honestPhaseFamily_excess_zero
plain-language theorem explainer
Honest phase family data produces sampled families whose annular excess vanishes at every mesh depth. Analysts establishing bounded excess controls for analytic traces cite this when converting a perturbation witness into a zero base case. The argument is a direct simplification that invokes the zero-excess statement already proved for the zeta-derived family.
Claim. Let $zfd$ be honest phase family data and let $N$ be a natural number. Then the annular excess of the mesh at depth $N$ of the sampled family obtained from $zfd$ equals zero.
background
The Honest Phase Budget Bridge module packages a perturbation witness for an actual phase family beyond completed-ξ symmetry. This witness supplies the quantitative Euler/Hadamard-style control that converts an honest phase family into bounded annular excess data on the sampled side. Annular excess measures the integrated deviation of the defect trace over annular regions at each mesh depth. The upstream zero-excess result for the zeta-derived family supplies the base case transferred to the honest setting.
proof idea
The proof is a one-line wrapper. It unfolds the family derivation field of the input data and directly applies the zero-excess theorem for the zeta-derived phase family at the supplied sensor, witness, and witness order.
why it matters
This supplies the zero-excess base case used by the downstream bounded annular excess theorem for the honest phase package. That theorem in turn advances the remaining step of upgrading excess control to the charge-zero conclusion required by AnalyticTrace. It closes the quantitative control step in the Recognition framework's handling of phase families, consistent with the eight-tick phase periodicity.
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