Crop census

• 2026-09-01 [Proved] Crop partition and anti-crop complement: classify puts every cell in exactly one of Out, Cut, In, so the keep-cut and strict crops bracket the boundary, and Shape::Anti flips In with Out fixing Cut, so the crop and the anti-crop under the complementary cut rule partition the filled set exactly; read off the definition and asserted both ways on all 118 printed configurations. Witness: mrlymath::shape, lab/rs/crop-counts.
• 2026-09-01 [Verified] The inscribed sphere never enters the level-1 sponge: census reads cells [0, 26, 1], the one In cell the empty centre and all 20 filled cells Cut, so the keep-cut crop keeps everything and the strict crop nothing; the inscribed octahedron holds no filled sponge cell fully inside through level 2. Witness: mrlymath::shape, lab/rs/crop-counts.
• 2026-09-01 [Proved] Exact dead zones of the inscribed crops: the carpet crop is empty for r < 1/6 under ball and diamond alike and the sponge diamond crop for r < 1/3, the central holes' inradii; the sponge ball's exact contact radius is sqrt(2)/6 = 0.2357, witnessed by the filled level-3 cell [13/27, 14/27] x [8/27, 9/27] x [8/27, 9/27] whose nearest point to the centre is (1/2, 1/3, 1/3), so the 1/24 sweep reads empty through r = 5/24 and first cuts at r = 6/24. Witness: crop.md, lab/rs/crop-counts.
• 2026-09-01 [Verified] Saturation and its one failure: the carpet ball crop holds all 4096 filled cells at level 4 from r = 17/24, the first sweep radius past the circumradius sqrt(2)/2, the sponge ball all 8000 at level 3 from r = 7/8, past sqrt(3)/2, and the sponge diamond never saturates in the sweep, reading in = 5356, cut = 1332 of 8000 at r = 1 since the cube's corners sit at L1 distance 3/2. Witness: lab/rs/crop-counts.
• 2026-09-01 [Verified] The strict inscribed diamond crop of the full side-2m grid holds exactly 2m(m-1) cells. Witness: mrlymath::shape.
• 2026-09-01 [Proved] A grid-aligned polytope crop is digit counting: walls on multiples of 3^-k keep exactly the cells with coordinates in integer intervals at level k, the count factors along digit positions as in mrlylab::press, and the crop adds nothing. Witness: crop.md.
• 2026-09-01 [Conjecture] The curved-slice dimension: the log_3 cut-ratio exponents of the inscribed circle on the carpet read 1.140, 0.909, 0.899, 0.951 and of the sphere on the sponge 2.166, 1.579, 1.705, hovering near the straight-slice yardsticks, the dimension minus one, 0.8928 and 1.7268 - yardsticks by analogy only, since Shmerkin 2019 and Wu 2019 cover intersections of xp- and xq-invariant line sets with p, q multiplicatively independent, not same-base carpet slices, straight or curved; five levels decide nothing. Witness: lab/rs/crop-counts, crop.md.
• 2026-09-06 [Proved] Over any triadic window r in [R, 3R) the mean of the crossing count C(r) is Theta(R^(log(fill)/log(base) - 1)), from C(r) = A(r) - B(r), the step bound 1 <= |x+1| - |x| <= sqrt(dim), the exact sandwich B(3R) - A(R) <= W(R) <= 2 (A(3R) - B(R)) and the bracket lemma B <= M <= A, with constants ((m-1)/2) G_min and (m-1) G_max. Witness: lab/rs/circle-crop mean lines, carpet r = 2187..6560 sum 13758140 inside [11019880, 22055720].
• 2026-09-06 [Proved] C_full(r) = Theta(r^(dim-1)): the shell bound above, and below C_full(r) >= (r/sqrt(dim-1))^(dim-1) from one crossing cell per orthant lattice point of the first dim-1 coordinates. Witness: lab/rs/circle-crop corner assert at every radius, band [2.000152, 2.037038] on the carpet at r = 27..6560.
• 2026-09-06 [Proved] Pointwise C(r) = Theta(r^(log(fill)/log(base) - 1)) holds if and only if Phi(r) = C(r) (3^dim/m)^level / C_full(r) is bounded above and below, level the least level with r < 3^level. Witness: lab/rs/circle-crop factor lines.
• 2026-09-06 [Proved] The digit transform route's budget is sum_(a != 0) |phi_level(a)| |S_r(a/3^level)| = O(r^(dim-1)), met term by term only if the l^1 mass grows by at most sqrt(3) = 1.7320508076 per triadic step, and the transfer step at the lattice is h(u) = (m + 3^dim - 1)/m = 2 at dim = 2. Witness: crop.md the transform route, and where it stops.
