Exact certificates for r(n) = e(n) = U(n), n = 9 to 31: regular triangle unions on the unit circle

This version v11 (November 2026) closes n = 26 to 31 at the ceiling: r(26) = 290, r(27) = 301, r(28) = 312, r(29) = 325, r(30) = 336 and r(31) = 347. With them, r(n) = e(n) = U(n) holds for every n from 9 to 31. No previously deposited certificate changes; the published v10 package remains in this record unchanged. For n triangles whose 3n vertices lie on a circle in the regular cyclic order, r(n) is the largest number of sides the boundary of their union can have; e(n) is the same maximum without the circle constraint (OEIS A375986). For n >= 5, e(n) <= U(n) = 11n - 14 + 2 floor((n+1)/3) (G. Alkauskas, arXiv:2510.22584v5, Theorem, p. 3), so a certified configuration with U(n) sides gives r(n) = e(n) = U(n). Where the sequence stands "Closed" means: a certified configuration attains the proven upper bound U(n). n certified V = U(n) status 9 to 25 U(n) closed (v1 to v10) 26 290 (was 289) closed 27 301 (was 300) closed 28 312 (was 311) closed 29 325 (was 322) closed 30 336 (was 333) closed 31 347 closed (first value) One folder per value. The deposit now has a folder values/n<n>/ for every n from 9 to 31 with the same layout: the certified witness, a second witness where one exists, the outputs of the exact checks, the blind checker's files (n = 26 to 31) and the lineage. Values 9 to 25 are byte-for-byte copies of the certificates published in earlier versions. How the new values were found. The 290 at n = 26 came from continuation on the conditions of one target exposure set, starting from a certified 289. From the 290, single-triangle insertions gave 301 and 312. The 325 at n = 29 came from the 290 by a local move at one chord, an exact one-vertex repair to a new certified 289, and an insertion of three consecutive triangles (+36). From the 325, single-triangle insertions gave 336 and 347. Each new value has two witnesses with the same exposure set: the configuration as found and an exact simplification of it with short coordinates. Independent checks. Every new value passed four exact checks that agree (including a standalone verifier that uses only the Python standard library and imports nothing from the search or certification tools), a check that deleting any one triangle leaves at most U(n-1) sides, a definition check and an exact genericity audit. Each was also checked in a blind check by a second, independently implemented exact checker (OpenAI ChatGPT, run separately from the search tools), called "Astra" in the file names, which received the configurations with decoys under random labels; its boundary lists agree item by item with ours, and its checker, rerun locally, gives identical hashes. Verification. One command, sh VERIFY.sh, checks all SHA-256 sums and re-verifies every witness with the packaged standalone verifier (Python standard library only), asserting V = U(n) for each n from 9 to 31. What is not claimed. The deposit certifies the values for n = 9 to 31 only; it does not contain a proof that r(n) = U(n) for all n. Notes. Produced with AI assistance (Anthropic Claude; OpenAI ChatGPT as a second, independently implemented toolchain) for code, search and drafting under the author's direction. Every claimed value rests on an exact certificate in this record, reproducible with the included verifier. --- Version v10 (October 2026) raises the lower bounds at n = 27, 28, 29 and 30 by one each, with eighteen new certified configurations, and updates the standalone verifier. No previously certified value changes. Where the sequence stands n = 9 to 25: r(n) = e(n) = U(n), seventeen consecutive values. n = 26: 289, ceiling 290. n = 27: 300, ceiling 301. n = 28: 311, ceiling 312. n = 29: 321, ceiling 325. n = 30: 333, ceiling 336. The new configurations. Each of the four certified 289-sided configurations at n = 26 was extended by inserting one triangle per rung: 289 to 300 to 311 to 322 to 333. This gives four independent chains with four certified configurations at each of n = 27, 28, 29 and 30, plus two further configurations at n = 30 from slots resolved on a rerun, so the n = 30 record is realized by six configurations. All eighteen are distinct arrangements. In 13 of the 18 steps every earlier coordinate is kept exactly; in the 5 steps where the new triangle enters at the seam of the parametrization, the earlier coordinates