theta

The Recognition Stability Audit (RSA) repeatedly maps the closed right half-plane $\mathrm{Re}(z)\ge 0$ into the closed unit disk $|\cdot|\le 1$ via a Cayley transform. This module is the purely algebraic core of that plumbing (see Recognition_Stability_Audit.tex).

Locally, $\mathrm{cayley}(z)=(z-1)/(z+1)$. The paper writes the same map in cost coordinates as $\theta(J)=(2J-1)/(2J+1)$. That is exactly $\mathrm{cayley}(2J)$. An explicit inverse $\mathrm{invTheta}$ is supplied alongside.

Upstream, Foundation.SchurPinch records a Herglotz-to-Schur Cayley in the same algebraic shape; the RSA file re-exports the half-plane form and the paper-facing $\theta$ as thin, self-contained training targets.

Pure definition. Unfold to the local Cayley: $\theta(J):=\mathrm{cayley}(2J)$. Expanding the local formula gives $(2J-1)/(2J+1)$ with no further lemmas.

This is the paper-facing entry point for every RSA argument that needs the half-plane-to-disk map in $J$-coordinates rather than raw $z$. Downstream it is consumed heavily across the audit and related geometry/calculus layers (order-forty use sites), including Primitive Recognition Calculus completeness targets that talk about $J$-cost modulus control, and dihedral-angle certificates that need a named angle/transform symbol in the same pipeline.

In the broader Recognition framework it is plumbing, not a forcing step: it does not itself force $\phi$, the eight-tick octave, or $D=3$. It does make the Schur/Herglotz disk language available wherever RSA stability bounds are stated, so later norm and invertibility lemmas (norm_theta_le_one_of_re_nonneg, invTheta_theta, theta_invTheta) can sit on a single shared symbol.