fill-polynomials

• Fits the odd-side fill polynomial of every design at dim 1..4 by exact Lagrange interpolation on the side index k = 1..dim+1, checks it out to k = 10, and counts the distinct polynomials against A129824.
• Walks the hyperoctahedral group to get the symmetry classes and counts how many classes their members split on the lower coefficients, and on the leading coefficient.
• Peels unit-linear factors (a n + 1) with integer slope a from every dim 4 signature and reports the degree-4 remainders that still split over Q into two quadratics.
• Factors every fill polynomial at dim 2..6 over Q and collects the discriminants of the quadratic factors, under two readings: every quadratic factor, and only a degree-2 peeled remainder.
• Rebuilds the seven census observables at dim 3 by literal cell and face counting at both side families, fits each on the side index n = 1..4 with n = 5, 6 held out, and calls a design locked when no observable normalizes to c n^k prod(a_i n + 1).
• Checks the grid-rendered fill count against the closed-form sum over filled corners on all 256 designs at dim 3, odd sides 3..13.
• Compares that lock against the origin-free path-or-edgeless predicate on all 256 designs.
• Domain run: all 2^(2^dim) designs at dim 1..4, all signatures at dim 2..6 (1053696 of them at dim 6), all 256 designs at dim 3 for the lock. Whole run is 90 seconds on 8 workers.

RUN

uv run python research/lab/py/fill-polynomials/fills.py

WITNESSES

• method.md:283 - distinct polynomials 4, 12, 64, 700 at dim 1..4, equal to A129824.
• method.md:285-286 - lower coefficients split 4 of 6 classes at dim 2, 20 of 22 at dim 3, 400 of 402 at dim 4; the leading coefficient splits 0 everywhere.
• the claims line (was DISCOVERIES.md:431) - the seven dim 4 signatures whose quartic remainder splits into two centered-polygonal quadratics: (1,0,1,0,1), (1,0,2,0,1), (1,1,2,1,1), (1,2,3,2,1), (1,3,2,3,1), (1,4,2,4,1), (1,4,5,4,1).
• the claims line (was DISCOVERIES.md:341) - the negative discriminants form a gapless run -3..-4 at dim 2, -3..-12 at dim 3, -3..-24 at dim 4, of lengths 2, 6, 12; the run reaches length 20 at dim 5, matching the page's prediction.
• the claims line (was DISCOVERIES.md:341) - the depth -63 is reached: at dim 6 the peeled-remainder reading is gapless from -3 to -63, length 31, so the page's dim(dim-1) = 30 at dim 6 is one short; under the wider reading of every quadratic factor the dim 6 run is length 80, to -160.
• the claims line (was DISCOVERIES.md:445) - 14 locked designs, 9 on the path clause 62, 94, 110, 118, 122, 124, 188, 218, 230 and 5 on the edgeless clause 128, 134, 146, 148, 150.
• The path-or-edgeless predicate and the census lock rule agree on 256 of 256 designs, checked here and carried nowhere else.
• method.md:275-279 - the two generators agree: the grid count matches the closed form on 256 of 256 designs at dim 3, over the six odd sides the census renders.