# Verification report for the focal antipedal centroid theorem The manuscript gives a general proof for ANY rectangle circumscribed about an ellipse, with the antipedal taken about either actual focus. The centroids are O and O+(F-O)/3. The outer-rectangle property of physical four-periodic elliptic billiards is known and credited, so the application proves both assertions in the exact N=4 row k406,b. General-even-period k407 and hyperbolic-caustic bowties are not claimed. ## General argument The ellipse matrix in an orthonormal rectangle frame is D=b^2 I+FF^T. The support distances satisfy h^2-f^2=k^2-g^2=b^2>0. Solving the four actual antipedal lines, (X-F) dot (Y-X)=0, gives (-f,g+2k), (f-2h,-g), (-f,g-2k), (f+2h,-g). All consecutive line determinants are positive. Translating by F gives a strictly convex axis-intercept quadrilateral with positive arms 2(h+f),2(h-f),2(k+g),2(k-g). Its area is 8hk; exact vertex and signed-area first moments give the two fixed physical points. No exceptional finite orientation is discarded and no division by zero is used. Reversed order and either focus obey the same formulas. ## Actual exact regression The portable checker uses only Python standard-library Fraction and constructs all intersections from their defining lines, independently of the simplified vertex formula. Root ran its finalized original successfully on 30 September 2026 with exit code 0. Reported counts: - 228 noncircular rational support frames and 5,472 corresponding both-focus, rigid-motion and orientation tests. - Three separate circle-limit frames and 72 separate extension tests. - Two distinct rational four-periodic configurations on the same fixed ellipse, not merely unrelated ellipses. - Six general rectangle/point cases, four parallel exclusions and a concurrent-line degeneration. - Wrong polar-definition and noncyclic bowtie guards. The asymmetric bowtie has nonzero signed area; no test infers simplicity from area. These are bounded regression tests, not proof-assistant verification or a substitute for the displayed general algebra. The tests verify ellipse incidence, support tangency, reflected ray directions, line incidence and exact centroids. The source ZIP's isolated portable rerun must pass before readiness is registered; its exact receipt is retained in the project publication checkpoint. ## Source and novelty limits The exact experimental row was checked in the original arXiv v11 Table 5 and the 2021 publisher Table 5. The signed-centroid definition was checked in source Section 2. The antipedal is formed from the original OUTER polygon, not from its inverse or a polar. A bounded current primary-source review plus a targeted addendum did not locate an explicit proof of both formulas. Nearby generic Poncelet centroid theorems, the outer PEDAL centroid theorem and the rectangle geometry remain credited prior art, not new contributions. Priority and possible equivalent specializations remain unconfirmed. This is an explicit proof of the listed assertion, not a first-ever or large deep-open-problem claim. ## Presentation and review status The final English source compiled successfully in the built-in Codex LaTeX compiler. The same source was exported to main.pdf using an already installed Tectonic engine with cached resources; no new TeX installation or native-UI export is claimed. The four exported pages were rendered and visually inspected by root. The final export log has no overfull, underfull, undefined-reference or warning matches. The author line contains only Alper Ferudun with the affiliation and contact in a footnote. AI assistance appears in body prose. All public artifacts are original and licensed CC BY 4.0; cited third-party PDFs are not included. This is self-audited and unrefereed. Independent human review, formal proof-assistant verification, absolute priority, Google indexing and DOI resolver activation are not inferred from publication readiness or eventual publication.