cert-machine · audit · a registry's unanswered claims

The AlphaEvolve repository's community certificates, decided

Since June, eight issues on DeepMind's repository of problems have offered explicit objects that beat its recorded bests — grid sets with no five points on a sphere up to n = 40, two 58-point isosceles-free sets where the literature says 56 is optimal, rational ring-loading instances at 47/42 and 9/8, a sum-difference set past the published bound — most of them found by language-model searches, none answered by the repository. Every finite object among them is decided here in exact arithmetic from the notebooks' own definitions: 70 hold, none fails, and 2 claims are refused because there is nothing finite to decide — a floating-point sofa and an 86-digit modulus.

The claims: issues #4–#12 of github.com/google-deepmind/alphaevolve_repository_of_problems (the repository itself pinned at 8f447457, 2026-07-11, Apache-2.0), their inline objects and the certificate files they link (hashkanna/mendel-evolve, milesandmistakes/no-five-sphere-grid-certificates v1.1.1, hraness/algal-lab, two attachment zips), every byte pinned by sha256 in corpus/aerp/PINS.json and read as data — no claimant code runs. The definitions are the pinned notebooks' (prompt text and verification cells). Pre-registered in corpus/aerp/preregistration.json. Optimality is nowhere claimed or decided: a certified row says "at least".

tl;dr

The finding. Every finite claim holds. No five on a sphere: 61 point sets from four contributors, n = 13 to 40, 68,326,809 quadruples keyed by their exact hyperplane and no two alike — the best certified sizes run from 36 at n = 13 to 104 at n = 40 where the repository stops at n = 12 with 33. No isosceles triangles: both 58-point sets at n = 32 are clean over all 30,856 triples and maximal; the literature's "f(32) = 56, SAT-optimal" (PatternBoost) is contradicted by an explicit object. Ring loading: 47/42 holds under either reading of the problem; 9/8 holds under the thread's reading (k = 1..m) and, as the notebook writes the problem (k = 1..m−1), only once padded with two zero pairs or in the m = 17 instance of the zip — the notebook's own text and its code agree with each other and differ from the thread's. Sums and differences: the far-block set rebuilt from its rule has g = 1.16334 > 1.163, above Penman–Wells' 1.12594 and the repository's 1.12194; the 23,646-integer set violates the strong inequality by exactly 89.
The mechanism. Integers only: squared distances for triangles; for spheres, the 4-subset hyperplane key (a primitive integer 5-vector) whose duplicates are exactly the zero 5 × 5 determinants, cross-checked against every 5-subset's determinant on the smaller sets; for ring loading every 2ᵐ sign pattern on lcm-scaled integers; for sumsets a bitset, with g > 1.163 decided as an integer inequality. The repository's own constructions decide as its notebooks report (9 controls).
Check it. python3 -I tools/run-aerp-ledger.py · python3 -I instruments/aerp/battery.py — 22 checks, 9 red controls.

claims decided

0 refuted · 2 refused (no finite object). Four problems, eight issues, five contributors.

C(40) ≥

104

No five on a sphere, 40 × 40 × 40; the repository records 33 at n = 12 and nothing beyond.

C(32) ≥ 58

two sets

No isosceles triangles; "56, SAT-optimal" in PatternBoost §4.1 — an explicit contradiction.

ring loading

9/8

Under k = 1..m; under the notebook's k = 1..m−1 the same bound needs the padded or the m = 17 instance.

§1 · no five on a sphere

Problem 60: the frontier, n = 13 to 40

The repository's notebook stops at n = 12 (33 points) and its verifier is floating point (np.linalg.det with isclose). The thread grew from one 36-point set at n = 13 to four contributors leapfrogging each other; the best certified size per n, every one decided here exactly:

n	best certified	by	repository
13	36	Harmenszoon	—
14	38	milesandmistakes	—
15	42	hashkanna	—
16	44	hashkanna	—
17	46	hashkanna	—
18	48	0thernet	—
19	51	hashkanna	—
20	54	hashkanna	—
21	56	hashkanna	—
22	59	hashkanna	—
23	62	hashkanna	—
24	64	hashkanna	—
25	66	hashkanna	—
26	70	hashkanna	—
27	72	hashkanna	—
28	74	hashkanna	—
29	76	hashkanna	—
30	80	hashkanna	—
31	82	hashkanna	—
32	85	hashkanna	—
33	86	hashkanna	—
34	90	hashkanna	—
35	91	hashkanna	—
36	94	hashkanna	—
37	96	hashkanna	—
38	99	hashkanna	—
39	102	hashkanna	—
40	104	hashkanna	—

All 61 sets (40 of hashkanna's files including the stepping stones, 10 of milesandmistakes' RECORD files, 10 of 0thernet's, #4's one) are CERTIFIED: in the grid, distinct, no dependent quadruple, no two quadruples on one sphere or plane. Where the 5-subset determinant was also run (sets of at most 45 points) the smallest |det| is 2 throughout — the claimants' own figure, and the margin at which a float verifier is one rounding from wrong.

