pick_gap
plain-language theorem explainer
The Pick gap at a complex point J is defined as one minus the modulus of the theta function evaluated at J. Researchers reformulating the Riemann Hypothesis via operator-theoretic spectral margins would cite this quantity as the basic distance to the unit circle. The definition is introduced by direct subtraction of the theta norm from unity.
Claim. Let $J$ be a complex number. The Pick gap at $J$ is $1 - |θ(J)|$, where $θ$ is the theta function from the BRF construction; the gap measures the margin by which the image lies strictly inside the unit disk.
background
The module recasts the Riemann Hypothesis as persistence of a positive spectral gap under the Pick operator applied to the relevant field $J$. The Pick gap itself is the quantity $1 - |θ(J)|$ that quantifies how far the image under $θ$ stays below modulus one. Upstream results supply the theta function via the BRFPlumbing module and concrete $J$ values via gap constructions in the Gap45 and Masses modules that arise from the Recognition Science phi-ladder.
proof idea
The declaration is a one-line definition that subtracts the complex modulus of the theta function at the input from one.
why it matters
This definition supplies the central margin fed into the Euler-region positivity result and the persistence property that directly implies the Riemann Hypothesis. It closes the link between real-part positivity of $J$ and exclusion of zeros off the critical line. Within the Recognition framework it supplies the analytic margin whose persistence aligns with J-uniqueness and the forcing chain to three spatial dimensions.
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