Foundation Schur Pinch Cayley Inv

The inverse Cayley map

The Cayley transform is a classical tool in complex analysis that maps the right half-plane, numbers whose real part is positive, onto the unit disk, numbers whose magnitude is less than one. The declaration cayleyInv defines the inverse of that map: given a point Θ in the disk, it returns the point (1 + Θ) / (2 * (1 - Θ)) in the right half-plane. This is the algebraic reverse of the forward transform (2H - 1) / (2H + 1), and it is a definition, not a theorem; the framework chooses this formula as the way back from the disk to the half-plane.

The definition matters because the framework's foundation paper F4 uses the pair of maps to translate between two kinds of functions. A Herglotz function has a real part that is never negative; a Schur function has a modulus that never exceeds one. The forward transform sends a Herglotz function to a Schur function, and the inverse map is what would bring a Schur function back. The framework proves that the forward direction works: if a function is Herglotz on a set, then its Cayley transform is Schur there, wherever the denominator is nonzero. The inverse definition is the formal counterpart to that result, ready for use when a bounded object needs to be studied through its half-plane representation.

In Recognition Science, this map is part of a template called the Schur pinch exclusion, which the framework uses to rule out poles in certain functions. The idea is to show a function is Herglotz, transform it to a Schur function, and then use the boundedness of the disk to force singularities away. The cayleyInv definition is the return ticket in that maneuver, though the framework's master pinch theorem as stated in the library simply assumes the pole set is empty rather than deriving it from the hypotheses. The definition itself establishes no fact about poles, singularities, or the Riemann Hypothesis; it only names the algebraic inverse of a standard map.

What the definition does not claim is as important as what it names. It does not assert that the inverse map is well-defined everywhere, because the denominator 1 - Θ can vanish. It does not prove that applying cayleyInv to a Schur function yields a Herglotz function; that would be a theorem about the map, and no such theorem appears in the pack. It also does not connect to the Riemann Hypothesis by itself. The framework lists the Schur pinch template as cited by RH work, but the cayleyInv definition carries no such result; it is a piece of vocabulary, not a proof.

/-- The inverse Cayley transform. -/
noncomputable def cayleyInv (Θ : ℂ) : ℂ := (1 + Θ) / (2 * (1 - Θ))

/-- **F4.1.3 (Herglotz-to-Schur)**: If f is Herglotz on D, then cayley ∘ f is Schur on D
    (wherever the denominator is nonzero). -/
theorem cayley_schur_of_herglotz {f : ℂ → ℂ} {D : Set ℂ}
    (hH : IsHerglotz f D) (hden : ∀ z ∈ D, 2 * f z + 1 ≠ 0) :
    IsSchur (cayley ∘ f) D := by
  intro z hz
  exact cayley_norm_le_one (f z) (hH z hz) (hden z hz)

What this page does not claim

The definition does not prove the inverse map is well-defined everywhere. The definition does not show that cayleyInv sends Schur functions to Herglotz functions. The definition carries no result about the Riemann Hypothesis.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/SchurPinch.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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