Complex Dimensions

• The complex dimensions of the four one-base designs, base 3 {0,2}, base 5 {0,2,4}, base 15 {0,4,10,14} and base 15 {0,2,4,10,12,14}: the 81 predicted poles s = dimension + 2 pi i m/ln(base), m = -40..40, checked against 1 - fill base^(-s), the numerator D(s) at each, and D(1) = 1 - fill/base.
• The two-ratio control with ratios 1/3 and 1/5: its dimension, its 21 Moran roots in Re [-3,3], Im [-40,40] by Newton from a grid, the winding number of 1 - 3^(-s) - 5^(-s) over that box, and the worst offset of the roots from each candidate spacing.
• Composition: alternating bases multiplies into the product base, checked as integers and as 256 exact intervals.
• The box count N(eps) of each cover, g(u) = ln N(e^-u) - dimension u on u in [ln(1/0.03), ln(1e6)] at 3000 points, its Blackman periodogram peak on a direct DFT grid, and the variance explained by folding u modulo ln 3, ln 5, ln 15 in 40 bins, for the four designs, the two-ratio control and a Thue-Morse aperiodic control.
• The inner tube V(eps) in closed form, M(eps) = eps^(dimension-1) V(eps) to u = 60, its swing per window, the periodicity defect M(eps) = M(eps/p), the decay of the two-ratio swing, and the Cantor limit profile 2^(1-dimension) (t^(dimension-1) + t^dimension) against the measured tube.
• carpet_tube.py: the Sierpinski carpet's tube by digit level, V(eps) = sum_m 8^(m-1) h(3^(-m), eps), the hole census cell by cell to level 5, the closed form against the level sum as exact rationals, the limit profile G(t) on its two branches with its seam, its extrema as quadratic roots, its swing, the level sum at u in [50, 60], and a level-6 distance transform equal to the closed form as rationals.

RUN

• uv run python research/lab/py/complex-dimensions/complex_dimensions.py
• About 15 seconds; prints only, writes nothing.
• uv run python research/lab/py/complex-dimensions/carpet_tube.py
• About 1 second; prints only, writes nothing, and asserts the closed profile against the direct sum.

WITNESSES

• dimensions.md:14-15 and dimensions.md:302 name two scratch passes; this study is the one pass that replaces both.
• dimensions.md:47-51 all 81 poles at m = -40..40 kill the denominator to 5e-14, minimum |D(s)| = 0.500000 for the Cantor design, D(1) = 1 - fill/base for all four.
• dimensions.md:62-65 dimension 0.630930, 0.682606, 0.511916, 0.661642 and omega 5.719202, 3.903963, 2.320188, 2.320188.
• dimensions.md:74-76 every periodogram peak within 1% of 2*pi/ln(base).
• dimensions.md:87-92 folding 0.192, 0.030, 0.016; 0.018, 0.509, 0.034; 0.007, 0.022, 0.667; 0.009, 0.030, 0.534; 0.082, 0.047, 0.038; 0.007, 0.012, 0.115.
• dimensions.md:106-113 the alternation equals base 15 {0,4,10,14} and the 256 intervals are identical.
• dimensions.md:158-164 dimension = 0.518370, 21 roots and winding number 21, real parts -0.699926 to 0.518370, offsets 0.43, 0.17, 0.38.
• dimensions.md:181-184 minimum 2.494975716 at t = 0.584963, maximum 2.583040469, swing 3.53%, measured tube within 4.9e-9 at the minimum.
• dimensions.md:186-191 swings flat from u = 15 to u = 60 (eps = 8.8e-27), periodicity at the own base to 1e-9 or better, the two-ratio swing decaying 3.79% to 0.42%.
• research/claims/:360 the poles of 1/(1 - fill base^(-s)) on one vertical line of period 2 pi/ln(base).
• dimensions.md:122 the sponge's hole boundary is not in the sponge: (1/2, 1/2, 1) at distance 1/6, (2/3, 1/2, 5/6) at distance 1/18; read off the digit rule, no generator.
• dimensions.md:126 dimension = 1.892789; every hole ringed by filled cells and the holes exhausting the complement at levels 3, 4, 5 (carpet_tube.py, HOLES).
• dimensions.md:129-130 the two branches of G(t), the level sum equal to the closed form at 96 exact rationals (carpet_tube.py, TUBE and PROFILE).
• dimensions.md:133 seam (44/35) 2^(2-dimension) = 1.354123517, ends 379/280, maximum 1.35561708227 (rounded up) at t = 0.429638415, minimum 1.3506702097 (rounded down) at t = 0.692137253, swing 0.36625%, the level sum within 1e-9 of G on u in [50, 60], the raster equal to V(21/729) as rationals (carpet_tube.py, PROFILE, MEASURED, RASTER).
• dimensions.md:135 the area of F_eps inside Gamma, 4 eps/3 - 4 eps^2 on (0, 1/6]: the level-1 term of the sum, read off the closed form; the theorem numbers are the arXiv version's.
• dimensions.md:137 the class profile G(t) = t^(dimension-dim) sum_j fill^(j-1) base^(-j dim) h(t base^j): at base 3, dim 2, fill 8 it is the two branches the script checks; no other member is run.