Claims · 12 dated claims on Sponge measurability, each with its tag and its witness; newest 2026-09-23.
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
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
ProvedT(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
VerifiedT(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
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
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
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
Verified The Menger sponge's Deep and T in closed form match rasters of the exact sponge distance at eleven radii over (1/6, sqrt(2)/6] and at 1/6, level by level to level 4 within 6.12e-3 relative, and give Deep(1/6) = 1.894504e-6 and Deep(1/8) = 4.827978e-8, inside the Monte Carlo 1.8882e-6 +- 1.4e-8 and 4.7432e-8 +- 2.9e-9. Witness: lab/rs/sponge-window; mrlyrs::math::three::sponge::deep and exact with their tests.
Verified The Menger sponge's periodic function p on a full period, from the exact tube in double precision: maximum 2.139869 at eps = 0.148577, minimum 2.122663 at eps = 0.081380, swing 0.8106 %, logarithmic mean 2.130510, falling across (1/6, sqrt(2)/6] from p(1/6) = 2.136706 to 2.122798. Witness: lab/rs/sponge-window.
Proved For every radius delta, with lambda = min(delta, 1/6), the corner volume Deep(delta) of the Menger sponge's plus lies in the crossing holes of the wall carpet, the holes whose open across-range contains lambda, one hole at a time, so Deep(delta) is a sum over crossing holes of one elementary arcsine integral P. Witness: dimensions.md, subsection "The tube past 1/6, and Deep in closed form".
Proved On [1/6, sqrt(2)/6] the Menger sponge's tube inside the plus is T(delta) = pi delta^2 - 8 sqrt(2) delta^3 - 4 delta^2 arccos(1/(6 delta)) + (2/3 + 16 c^2) c + 2/9 - 24 sum_m 2^(m-1) P_m(delta) with c = sqrt(delta^2 - 1/36), elementary on [sqrt(10)/18, sqrt(2)/6] and on each [delta_(m+1), delta_m), delta_m = sqrt(9^m + 1)/(6 * 3^m), the breakpoints accumulating only at 1/6. Witness: dimensions.md, subsection "The tube past 1/6, and Deep in closed form".
Proved For the Menger sponge Deep(1/12) = 0, since 3 lambda = 1/4 = 0.0202... in base 3 has no digit 1, so T(1/12) is the arcsine hole sum alone, and T(1/6) = (pi + 8)/36 - sqrt(2)/27 - 24 sum_m 2^(m-1) P_m(1/6) exactly. Witness: dimensions.md, subsection "The tube past 1/6, and Deep in closed form".