pascal-shear

• The level-level cell set of the 2D design with code 7, tile [[1, 1], [1, 0]], is {(i, j) in [0, 2^level)^2 : i AND j = 0}.
• Builds that set by literal Kronecker powers for level = 1..9 and counts 3^level cells, then recounts 3^level by the digit sum Sum_i 2^(level - popcount(i)) for level = 1..14.
• Checks Kummer directly: v2(C(i+j, i)) against the base-2 carry count, on exact binomials for 0 <= i, j < 128.
• Rebuilds Pascal mod 2 from the additive recurrence alone, rows 0..1023, and compares entry k of row n against k AND (n - k) = 0.
• Checks the shear (i, j) -> (i, i + j), determinant 1, as a set bijection onto the odd Pascal entries at level = 6.
• Confirms row sums equal 2^popcount(n) to n = 1023, and that inside a level-8 square the antidiagonal count is 2^popcount(n) below 2^level and strictly smaller above.
• Compares both OEIS b-files against the recomputed triangle, index column included, A047999 from the copy kept here and both from the live files when the network answers.

RUN

• uv run python research/lab/py/pascal-shear/pascal_shear.py
• Full domain, no arguments; about two seconds plus the two b-file reads.
• a047999.txt is the whole b-file, 10585 terms. b001316 has 50001 terms and is too large to keep here, so that count needs the network; without one the run still checks A047999 and exits 0.

WITNESSES

• cuts.md:181-182: level sets to level = 9 with 3^level cells each, 3, 9, 27, ..., 19683, exact binomials for i, j < 128, 16384 of them with 0 faults.
• cuts.md:182-183: Pascal mod 2 from the additive recurrence to row 1023, 524800 entries, 0 mismatched cells.
• cuts.md:183-184: the shear checked as a bijection, 729 cells onto 729 odd entries at level = 6.
• cuts.md The same page, one dimension down: 50001 terms of A001316 against 2^popcount(n), 0 differences, only when the live b-file read succeeds.
• cuts.md:185-186: A047999 is checked against the triangle with 0 differences, but at 10585 terms, rows 0..144, not the 8256 the line says; 8256 is a 128-row prefix, and REFS.md:176 carries the same 10585.
• REFS.md:175: b001316, 50001 terms. REFS.md:176: b047999, 10585 terms, rows 0..144.
• README.md:49: A047999 with antidiagonal populations A001316, both b-files checked term for term.