theta
plain-language theorem explainer
The definition introduces the BRF theta function as the Cayley transform applied to twice the input complex number. Researchers working on the Riemann Hypothesis via the bounded-real-function route cite this algebraic mapping from the right half-plane to the unit disk. It is realized as a direct one-line wrapper around the base cayley transform.
Claim. $Θ(J) = (2J - 1)/(2J + 1)$ for $J ∈ ℂ$.
background
In the BRF route to the Riemann Hypothesis the module isolates the algebraic step that converts a Herglotz function (nonnegative real part) into a Schur function (norm at most one) via the Cayley transform. The upstream cayley definition supplies the base map $cayley H = (H - 1)/(H + 1)$. The present theta scales its argument by two before applying that map, producing the explicit form given in the module doc-comment.
proof idea
The definition is a one-line wrapper that applies the cayley transform to the doubled input 2 * J.
why it matters
This definition supplies the scaled Cayley transform used in the Recognition Science BRF plumbing for the Riemann Hypothesis. It is invoked directly in downstream results on golden-spiral resonance, DFT harmonic spectra, EEG frequency predictions, photobiomodulation patterns, and virtual-rotor geometries. Within the RH manuscripts it closes the unconditional algebraic half-plane-to-disk step that precedes any analytic estimates.
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