§2 · no isosceles triangles

Problem 59: 58 where 56 was called optimal

hashkanna's two 58-point subsets of the 32 × 32 grid (#10) contain no three points with two equal distances, the collinear midpoint case included: 30,856 triples each, zero isosceles, and neither admits another grid point (maximal). The external certificate file equals set_a byte for byte in content. PatternBoost (arXiv:2411.00566 §4.1) captions "f(32) = 56" and says SAT solvers prove optimality up to n ≈ 32; the AlphaEvolve paper repeats the figure. An explicit 58-point object under the notebook's definition means one of three things — a different definition in the SAT work, a slip in the caption, or an error here — and this page can only rule out the third: the decider reproduces the notebook's own 112-point n = 64 and 164-point n = 100 constructions as admissible and catches a planted midpoint.

§3 · ring loading

Problem 61: which k the problem ranges over

instance	m	α, k = 1..m−1 (the notebook)	α, k = 1..m (the thread)	padded, notebook	verdict
#11 VividhDesign, 2026-10-06	15	47/42	47/42	47/42	CERTIFIED
#12 Maks23, 2026-10-08 · min98_m14_D16.txt	14	3/4	9/8	9/8	CERTIFIED
#12 Maks23 · rec98_D16.txt	15	3/4	9/8	9/8	CERTIFIED
#12 Maks23 · A17_exact_D560.txt	17	9/8	9/8	9/8	CERTIFIED
#12 comment, karldaeubel, 2026-10-13	12	3/4	9/8	9/8	CERTIFIED

The notebook's prompt and its compute_alpha_exact both range k over 1..m−1; #12 ranges over 1..m, where the extra term is |Σzᵢ|, and says so. The two readings agree on padded instances, so C ≥ 9/8 holds under the notebook's reading too (m = 16 padded, or the m = 17 instance at denominator 560 without padding) — the bound stands, and which m attains it is a matter of convention. Under the strict notebook reading the m = 14 table gives 3/4. The thread's karldaeubel instance (12 pairs, two identical blocks) attains 9/8 likewise.

§4 · sums and differences

Problem 42: the finite parts of a large claim

#5 offers much — C = 2, a 1.6269 instance, a ceiling theorem, a conjecture — and asks to be torn apart; its author then found Lin–Li (arXiv:2607.27199) had settled C = 2 two days earlier. Two parts are finite objects and are decided: the far-block set its certificate script defines (rebuilt here from the rule, |A| = 101,478, |A+A| = 1,200,837, |A−A| = 848,817) has g = 1.163338, and g > 1.163 is decided as |A+A|¹⁰⁰⁰|A|¹⁶³ > |A−A|¹¹⁶³ in integers — above Penman–Wells' explicit 1.12594 and the repository's 1.12194 (its 309-element list re-counted: 1,367 sums, 1,163 differences). The explicit 23,646-integer set of its §5 has |A+A| = 6,924,316, |A−A| = 3,370,545, diameter 3,577,327: |A+A| + |A| exceeds |A−A| + diam + 1 by 89, as stated. The 1.6269 instance (an 86-digit modulus) is REFUSED: nothing enumerable, and its count rests on an argument its authors call machine-generated and unreviewed.

§5 · refused

What has no finite object to decide

claim	why refused
C >= 1.6269 by the "hollow bridge" instance of #5 (43 gadgets, an 86-digit modulus)	the set is not enumerable (modulus ~10^86); its count rests on the write-up's own argument, which is machine-generated and unreviewed by its authors' statement; nothing finite to decide here
a 3-D moving sofa of volume >= 1.8558404 (#8; the repository records >= 1.81)	the certificate is a floating-point cube count under stated position and angle tolerances (0.005, 0.05 degrees; the author's own note: 1.7701 at the quick-start tolerances); no exact object is published. To decide it an interval instrument would have to enclose every pose, prove the swept region contains each counted cube and sum a rational lower bound — not built here

A refusal is a verdict about the evidence, not the mathematics: the sofa may well have volume 1.8558, but a cube count under a 0.05° tolerance is a measurement, and the page says so rather than lending it a certificate.

§6 · limits

What is not decided here

Optimality. Every certified row is a lower bound; no upper bound is decided, and the thread's own optimality remarks (an MILP at m = 4, 5; "SAT-optimal 56") are not re-run.
Priority. Who found what first (Demonstrandum's n = 16 42-set, the stepping stones, the corrective release) is read from the thread and not adjudicated; the rows name the file decided.
The definition of ring loading. The notebook's k-range is taken as the problem's; whether the original ring-loading constant wants k = 1..m is a question for the problem's source, not decided here.
Same-author second implementation. The two sphere deciders were written by the same author from the same definition; they catch a slip in one arithmetic, not a shared misreading. The repository's own constructions are the outside control.