete_phase : ℝ := ticks * π / 4
proof body
Definition body.
184
185/-- **THEOREM**: Discrete and continuous phases align at multiples of 8. -/