Verified search, in which a language model proposes programs, a hard evaluator scores them, and selection keeps the best, has recently moved mathematical records; controlled ablations of the proposer-side components remain rare. We instrument a minimal FunSearch-style loop at office scale (a 30B local model on a laptop, 120-600 verified samples per run) with three operator packages: a schematic notebook the model writes and carries instead of verbatim elites, a named obstacle, and behavioural repulsion from constructions already found. On nine construction problems from a public repository, the complete 2^3 factorial with two replicates favours the primary contrast in a nominal two-stage analysis: the composition closes more of the seed-to-record gap (+0.196; nominal pooled p=0.023, stage-combination p~0.08; median per-problem effect +0.045). Repulsion raises construction-hash diversity everywhere (p=0.0039; partly a manipulation check). The factorial finds no positive memory-by-repulsion interaction (bounded to about +/-0.04); the gain decomposes additively, and memory+repulsion is the only arm that never collapses (0 of 18 runs), within 0.025 of the full composition. A frontier proposer under the identical loop reaches in tens of samples what the local model does not in hundreds; in single scoping runs its gains arrive without the operators. The search stalls after closing ~92% of the gap on the flagship problem, and the registered family-hint test gives the stall its first reading: named in words, the reference family is adopted and loses; handed as code, it is optimized, but our best finite-grid implementation remains below the plateau reached unaided. The loop transported and optimized the idea it was handed; no unaided run produced it. We release the harness, every candidate, and the dated pre-registrations.
Figures & tables
frac (pooled over 2 replicates)
A
B
C
D
BC
BD
CD
E
circle packing 26
0.703
0.974
0.976
0.000
0.989
0.752
0.980
0.992
ring loading 15
0.444
0.767
0.808
0.808
0.731
0.808
0.808
0.808
beat-the-average
0.181
0.384
0.519
0.608
0.661
0.789
0.540
0.709
autocorrelation C1
0.059
0.156
0.041
0.202
0.014
0.212
0.000
0.104
isosceles-free 642
0.560
0.560
0.560
0.399
0.560
0.560
0.560
0.560
sum-difference III
0.392
0.329
0.253
0.158
0.314
0.327
0.069
0.362
Table 1: The complete 23 factorial over nine problems, pooled over two independent replicates (120 verified samples per run). frac = fraction of the seed-to-record gap closed. E has the best mean and is best or tied-best in five of nine problems; D alone is bimodal: best on wide valid manifolds (pmfs, sequences), catastrophic where validity is fragile (packing geometry, constrained integer sets); BD is the only arm that never collapses, CD the one that collapses most; two problems (isosceles-free, 180! ) are family plateaus where every arm meets the same wall.
Figure 1: The primary contrast, problem by problem: E − A on the fraction of the seed-to-record gap closed (pooled over two replicates). Small grey dots are the two replicates (lines join them); the large dot is their mean. The pooled mean is +0.196 ; the median, +0.045 ; leave-one-problem-out means stay between +0.153 and +0.224 . Within-problem replicate spread is large on several problems (circle packing: +0.01 and +0.56 ), which the problem-level test does not see.
Figure 2: The completed 23 at a glance (cell means over nine problems, two replicates). Memory-on cells (blue) hold the four best gap fractions; D-on cells (circles) hold the highest construction-hash diversity; the two axes barely interact ( βM×R≈0 ). Collapse counts in parentheses: memory+repulsion is the only arm that never collapses; agenda+repulsion collapses most.
Figure 3: Beat-the-average under the same loop (arm E): fraction of the seed-to-record gap closed per verified sample. The local 30B at two grid resolutions; Claude Opus 5; dotted lines mark AlphaEvolve’s reported value (0.3890) and the best known (0.400695, Bellec–Fritz).
condition
seed 11
seed 12
seed 13
mean
finals in-family
no hint
0.3496
0.3607
0.3463
0.3522
0/3
generic hint
0.3512
0.3381
0.3086
0.3326
0/3
structural hint
0.3284
0.3383
0.3140
0.3269
3/3
crude in-family seed
0.3554
0.3679
0.3520
0.3584
2/3
strong in-family seed
0.3762
0.3739
0.3743
0.3748
3/3
Table 2: Family-hint test: final best per seed. In-family membership by the preregistered family-signature classifier (at most 50 support atoms, mass ≥0.3 on {0..3} , atoms in ≥3log10 distance-from-top bands; the structural hint instructs exactly this structure, so the classifier is partly a manipulation check of instruction-following). With three seeds per condition every contrast is descriptive (the smallest one-sided sign-flip p at n=3 is 0.125).
battery
registered
data already seen
primary
status
F ( 5×6×2 )
2026-08-22
none
E > A frac
p=0.0625 (floor)
F-ext ladder
2026-08-23
F
budget/grid/proposer
descriptive
F-ext2 ( n=9 )
2026-08-24
F, ladder
E > A frac
p=0.023 combined
23 completion
2026-08-24
F, ext2 partial
M × R >0
rejected ( p=0.47 )
Frontier scoping
2026-08-26
all above
estimation only
operators ≈ null
Family hint
2026-08-27
all above
optimize-once-given
yes as code, no as words
Table 3: Order of registrations and looks. The nine-problem primary analysis is a two-look sequence (discovery + prospective extension), not an independent confirmation.