README.md

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base-q-anf

  • Builds the base normal form of a design: the inverse Vandermonde on 0..base-1 applied axis by axis over GF(base), and the integer Mobius transform along the product of chains.
  • Proves both round-trip on every value table by the matrix identities T E = E T = I over GF(base) and B M = I over Z, which settles base 3 at dim 3 without touching its 2^27 designs.
  • Sweeps every base-2 design at dim 2, 3, 4 against the classical XOR ANF and the signed subset real ANF, recording the variable reversal between the two labelings.
  • Sweeps all 512 base-3 designs at dim 2 for both round-trips and both degree histograms.
  • Evaluates the four historical rules at base 3, dim 2, 3, and the Sierpinski carpet and Menger sponge as actual shapes.
  • Shows the Vandermonde is never invertible for a composite base.
  • Domain run: exhaustive at every (base, dim) named above; base 3 at dim 3 rests on the identity alone. Whole run is 10 seconds on one core.

RUN

uv run python research/lab/py/base-q-anf/anf.py

WITNESSES

  • complexity.md:296 - at base 2 the inverse Vandermonde is [[1,0],[1,1]].
  • complexity.md:297-299 - the Vandermonde is not invertible mod base at base 4, 6, 8, 9, 10, 12, and is at every prime through 13.
  • complexity.md:305-306 - T E = E T = I at base 3, dim 3 covers all 3^27 value tables, so all 2^27 designs.
  • complexity.md:308-311 - all 16, 256, 65536 base-2 designs at dim 2, 3, 4 round-trip with 0 failures and match the XOR ANF with 0 differences after variable reversal.
  • complexity.md:312-315 - 512 base-3 designs at dim 2, both round-trips 0 failures, GF(3) degree histogram -1: 1, 0: 1, 2: 24, 3: 144, 4: 342, no degree 1.
  • complexity.md:326-329 - GF(3) degrees void 2, 4, tree 2, 4, carpet 3, 6, net 4, 6.
  • complexity.md:332-336 - Sierpinski carpet 8/9 cells, GF(3) degree 4; the carpet row keeps 3/9; the dim 3 row keeps 4/27; Menger sponge 20/27, GF(3) degree 6, the ceiling dim (base-1).
  • complexity.md:341-342 - at base 3, dim 2, the tree has GF(3) degree 2 and integer degree 1; the void GF(3) degree 2 and integer degree 4.
  • complexity.md:646-647 - the provenance lines this study replaces.