README.md

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flake-band-gap

  • Builds the base-2 code-23 flake at any level: 4^level cells, face adjacency by plus-one lookup on each axis, rooted by breadth-first layers from the origin cell.
  • Asserts the face graph is a tree: 4^level nodes, 4^level - 1 edges, every node reached.
  • Counts eigenvalues below a shift by Sylvester's law on the leaf-to-root elimination of Lap - shift*I, never forming a matrix, and bisects both band edges: lo(level) in (1, 3) against the target 3*4^(level-1), and the top of the spectrum hi(level) in (4, 8) against 4^level.
  • Repeats the elimination over Fraction at shifts 2, 4, 4 - 1e-9, 4 + 1e-9, then back-substitutes the null vector at 4, clears it to integers, and takes (Lap - 4I)v in exact integer arithmetic.
  • One generator. Float bisection reaches level 11; the exact counts reach level 6 and the integer null vector level 4.

RUN

uv run python research/lab/py/flake-band-gap/flake.py
uv run python research/lab/py/flake-band-gap/flake.py 11 6 4
  • Both commands are the same run; the arguments are top exact_top vector_top. Under ten seconds on one core.
  • The source ran the float sweep to level 11 and the page table stops at level 10; this study prints both. At level 12 the bisection has spent its double precision and the ratio column stops rising, so 11 is the top it certifies.

WITNESSES

  • README.md:51 lower edge 1.000000, 1.827520, 1.975680, 1.996862, 1.999605, 1.999950, upper edge exactly 4 and simple, 3*4^(level-1) below 2, c near 12.9868.
  • the claims line (was DISCOVERIES.md:62) the same edge list, and the top of the spectrum 3 + sqrt(5) = 5.2360679775 at level 2 climbing to 5.7090316570 at level 6.
  • complexity.md:451-456 the edge table 1.000000, 1.827520, 1.975680, 1.996862, 1.999605, 1.999950, upper edge 4, zero strictly inside.
  • complexity.md:465-468 4 is not the top: the spectrum climbs to about 5.7090, split 3*4^(level-1) below 2 and 4^(level-1) at or above 4.
  • complexity.md:480-489 the defect table to level 10, ratios 5.7978, 7.0920, 7.7494, 7.9384, 7.9850, 7.9963, 7.9991, 7.9998, 7.9999, scaled defect 12.986750.
  • complexity.md:494 c near 12.9868: the run prints 12.986773 at level 11.
  • complexity.md:621 upper edge exactly 4: root pivot 0, no earlier zero pivot, residual 0, jump of one at 4.
  • The source's stored float values 4.000000000000001 and 3.9999999999999893 are a dense solver's noise; this study prints the exact upper edge instead and carries no such constant.
  • complexity.md:639 names the source generators and carries no number.