# 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.