Sponge Tube

• The Menger sponge through Kombrink, Pearse and Winter 2016, Corollary 3.2, with the open unit cube as the feasible open set: p(eps) = eps^(D-3) (20/27 + sum_(l >= 0) (27/20)^l T(eps/3^l)) for eps in (sqrt(2)/18, 1/6], where T(delta) is the volume of the sponge's delta-neighbourhood inside the plus of seven removed cubes and D = log 20 / log 3.
• T(delta) on (0, 1/6] in closed form: the centre cube's edge tube pi delta^2 - 8 sqrt(2) delta^3, and per arm four wall carpets, T = pi delta^2 - 8 sqrt(2) delta^3 + 8 delta^2 + 48 (V1 - A1) - 24 Deep, with V1 half the wall's integral of sqrt(delta^2 - d^2) over the carpet's holes, A1 the same integral over the strip within delta of the wall's edge, both hole sums of arcsine closed forms with a proved geometric tail, and Deep the volume of the set deep in holes of two perpendicular walls at once, bounded above by an enumeration of hole pairs to level 7 and a counting bound beyond.
• Every printed interval is an enclosure: the closed forms run in mpmath interval arithmetic at 133 bits (sqrt, pi, atan2 for the arcsine, exp and log for the power), the truncation tails of the level sums, the Deep bound and the tail of the criterion's series over levels above 40 are exact rationals added to the endpoints, and the script asserts the hard bounds V1 <= delta/18 and A1 <= delta^2/3 at every radius; the raster is a float check only. mpmath's interval + - * / sqrt and integer powers are exactly directed, while pi, atan2, exp and log are guard-bit approximations rounded outward, a drift below 1e-30 absorbed by the printed rounding of at least 5.6e-9.
• A raster check: distances from the wall carpets and the centre edges on 120^3 cells per cube, bracketed by the half diagonal, contain the closed form at delta = 1/8 and 1/12.
• checks.py is the second verb: four independent checks of the lemmas and the certificate, none part of it, each printing its own numbers: oracle (exact distances to the level-4 prefractal through a KD-tree with a completeness test, against the 24 level-4 wall carpets and against the own-walls or centre-edges reduction), montecarlo (seeded Monte Carlo of T at three radii through the reduced distance, a lemma-free bracket of T from the prefractal, and V2 at 1/12), seeded (Monte Carlo of Deep on its confinement box and of V2 at 1/6 and 1/8, seed 20260921), recompute (the tube, the Deep bound and the three bands from the closed forms of the paper at 60 digits with their own column recursion, hole integrals checked against quadrature, plus V1 and A1 at 1/12 at 200 digits with the naive a1 and on a 3000^2 midpoint grid).

RUN

• uv run python mrlyprod/research/lab/py/sponge-tube/sponge_tube.py
• About 46 to 50 seconds; prints only, writes nothing, asserts the raster brackets and the positive gap.
• uv run python mrlyprod/research/lab/py/sponge-tube/checks.py runs the four checks, about 125 seconds in all: oracle 4 s, montecarlo 82 s, seeded 29 s, recompute 9 s; append one name for one check. Needs numpy, scipy and mpmath; prints only, writes nothing.

WITNESSES

• dimensions.md, section "The sponge, through the criterion without pluriphase": T(1/8) in [0.234186414, 0.234701259] and T(1/12) in [0.180947086, 0.180947093], raster bands [0.23229, 0.23708] and [0.17665, 0.18531]; T(1/6) = (pi + 8)/36 - sqrt(2)/27 - 24 Deep in [0.256188319, 0.257110405].
• The same section: p(1/12) in [2.122718, 2.122723], p(1/8) in [2.134668, 2.135742], p(1/6) in [2.135019, 2.136794], p(1/6) - p(1/12) >= 0.012296, relative swing at least 0.5792 %, the Deep bound 3.84e-5 at delta = 1/6, 2.15e-5 at delta = 1/8 and 2.42e-10 at delta = 1/12.
• The paper papers/sponge-measurability.md, Fact 4.4 and Fact 5.3: the same bands; its Section 6, every number: checks.py oracle prints 160000 cubes, 12288 wall faces, 22844 points with 2844 near a face, the differences 0.0 and 2.8e-17, and the centre distance to 8.3e-17; checks.py montecarlo prints the brackets [0.23190, 0.23467] and [0.17601, 0.18090] at 1e6 points, T(1/12) = 0.180952 +- 0.000056, T(1/8) = 0.234691 +- 0.000036, T(1/6) = 0.257060 +- 0.000011 at 4e7 points, and V2(1/12) = 0.00231180 +- 5.6e-8 against 0.00231178; checks.py seeded prints Deep = 1.8882e-6 +- 1.4e-8 against the bound 3.8420e-5 at 1/6 and 4.7432e-8 +- 2.9e-9 against 2.1452e-5 at 1/8, with V2 at both radii against 2 A1 - delta^2/3 + Deep; checks.py recompute prints the column counts validated to level 6, the hole integrals against quadrature to 1e-24 or better, V1, A1, the Deep bound and T at the three radii, the three bands to twelve digits, the series tails, and V1(1/12), A1(1/12) at 200 digits with the naive a1 and on the grid.