Unequal Split

• The patch: one child of side 1/2 in the corner of the unit square and five of side 1/3, the letter drawn on the 6-grid; its open set condition as exact rationals, its dimension solving 2^(-s) + 5*3^(-s) = 1 by PARI, the level render's fill (9 + 4k)^level on side 6^level to level 4, and the census of cells by size 2^-a 3^-b against C(a+b, a) k^b to word length 7, and the level-2 render against the Kronecker square of the tile.
• Its complex dimensions: the zeros of 1 - 2^(-s) - k 3^(-s) for k = 1..27 in the box Re [D_l - 1/4, D + 1/4], Im [-60, 60], seeded by PARI polroots of the lattice approximant 1 - x^41 - k x^65 and polished by Newton, the count certified by the winding number of the boundary with a Lipschitz margin; the first ones at k = 5; the printed root of the 2-3 nonlattice equation against its lattice approximant; the roots at D + 2 pi i q/ln 2 for the convergents p/q of log2 3 at 60 digits against the law P Q (ln 2)^2 theta^2/(2 f'(D)^3).
• The count of cells N(r) = #{cells of side >= r} as exact integers to r = e^(-300), the sizes sorted by exact integer comparison, the renewal identity at every breakpoint, N(r) r^D against its limit 1/(D f'(D)) over windows with its swing, the carpet as the lattice control, the folded variance at ln 2, ln 3, ln 6, and the periodogram of ln N(e^(-u)) - D u against the complex dimensions and their residues.
• The spin mass mu(B(c, r)) of the natural measure about three corner fixed points, enclosed by cells: the identity M(r) = ratio^D M(r/ratio), the ripple's swing against its enclosure, and the folded variance at ln 2, ln 3, ln 6.

RUN

• uv run python mrlyprod/research/lab/py/unequal-split/unequal_split.py patch, about 1 second.
• uv run python mrlyprod/research/lab/py/unequal-split/unequal_split.py poles, about 5 seconds.
• uv run python mrlyprod/research/lab/py/unequal-split/unequal_split.py count, about 1 second.
• uv run python mrlyprod/research/lab/py/unequal-split/unequal_split.py ripple, about 1 second.
• Every verb prints only and writes nothing; PARI runs through gp -q.

WITNESSES

• dimensions.md The unequal split, The patch: the open set condition, dimension = 1.778602507, the render's fill 29^level and log(29)/log(6) = 1.879323585, the census of cells (patch).
• dimensions.md The unequal split, Its complex dimensions: 21 zeros and winding number 21 at every k = 1..27, the margins of at least 161, the gap 0.0116 from D_l, the floored offsets 0.470, 0.058, 0.461, the first eleven at k = 5, the root of the 2-3 nonlattice equation, the convergent law to 7 digits at q >= 665 (poles).
• dimensions.md The unequal split, The count of cells: 59448 sizes, the renewal identity, 1/(D f'(D)) = 0.573459971, swing times sqrt(U) between 1.486 and 1.546 on the six printed windows, folds 0.003, 0.005, 0.004 against the carpet's 0.999 at ln 3, the ten peaks within 0.001 of a complex dimension and within 1.3% of its residue amplitude (count).
• dimensions.md The unequal split, What the detectors see: the identity at 96 shifted radii per centre, swings 0.23906, 0.15388, 0.17843 against enclosures at most 8.2e-5, at least 2661-fold, folds 1.000 at the centre's own period (ripple).