zetaDerivedPhaseData
plain-language theorem explainer
zetaDerivedPhaseData extracts the continuous phase assignment on the nth sampled circle of radius r0/(n+1) from a quantitative local factorization of a meromorphic function. Researchers constructing defect phase families for zeros of the zeta function cite this when matching charges to DefectSensor requirements. The definition is a one-line wrapper that projects the ContinuousPhaseData value from the bundled construction.
Claim. Given a quantitative local factorization $Q$ of a meromorphic function with order $k$ and a positive integer $n$, the phase data is the continuous argument function $Θ:ℝ→ℝ$ on the circle of radius $r_0/(n+1)$ centered at the factorization point such that the winding number satisfies $Θ(2π)-Θ(0)=-2πk$.
background
The Meromorphic Circle Order module bridges meromorphic-order machinery to the RS annular cost framework. A meromorphic function admits a local factorization $f(z)=(z-ρ)^n g(z)$ with $g$ holomorphic and nonvanishing at $ρ$. On a small circle around $ρ$ the phase charge of $f$ equals $-n$, since the power term contributes $-n$ while the regular factor contributes zero charge (via holomorphic_nonvanishing_zero_charge and charge_additive).
proof idea
The definition is a one-line wrapper that returns the ContinuousPhaseData component of zetaDerivedPhaseDataBundle for the supplied quantitative local factorization and index $n$.
why it matters
This definition supplies the phase data consumed by zetaDerivedPhaseFamily and zetaDerivedPhasePerturbationWitness. It realizes the charge extraction from meromorphic_phase_charge for the reciprocal zeta function, where order $-m$ at a zero of multiplicity $m$ produces charge $m$ matching the defect sensor. The construction supports the transition from local analytic factorizations to sampled increments in the RS cost framework.
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