First-known no-isosceles grid sets for open n values

Numaro Autoresearch Team

Plain language

What this result means

This result matters because the rule is local and exact. A set is valid only if no chosen point sees two other chosen points at the same squared distance. That one repeated distance would immediately create an isosceles triangle.

AlphaEvolve published strong sets at n=64 and n=100, but not the open n values reported here.
The 54 claimed rows are the evolve_sweep records in RECORDS.jsonl.
The largest reported sets are n=77 and n=79, each with 116 selected points.

Visual notes

How to read the result

Line chart of no-isosceles grid set sizes for open n values, with AlphaEvolve n=64 and n=100 shown as hollow reference points. — **Open grid sizes**White points are Numaro first-known rows. Hollow points are AlphaEvolve's published n=64 and n=100 rows, which are not beaten.

Scatter plot of the 116-point no-isosceles set in the 77 by 77 grid. — **Actual n=77 set**The 116 points are rendered from RECORDS.jsonl. The dashed midlines show the symmetry-preserving search space.

Distance-ring visualization around one selected point in the n=77 no-isosceles set. — **Distance test**For one anchor, each squared distance is unique. The checker repeats the same exact test around every selected point.

Result table

Fifty-four first-known no-isosceles grid values for open n values.

Cell	Baseline	Numaro	Delta	Note
n=22	none listed	36	first-known	first claimed row
n=23	none listed	40	first-known	integer-coordinate set
n=24	none listed	40	first-known	integer-coordinate set
n=25	none listed	40	first-known	integer-coordinate set
n=26	none listed	42	first-known	integer-coordinate set
n=27	none listed	44	first-known	integer-coordinate set
n=29	none listed	48	first-known	integer-coordinate set
n=31	none listed	52	first-known	integer-coordinate set
n=32	none listed	52	first-known	integer-coordinate set
n=33	none listed	52	first-known	integer-coordinate set
n=35	none listed	56	first-known	integer-coordinate set
n=36	none listed	56	first-known	integer-coordinate set
n=37	none listed	64	first-known	integer-coordinate set
n=38	none listed	60	first-known	integer-coordinate set
n=39	none listed	60	first-known	integer-coordinate set
n=40	none listed	64	first-known	integer-coordinate set
n=41	none listed	64	first-known	integer-coordinate set
n=42	none listed	64	first-known	integer-coordinate set
n=43	none listed	68	first-known	integer-coordinate set
n=44	none listed	68	first-known	integer-coordinate set
n=45	none listed	72	first-known	integer-coordinate set
n=46	none listed	70	first-known	integer-coordinate set
n=47	none listed	72	first-known	integer-coordinate set
n=48	none listed	74	first-known	integer-coordinate set
n=49	none listed	76	first-known	integer-coordinate set
n=50	none listed	78	first-known	integer-coordinate set
n=51	none listed	80	first-known	integer-coordinate set
n=52	none listed	80	first-known	integer-coordinate set
n=53	none listed	84	first-known	integer-coordinate set
n=54	none listed	80	first-known	integer-coordinate set
n=55	none listed	84	first-known	integer-coordinate set
n=56	none listed	84	first-known	integer-coordinate set
n=57	none listed	88	first-known	integer-coordinate set
n=58	none listed	92	first-known	integer-coordinate set
n=59	none listed	92	first-known	integer-coordinate set
n=60	none listed	92	first-known	integer-coordinate set
n=61	none listed	92	first-known	integer-coordinate set
n=62	none listed	92	first-known	integer-coordinate set
n=63	none listed	96	first-known	neighbor of published n=64
n=65	none listed	96	first-known	neighbor of published n=64
n=66	none listed	100	first-known	integer-coordinate set
n=67	none listed	104	first-known	integer-coordinate set
n=68	none listed	104	first-known	integer-coordinate set
n=69	none listed	104	first-known	integer-coordinate set
n=70	none listed	104	first-known	integer-coordinate set
n=71	none listed	112	first-known	integer-coordinate set
n=72	none listed	112	first-known	integer-coordinate set
n=73	none listed	112	first-known	integer-coordinate set
n=74	none listed	112	first-known	integer-coordinate set
n=75	none listed	112	first-known	integer-coordinate set
n=76	none listed	112	first-known	integer-coordinate set
n=77	none listed	116	first-known	largest value in table
n=78	none listed	112	first-known	integer-coordinate set
n=79	none listed	116	first-known	largest value in table

Method

How it was found

The campaign improved the generator of point sets, not just one fixed set. The best rows came from an evolved construction that keeps symmetry while adding and removing whole orbits of points.

Loaded and checked the published AlphaEvolve n=64 and n=100 reference sets for calibration.
Observed that direct point-level search plateaued at lower density.
Evolved construction parameters for symmetry group, axis, ruin fraction, local search, and candidate count.
Kept the 54 evolve_sweep rows whose saved point sets pass the exact no-isosceles check.

Verification

How it was checked

For each selected point P, the checker computes the squared distance from P to every other selected point. If a squared distance appears twice, P and those two points form an isosceles triangle, so the set fails. All 54 reported evolve_sweep rows pass this exact integer check.

Scope

What is not being claimed

These are first-known lower bounds for open n values, not optimality proofs. They are not beats of AlphaEvolve's published n=64 and n=100 values. The missing open sizes in this sweep are n=28, n=30, and n=34; n=64 already had a published AlphaEvolve row.

References

Baseline sources

Citation

How to cite

Numaro Autoresearch Team. "First-known no-isosceles grid sets for open n values." Numaro Research Report NUMARO-2026-010, 2026.

@techreport{numaro2026NoIsoscelesGrid,
  title = {First-known no-isosceles grid sets for open n values},
  author = {Numaro Autoresearch Team},
  institution = {Numaro},
  number = {NUMARO-2026-010},
  year = {2026},
  url = {https://numaro.tech/research/no-isosceles-grid-2026/}
}