A solver written during one self-improvement run exceeded 21 best-known values in the 10 August 2026 Packomania snapshot. A separate sweep used the extracted solver unchanged across N=2 through N=93 unequal circles in a unit square, maximizing their total radius while preventing overlaps.
Across 92 sizes, the sweep produced 21 improvements, 45 ties, and 26 shortfalls, with no infeasible packings. Improvements occurred at N=50–55, 62, 63, 66–69, 71, 72, 77, 80, 82–85, and 87, with absolute margins from 2.3e-5 at N=55 to 2.6e-3 at N=51. These are dated comparisons with Packomania, whose entries continue to change.
This followed an earlier solver’s 27-circle improvement. The sweep does not establish whether the agent’s disposition or permission to rewrite values caused the improvement.
What changed during the run
The run used an expansive disposition and unrestricted permission to rewrite it. Ten substantive iterations took 117 minutes and $28.83 in API spending, using one agent seed. The run used no web assistance or copied external solver code. The API figure is not a complete CPU-compute budget or a matched comparison with other systems. The final records combine several search passes: a base sweep taking roughly seven hours on ten CPU cores, followed by targeted searches with larger per-seed budgets. The reproduction instructions separate those costs from agent development.
The decisive solver change appeared at iteration two. Iteration one built a multistart search that finished 3.5e-4 below the N=54 reference. The next iteration included centre movements in a linear-programming step and exceeded that value.
Later iterations failed to improve the N=54 incumbent: iterations three and four accepted no moves; iteration five found no improvement in 504 fresh starts; iterations eight through ten tested changes that lost score. The run also generalized the solver to other circle counts. The eventual 21-size result came from subsequent search passes rather than 21 separate breakthroughs inside the agent loop.
How the solver works
With circle centres fixed, choosing the best radii is a linear program. Each radius must fit between its centre and the square’s walls, and each pair must sum to no more than the distance between its centres. This formulation appears in Eppstein’s work. It solves radius selection for a proposed set of centres, leaving centre placement as the outer search problem. Numerical solutions still require feasibility checks.
The new solver used sequential linear programming to move the centres. A linear lower bound on centre distance supplies conservative non-overlap constraints, allowing feasible improvements toward a locally jammed packing, typically in about a second. A second stage perturbed those packings and accepted some small losses under a tightening threshold, retaining the best packing throughout.
The ingredients have established precedents: Eppstein’s radius optimization; convex-concave linearization associated with Shen, Diamond, Gu and Boyd; monotonic basin hopping developed by Addis, Locatelli and Schoen for disk packing; and Dueck and Scheuer’s threshold accepting. The result demonstrates autonomous assembly and tuning of known methods, without establishing a new optimization technique.
Independent checks and a changing reference table
A separate checker recomputed circle counts, containment, pairwise separation, and total radii from saved coordinates, sharing no code with the solver. All twenty-one improvements were strictly feasible under that check.
At N=54, the smallest wall clearance was 1.0e-11 and the smallest circle-to-circle gap 1.8e-11, compared with a 9.0e-4 improvement over the historical reference. The improvement is seven to eight orders of magnitude larger than those clearances. Feasibility and the objective value are independently checkable properties of the saved packing.
The reference table combines older contributed packings with recently optimized entries. These twenty-one improvements concerned the older group; they did not exceed the recent N=26 entry credited to Haowei Lin or the roughly 25 improvements Everett Dutton contributed on 1 August. At N=26, this solver was 2.5e-10 below the reference, counted as a numerical tie.
Between the 3 and 10 August snapshots, 43 of the first 100 entries increased. Fifteen of the twenty-one improved sizes had their reference raised during that week, and the new packings exceeded those raised values. Conversely, the earlier N=27 result had already entered the table: rescoring the original solver against the 10 August snapshot gave zero improvements, 28 ties, and 70 shortfalls. It now tied its own contribution.
A follow-up produced no additional records
A second run began from a byte-identical copy of the finished solver. It tested whether packings found at other sizes could provide useful starting points for further records. The run cost $24.39 and found no additional record crossings.
At N=44 it finished 3e-11 below the reference, a numerical tie. An apparent N=50 improvement reproduced a packing already held by the first solver; 40 of 40 warm-start rounds returned the same value. Tests at three targeted sizes produced no crossings. Transplanting parts of other packings improved some candidates without crossing a reference value; a final ten-arm attempt also yielded zero of ten. The predictions preceded their measurements.
The last five iterations generated 48 endpoints, which reduced to 24 distinct packings after treating rotations and reflections as equivalent. These counts suggest that the tested generators repeatedly revisited existing local optima. Additional independent seeds did move some sizes from below-reference to above-reference values; the tested recycling strategy added none.
Each self-improvement experiment used one seed, so the contrast between a productive first run and a saturated follow-up remains one observation. The twenty-one independently checkable packings exceeded a dated reference table. They are best-known-value improvements for that comparison, with no claim of global optimality or permanent record status.