Constructions and exact checkers for "AI-Run Research on Nenya Discovers the Headroom Law, Recursively Improves Its Own
Harnesses, and Sets 325 Mathematical Records" (QWNTL Labs, October 2026) and its companion "Past the Frontier, Exactly:
Long-Horizon Agents Set New Best-Known Bounds and Record Circle Packings". Paper: https://qwntl.com/frontier

All construction files are plain text; lines starting with '#' are comments; every decimal is an exact rational.

erdos-A51.txt       3,584 heights a_i in [0,1] with sum exactly 1792 = n/2: a step function h on [0,2] (cell width 2/n),
                    integral 1. M <= (2/n) max_k sum_i h_i (1 - h_{i+k}) = 97499325154129259661263000479811180603/(256*10^36).
erdos-n2560.txt     2,560 heights in [0,1] with sum exactly 1280 = n/2: the re-grid onto 2,560 cells, scored as above.
                    M <= 487496381280743038886397039070859359383/(128*10^37) = 0.38085654787558049912999768677...
c1-A43.txt.xz       262,144 nonnegative heights on [-1/4,1/4]. C1 <= 2n max_k c_k / S^2, c_k = sum_{i+j=k} a_i a_j,
                    S = sum a_i; = 1.50274043592276697640985...
c3-n102400.txt.xz   102,400 signed heights on [-1/4,1/4]. Signed constant: 2n max_k c_k / S^2 = 1.44888247044528093575...
                    Not an Einstein Arena-eligible input (more than 100,000 values); derived from MILO's published coarse
                    vectors. Its absolute-value score max|f*f|/(int f)^2 is 5.827..., so it says nothing about that variant.
c3-n97000.txt.xz    97,000 signed heights on [-1/4,1/4]: the third re-grid, within the 100,000-value limit.
                    Signed constant: 2n max_k c_k / S^2 = 1.44886703134291290097000893933...
minratio14_3d.txt   14 points 'x y z'; (d_max/d_min)^2 = 4.16578347458007331688... (a polish of FICO's configuration).
csqv.tar.xz         csqv/csqvN.txt for the 306 sizes: N lines 'x y r', unit square [0,1]^2 in the corner frame (subtract
                    1/2 for Packomania's centred frame). Feasible as written: r > 0, r <= x, y <= 1 - r and
                    (xi-xj)^2 + (yi-yj)^2 >= (ri+rj)^2 for every pair, all exactly.
csqv_values.txt     N, exact sum of radii, previous best located (as of 2026-10-09), margin, run.
verifier/           verify_all.py runs the checkers below on these files and compares with the paper's values:
                      cd qwntl-frontier-constructions && python3 verifier/verify_all.py          (everything)
                      cd qwntl-frontier-constructions && python3 verifier/verify_all.py --quick  (analytic files, three packings)
                    Needs Python >= 3.9 and libgmp. The checkers are copied unchanged (hashes in SHA256SUMS):
                      audits_crossfamily/exact.py  cross-family auditor (GPT-6 Astra): Erdos and C1, exact GMP
                                                   convolution of carry-free packed integers, plus direct integer sums
                      audit_geo/exact.py           cross-family auditor: decimal parser, all-pairs check, 14-point ratio
                      audit_geo2/exact_all.py      cross-family auditor: integer grid check with a completeness proof
                      audit_c3/exact.py            cross-family auditor: signed and absolute C3 scores
                      builder/verify_indep.py      the geometry agent's own independent packing verifier
                      builder/verify_minratio.py   the geometry agent's own 14-point verifier
                    verify_all.py itself was written by the paper's authors.
analysis/           boundary_law.py: the paper's boundary-law analysis of the 306 packings (hexagonal deficit bands,
                    margin against the previous entry's excess deficit, leave-out forecast), read from csqv_values.txt:
                      cd qwntl-frontier-constructions && python3 analysis/boundary_law.py
SHA256SUMS          sha256 of every file above (uncompressed); construction hashes equal the frozen claim manifests.
