Complex dimensions

• 2026-08-28 [Conjecture] Every one-base design is lattice, so the Lapidus-Maier machinery is not out of reach but empty: zeta_level(s) = 1/(1 - fill base^(-s)) puts the complex dimensions on one vertical line of period 2 pi/ln(base), and (ISP)_dim, the Riemann hypothesis in the language of fractal strings, has no content on the degenerate case, several incommensurable ratios being the change that gives it content; the lattice half is checked on this tree's own poles and folding tables, the (ISP)_dim half is a literature reading not yet checked at source, and the dichotomy is a theorem for strings only, so dimension two and above is open outside the pluriphase class, inside which the carpet and the interior-hole designs are settled. Witness: lab/py/complex-dimensions.
• 2026-09-06 [Proved] The Sierpinski carpet is not Minkowski measurable: its complement in the open square is the disjoint union of 8^(m-1) open squares of side 3^(-m) whose boundaries lie in the carpet, the tube is the exact hole sum, and eps^(log 8/log 3 - 2) V(eps) -> G(t) with G = t^(log 8/log 3 - 2)(1 + 4t/5 - 4t^2/7) on [1/3, 1/2) and t^(log 8/log 3 - 2)(9/8 + 3t/10 - t^2/14) on [1/2, 1), C^1 at the seam, 379/280 at the ends, maximum 1.35561708227 at t = 0.429638, minimum 1.3506702097 at t = 0.692137, swing 0.3662%; a corollary of Kombrink, Pearse and Winter 2016 Theorem 1.1(ii), whose hypotheses are verified for the carpet with the open square, the profile and the elementary proof being the addition. Witness: lab/py/complex-dimensions carpet_tube.py, dimensions.md measurability with its hypotheses.
• 2026-09-06 [Proved] Every one-base design at base >= 3 in dim >= 2 removing at least one digit vector, all removed vectors interior and pairwise differing by at least 2 in a coordinate, is not Minkowski measurable: base^(dim-1) < fill < base^dim so log(fill)/log(base) is never an integer, G(t) = t^(log(fill)/log(base) - dim) sum_j fill^(j-1) base^(-j dim) h(t base^j) > 0, and t^(dim - log(fill)/log(base)) G is a polynomial on [1/base, 1/2); includes the parity carpets at every odd base and dimension. Witness: dimensions.md measurability with its hypotheses (proof), lab/py/complex-dimensions at base = 3, dim = 2 only.
• 2026-09-06 [Refuted] The hole-sum route for the sponge: the level-1 plus hole's 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), the tube inside the hole is not its parallel volume, and the pluriphase theorem does not reach the sponge with the open cube; the sponge stays Conjecture. Witness: dimensions.md measurability with its hypotheses, qualification 2, read off the digit rule.