• 2026-09-06 [Proved] At dim = 2 the crossing shell is exactly 2r + 1 cells at every integer r >= 1, by the telescoping lo_i = hi_(i+1) of the column intervals with hi_(r+1) := 0; the same count at real radius is 2 floor(R) + 1, so the level-j boxes meeting the shell number at most 2 floor(r/3^j) + 1 and one holds at most 2 * 3^j crossing cells. Witness: lab/rs/circle-crop, asserted at every radius of every carpet level to r = 19682.
• 2026-09-06 [Proved] The fraction p_j(r) of crossing cells whose base-3 digit vector at position j is one the design omits obeys p_j(r) <= 2 * 3^j (2 floor(r/3^(j+1)) + 1)/(2r + 1) < 2/3 + 3^j/r at dim = 2, capping every position with 3^j <= r/30 at 0.7 uniformly in r; it does not transfer to Phi, since the sharpest bound the marginals alone support is Frechet-Hoeffding, C >= C_full (1 - sum_j p_j), and sum_j p_j reaches 1.349974 at level = 8 on the carpet and 1.627693 at level = 5 on the sponge. Witness: lab/rs/circle-crop digits and digitrate lines, asserted in exact integers.
• 2026-09-06 [Verified] The crossing shell's digits are equidistributed away from the top of the scale: carpet window means 0.111086, 0.111086, 0.111068, 0.111063, 0.111141, 0.109478, 0.109295, 0.166786 at r = 2187..6560 against 1/9, the whole departure in the top three positions and locked to level - j, fine positions inside [0.108363, 0.111141] and the scaled drift (p_j - 1/9) 3^k/3^j inside [-0.111806, 0.063806]; sponge 0.259211, 0.259237, 0.259663, 0.256864, 0.286061 against 7/27 with fine positions inside [0.259103, 0.259237], wholly below the null on three readings. Witness: lab/rs/circle-crop digitrate and digittotal lines.
• 2026-09-06 [Verified] Pairwise digit dependence in the crossing shell is bounded per pair and falls off with the gap: consecutive ratios 1.000165, 1.000219, 1.000521, 0.998543, 0.992613, 1.076698, 0.956158 and gap-two 1.000298, 1.000022, 1.000203, 1.000221, 1.003261, 1.005671 on the carpet at level = 8, over every window [0.939130, 1.714286] and [0.988460, 1.126957], sponge [0.954573, 1.151415] and [0.997413, 1.019127]. Witness: lab/rs/circle-crop digitpair and digittotal lines.
• 2026-09-06 [Verified] The pointwise factor widens with decelerating drift: carpet maxima rise 1.125000 to 1.518945 by increments 0.140625 down to 0.000808 over eight windows r = 1..6560, minima in [0.588115, 0.900000]; sponge maxima 1.350000 to 1.673315 over five windows r = 1..242. Witness: lab/rs/circle-crop factor lines.
• 2026-09-06 [Verified] The window multiplicity kappa = W(R)/((m-1) M(R)) brackets to [1.247746, 1.248322] on the carpet at r = 2187..6560 and to [1.417534, 1.445977] on the sponge at r = 81..242, still climbing there; the [1, 2] bound is asymptotic, the exact slack being (C(3R) + C(R))/((m-1) M(R)). Witness: lab/rs/circle-crop mean lines, form_low = -4.744629 at r = 1..2.
• 2026-09-06 [Verified] The l^1 mass Lambda_level reads 1.000000, 3.585973, 9.637999, 23.736907, 56.547512, 132.884543 at level = 1..6 with log_3 step 0.777708, forcing a term-by-term cost r^1.277708 against a budget r^1 and an error no better than r^1.170497, worse than the exact floor Lambda_level >= 2^level - 1 gives; closing it needs |S_r| = O(r^0.222292), below the square-root floor. Witness: lab/rs/circle-crop transform mass lines.
• 2026-09-06 [Conjecture] Phi = ind * Psi exactly with ind = prod_j (1 - p_j) (3^dim/m)^level and Psi the dependence correction; carpet means settle at 0.942104 and 1.005714 while the global brackets [0.542697, 1.515753] and [0.793296, 1.374208] still widen with decelerating drift, so ind and Psi bounded is sufficient for the pointwise C(r) = Theta(r^(d-1)) and is the whole of what is left. Witness: lab/rs/circle-crop digits and digittotal lines over eight carpet and five sponge windows.
• 2026-09-06 [Refuted] That two is the minimum of the transfer step h: h at the level-5 triadic point (155/243, 155/243) is 1.951261. Witness: lab/rs/circle-crop transform step line.