# 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.