# Verification and scope

The manuscript proves a complete result within an explicitly singular class, not unconditional closure of the original AIM question. Verification is a detailed originating-researcher self-audit with exact regressions. It is not independent peer review or proof-assistant certification.

## Analytic proof checks

The six bar lengths follow from explicit trigonometric identities for every parameter in the stated interval. All four nonedge squared-distance functions have strictly positive derivatives at positive parameter. The reflection cannot persist because the two apex distances differ. The configuration germ has two transverse analytic branches; the initial rigidity rank is five, giving two nontrivial infinitesimal degrees of freedom.

Three signed-area inequalities place D strictly inside ABC for all positive parameters. The other two bars lie above AB. This proves the positive-time opening is noncrossing, while preserving the qualification that the initial bars overlap.

The five-vertex lower bound exhausts connected simple graphs on three and four vertices. Trees have the wrong real dimension. Full-span four-cycles have independent rigidity rows. For the triangle with pendant edge, even a collinear triangle is locally fixed as a real configuration set, and the pendant circle has a regular distance image. Denser graphs are locally finite modulo congruence. This argument does not confuse a degenerate triangle's rigidity rank with its real local dimension.

The positive theorem uses a nonnegative tangent direction of the regular distance image, then the identity property of a finite-order increasing interval homeomorphism. It does not assume a nonzero initial time derivative. Analytic continuation and the full-span finite-group moving-frame argument are proved separately with their respective hypotheses.

## Exact regressions

- 251 rational half-angle parameter samples, using exact quadratic extensions of the rationals.
- 1,506 bar-length and zero-derivative checks and 2,510 all-pairs derivative checks.
- 250 symmetry-breaking and failed-averaging negative controls.
- 750 strict interior-orientation checks at 250 positive parameter values, with an initial-collinearity control.
- A census of all 43 connected labeled simple graphs on two through four vertices.
- 378 full-span four-cycle rank checks, three collinear-cycle controls, and one collinear-triangle-with-pendant rank control.

The checkers run with and without Python optimization and give byte-identical outputs. Finite tests supplement the interval and arbitrary-placement proofs; they do not replace them. No floating-point tolerance is used.

## Manuscript and attribution

The seven-page English source compiled with both the built-in document compiler and Tectonic. Every final page was rendered and visually inspected, including the linkage diagram and references. The author line contains only the author's name; affiliation and contact information are in a footnote. AI assistance is disclosed in the body.

Classical conic four-bar theory, the prior symmetry results and the dataset report are credited. No new four-bar classification, initially noncrossing counterexample, regular-1DOF counterexample, independent validation or absolute priority is claimed.