Foundation Schur Pinch Phase Le Half Pi Re Nonneg

A formal tautology

The declaration phase_le_half_pi_re_nonneg in the framework's machine-checked library of formal theorems is a statement about complex numbers, the numbers with a real part and an imaginary part. The theorem says: if a complex number's real part is non-negative, then that real part is non-negative. The proof is immediate, because the conclusion is exactly the hypothesis. It is a tautology, a formal restatement of the assumption.

In plain terms, the declaration establishes nothing beyond itself. It does not connect a bound on a complex number's argument, the angle it makes with the positive real axis, to the sign of its real part. The neighboring theorem phase_lt_half_pi_re_pos does that work: it proves that a nonzero complex number with argument strictly less than π/2 has a strictly positive real part. The weak-form declaration, by contrast, simply mirrors its own input.

Within the framework, this kind of declaration is a formal placeholder. The library's authors state a weak version of a phase-cap lemma so that later material can cite a non-strict inequality without re-proving it. The substantive mathematics, the Herglotz positivity condition and the Cayley–Schur pinch that excludes poles, sits in the stronger theorems around it. The weak form is a bookkeeping step, not a result.

What the declaration does not claim is the interesting part. It does not assert that a small argument forces a non-negative real part; that direction is the strong theorem's job. It does not say anything about functions, only about single complex numbers. And it certainly does not imply the Riemann Hypothesis, even though the framework's paper F4 lists the phase-cap material as a cited component of its RH equivalence. The tautology itself carries no such weight.

/-- **F4.2.1 (weak form)**: Re z ≥ 0 when |arg z| ≤ π/2 (non-strict). -/
theorem phase_le_half_pi_re_nonneg (z : ℂ) (hre : 0 ≤ z.re) : 0 ≤ z.re := hre

/-- **F4.2.1**: A nonzero complex number with argument strictly less than π/2
    has strictly positive real part. -/
theorem phase_lt_half_pi_re_pos (z : ℂ) (hz : z ≠ 0) (harg : |z.arg| < π / 2) :
    0 < z.re := by
  have hpos_or_zero : 0 < z.re ∨ z = 0 :=
    (Complex.abs_arg_lt_pi_div_two_iff).1 harg
  rcases hpos_or_zero with hre | hz0
  · exact hre
  · exact (hz hz0).elim

/-- **F4.2.1 (weak form)**: Re z ≥ 0 when |arg z| ≤ π/2 (non-strict). -/
theorem phase_le_half_pi_re_nonneg (z : ℂ) (hre : 0 ≤ z.re) : 0 ≤ z.re := hre

What this page does not claim

A small argument forces a non-negative real part; that direction is the strong theorem's job. The declaration says anything about functions; it concerns only single complex numbers. The tautology implies the Riemann Hypothesis, even though the framework lists the phase-cap material as a cited component of its RH equivalence.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: