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