are kept up to an exact rational rotation of the circle, and all parent points are recovered after that rotation. What was tested.For every parent in the chains, every insertion gain above +11 was examined at every slot, including the seam, with exact linear certificates and, where those left a slot open, exact one-triangle solves. None succeeded; every step gained exactly 11 sides. This is a statement about these parents at their fixed coordinates, not a bound on r(n): raising n = 27 to 30 further requires different parents, most directly a 290-sided configuration at n = 26. Thinness. The new configurations are thinner than their parents, with exact minimum separations from about 3e-13 down to about 2e-31. Certificates are valid at any exact positive separation; each certificate records its exact minimum separation as a robustness label, not as an acceptance test. Verifier update. The standalone verifier (Python standard library only) now lifts Python's default limit on the number of digits in integer conversion, which would otherwise reject certificates with coordinates of more than 4,300 digits. Its output is byte-identical to the previous version's on every previously deposited certificate; the previous version is kept in the package for comparison. ---- Version v9 (October 2026) Adds four certified configurations with 289 sides at n = 26 and two further arrangement classes of the 255-sided maximum at n = 23. No previously certified value changes. Where the sequence stands n = 9 to 25: r(n) = e(n) = U(n), seventeen consecutive values. n = 23: 255 = U(23), now in four distinct arrangement classes. n = 26: 289, ceiling 290 (new in v9; previously 288). n = 27: 299, ceiling 301. n = 28: 310, ceiling 312. n = 29: 321, ceiling 325. n = 30: 332, ceiling 336. The n = 26 configurations. Four certified configurations with 78 corners and 211 exposed crossings. Two were found by insertion of one triangle into certified 277-sided configurations at n = 25; two by inserting three consecutive triangles into the certified 255 at n = 23. All four are extremely thin in their original coordinates (minimum parameter gaps from about 1e-20 down to 1e-59); separation-polished versions with the same boundary and exposure set (minimum separation about 7e-13) are deposited alongside the originals. Certificates are valid at any exact positive separation; the separation is recorded as a robustness label, not used as an acceptance test. The two new n = 23 classeswere found by direct random-start search with exact acceptance. With them, r(23) = 255 is realized by four arrangement classes, which share no single boundary word. --- Version v8 (September 2026) closes n = 23. With it, r(n) = e(n) = U(n) holds for every n from 9 to 25: seventeen consecutive values. Where the sequence now stands:- n = 9 .. 25: r(n) = U(n); n = 23: **255 = U(23)**, closed 2026-09-27- n = 26: 288, ceiling 290- n = 27: 299, ceiling 301- n = 28: 310, ceiling 312- n = 29: 321, ceiling 325- n = 30: 332, ceiling 336 The n = 23 configuration. It was found by direct search, not by insertion: a 5-minute cold C++ search. It has 69 corners and 186 exposed crossings, and its exact minimum separation is 1.549e-7 after the standard margin polish. Two certificates are deposited, in two distinct arrangement classes. Why n = 23 took this long (the acceptance-rule post-mortem). Every search from n = 18 on accepted a configuration only if a floating-point test found all boundary vertices at least 1e-5 apart. The best configurations are thinner thanthat: n = 23 summits sit at 4e-8 .. 2.2e-7, and so do several already-certified records, including r(22) = 242. With the float guard at 1e-9 and every thin stateescalated to exact certification, the mean of 5-minute cold searches rose from 246.2 to 251.2, and a 255 appeared after 207 colds. The exact certifiers werealways the arbiter; the guard only decided what reached them. Parameterizing the guard changes no previously certified value (re-gate over all 37 certified states,n = 9..25). --- Version 7 (September 2026) This version (v7) extends the record from n = 21 to n = 30. It closes three further rungs at the proven ceiling (n = 22, 24 and 25) and gives the first values ever recorded for n = 24 through n = 30. Every configuration is deposited as exact rational parameters with the code to check it. --- 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.