# Sponge measurability

- 2026-09-21 [Verified] Kombrink, Pearse and Winter 2016, Theorem 3.1 and Corollary 3.2, characterise Minkowski measurability of a nontrivial lattice self-similar set under the open set condition by the constancy of `p(eps) = eps^(D-d) sum_l r^(l(D-d)) lambda_d(F_(r^l eps) meet Gamma)` for any strong feasible open set with the projection condition, with no pluriphase hypothesis. Witness: read at source, arXiv 1501.03764, Section 3
- 2026-09-21 [Proved] For the Menger sponge with the open unit cube: the projection condition holds by coordinatewise folding, the distance from a point of the closed plus to the sponge equals its distance to the 24 wall carpets by coordinatewise clamping, a point of an arm is nearest to its own four walls and a point of the centre cube to the cube's edges, and the plus is covered at radius `sqrt(2)/6`. Witness: dimensions.md, sponge section, digit-rule lemmas
- 2026-09-21 [Proved] `T(delta)`, the volume of the sponge's `delta`-neighbourhood inside the plus, equals `(pi + 8) delta^2 - 8 sqrt(2) delta^3 + 48 (V1 - A1) - 24 Deep` on `(0, 1/6]`, `V1` and `A1` hole sums of the arcsine integral `int 4 (s - 2t) sqrt(delta^2 - t^2) dt` over the wall carpet and over its edge strip, `Deep` in `[0, 3.84e-5]` at `delta = 1/6`. Witness: lab/py/sponge-tube, TUBE
- 2026-09-21 [Verified] `T(1/8)` in `[0.234186414, 0.234701259]` and `T(1/12)` in `[0.180947086, 0.180947093]`, inside the raster brackets `[0.23229, 0.23708]` and `[0.17665, 0.18531]` from `120^3` cells per cube. Witness: lab/py/sponge-tube, TUBE
- 2026-09-21 [Proved] The sponge's periodic function on `(sqrt(2)/18, 1/6]`: `p(1/12)` in `[2.122718, 2.122723]`, `p(1/8)` in `[2.134668, 2.135742]`, `p(1/6)` in `[2.135019, 2.136794]`, so `p(1/6) - p(1/12) >= 0.012296`, relative swing at least `0.5792 %`, every step in interval arithmetic at 133 bits with the level tails, the `Deep` bound as exact rationals and the series tail as an interval. Witness: lab/py/sponge-tube, PERIODIC
- 2026-09-21 [Proved] The Menger sponge, `D = log 20 / log 3`, is not Minkowski measurable, and `lambda_3(F_eps) = eps^(3-D) p(eps) (1 + o(1))` with `p` multiplicatively 3-periodic and non-constant; computer-assisted through the two-phase band and Corollary 3.2. Witness: dimensions.md, sponge section, and lab/py/sponge-tube
- 2026-09-21 [Proved] At `delta = 1/6` the edge strip is the half wall, so `A1 = V1` and `T(1/6) = (pi + 8)/36 - sqrt(2)/27 - 24 Deep`, in `[0.256188319, 0.257110405]`. Witness: lab/py/sponge-tube, TUBE