anomaly_zero
Zero charge yields a vanishing U(1) cubic anomaly coefficient in the Recognition Science QFT setup. Gauge theorists examining anomaly cancellation would cite this as the foundational trivial case. The proof reduces directly via definition unfolding followed by ring normalization.
claimThe anomaly coefficient for a U(1) gauge theory, defined by $c(Q) = Q^3$, satisfies $c(0) = 0$.
background
The module treats quantum anomalies as consequences of 8-tick phase mismatches between continuous classical symmetries and discrete time structure. The coefficient itself is introduced as the cube of the charge, so that the anomaly is proportional to $Q^3$. Upstream results supply the coefficient definition together with supporting structures from empirical programs, simplicial edge lengths, and spectral emergence on the D-cube.
proof idea
The proof is a one-line wrapper that unfolds the coefficient definition to obtain $0^3$ and then applies the ring tactic to confirm equality with zero.
why it matters in Recognition Science
This base case anchors anomaly computations inside the Recognition Science framework, where anomalies arise from 8-tick phase mismatches (T7). It precedes the module's description of the chiral anomaly and supplies the zero-charge instance needed for later cancellation arguments. The result aligns with the forcing chain's discrete time structure and the requirement that gauge anomalies must cancel.
scope and limits
- Does not compute the coefficient for nonzero charges.
- Does not derive the full chiral anomaly equation or its divergence term.
- Does not address cancellation conditions across multiple fermion representations.
- Does not connect the coefficient to explicit particle processes such as neutral pion decay.
formal statement (Lean)
150theorem anomaly_zero :
151 u1CubeCoeff 0 = 0 := by
proof body
Term-mode proof.
152 unfold u1CubeCoeff
153 ring
154
155/-! ## Chiral Anomaly Description -/
156
157/-- The chiral anomaly (Adler-Bell-Jackiw):
158 The axial current is NOT conserved: ∂_μ J⁵^μ = (α/π) E·B -/