Exact certificates for r(n) = e(n), n = 9–21: regular triangle unions

Abstract

Version 6 (Augustus 2026) adds r(21) = e(21) = 231, closing the hardest rung to date after a five-day resistance documented in CHANGELOG_v6.md; the pattern now holds for thirteen consecutive values. -- Version 5 (August 2026) adds r(19) = 207 and r(20) = 220, extending r(n) = e(n) = U(n) to twelve consecutive values; the n=20 census was dual-computed by the established Python pipeline and a gate-validated native kernel with exact agreement. --- Version 4 (August 2026) closes the question left open in v3: r(18) = e(18) = 196, certified circle-inscribed and independently verified. The pattern r(n) = e(n) holds continuously for n = 9 through 18; the apparent separation was a search-capability artifact, documented in CHANGELOG_v4.md. --- Version 3 (August 2026) adds three results: r(17) = e(17) = 185, extending the circle-inscribed series to nine consecutive values meeting the proven combinatorial ceiling. r(18) >= 195, an exact circle-inscribed certificate one below the ceiling of 196. e(18) = 196: the first FREE-PLANAR certificate in this series, consisting of 54 rational point coordinates not on a circle, 196 sides meeting the ceiling, with a new decisive assertion verified in exact integer arithmetic: REGULARITY, i.e. the boundary cycle visits the 54 triangle corners with labels 0..17 repeated exactly three times. Consequently the sequence A375986 extends to a(18) = 196, attained off-circle, while the best known circle configuration at n = 18 has 195 sides: whether r(18) = 195 < e(18), which would be the first separation of the circle-restricted and regular quantities, or r(18) = 196, is open and under active search. All three new certificates passed the same five-tier verification standard as v1/v2 (two independently written exact-arithmetic verifiers, two execution environments, zero floating point in any decisive predicate); the three independent verifiers are included with SHA-256 hashes in CHANGELOG_v3.md. See CHANGELOG_v3.md for details and candid provenance notes. ----- Version 2 (August 2026) extends the results to n = 16: exact certificates for r(13)=137, r(14)=150, r(15)=161, r(16)=172 are added, each verified to the same standard as v1 (two independently written exact-arithmetic verifiers, two environments, zero floating point). See CHANGELOG_v2.md for details. The sequence A375986 now reads 3, 12, 22, 33, 45, 56, 67, 80, 91, 102, 115, 126, 137, 150, 161, 172. Summary This deposit contains explicit, exactly-verifiable configurations answering and extending open questions from: G. Alkauskas, Regular triangle unions with maximal number of sides, arXiv:2510.22584 (v5, April 2026). For n triangles inscribed in the unit circle with their 3n vertices in cyclic arrangement (a regular union, in the paper's sense), r(n) denotes the maximal number of sides of a union that is a simple polygon. The paper proves the combinatorial ceiling e(n) ≤ 12n − 12 − γ(n+1) with γ(n+1) = n + 2 − 2⌊(n+1)/3⌋, poses "prove rigorously that r(9) = 90" as Open Question 2, and asks in Question 3 to improve the bound r(n) ≥ 10n − 7. Main results certified here: r(9) = e(9) = 91 — answering Open Question 2 in the opposite direction to the conjecture; r(10) = e(10) = 102, r(11) = e(11) = 115, r(12) = e(12) = 126 — three new exact values of the sequence e(n) (cf. OEIS A375986: 3, 12, 22, 33, 45, 56, 67, 80, 91, ...), each meeting the proven ceiling; consequent data for Open Questions 6 and 7: the observed increments are 11, 13, 11 (exactly the ceiling increments; no increment of 14), consistent with limsup e(n)/n = 35/3. The certificates Each certificate (certificates/r{n}_exact_certificate.json) is a list of 3n rational numbers t, in increasing order. The corresponding vertex is P(t) = ((1 − t²)/(1 + t²), 2t/(1 + t²)), which lies exactly on the unit circle for rational t. Increasing t corresponds to circular order (wrapping through (−1, 0)); the vertex at position j belongs to triangle j mod n. The claim per certificate: the union of the n closed triangles is a simple polygon with exactly S sides (S = 91, 102, 115, 126), all 3n corners on its boundary in circular order. Verification Two independently written verifiers are included; both use only Python's standard-library fractions.Fraction — no floating point enters any decisive predicate: verifiers/exact_certifier_pipeline.py — the author-side certifier; verifiers/independent_verifier_generalized.py — an independent verifier written from scratch by OpenAI's ChatGPT on request, covering all four certificates. It additionally checks: no coincident vertices, no degenerate triangles, no vertex on a foreign edge, no collinear foreign edges, no endpoint/tangent contacts, no three concurrent edges, boundary graph 2-regular with a single component, no collinear boundary nodes, all corners genuine polygon vertices in circular traversal order, and connectedness of the triangle-interior overlap graph. verifiers/independent_verifier_n9.py is its original n = 9 version. Both verifiers were cross-executed in two separate environments with identical output. To verify yourself: python3 verifiers/independent_verifier_generalized.py (Python ≥ 3.9, no dependencies; runtime seconds to minutes). Method and provenance The configurations were found with substantial help from AI systems (Anthropic's Claude; independent verification code by OpenAI's ChatGPT). Blind numerical search over circle configurations reliably plateaus just below sharp optima (reproducibly 44/45 and 77/80 on the paper's known Pentastar/Octastar values, which may explain the experimental value 90 at n = 9 reported in the paper). The successful approach was combinatorics-first, built on the paper's own triangulation-shift tool: (1) exhaustively enumerate maximal-weight triangulation shifts of the (n+1)-gon; (2) compile each champion into its full boundary word (the compiler reproduces the paper's 79-edge worked example symbol-for-symbol and its Pentastar/Octastar structure); (3) solve the geometric realization on the circle guided by the target word; (4) inflate degeneracy margins, round to rational circle points, and certify exactly. search_code/ contains the complete pipeline. License Code: MIT. Data (certificates) and accompanying text: CC BY 4.0. If you use these certificates or values, please cite this deposit and arXiv:2510.22584.