# Circle Crop - Counts a design's filled cells inside a ball and measures the error against the density and against the design's own self-similar main term: the circle-count section of [crop](../../../notes/crop.md). - The designs are code 7 at `dim 2` (the carpet, `mrlymath::bang::factory::create(7, 3, 2, 2, level)`) and code 23 at `dim 3` (the sponge, `create(23, 3, 3, 2, level)`). - One convention throughout: cells are indexed `x` in `[0, 3^level)^dim`, a cell counts when its centre `x + 1/2` lies in the closed Euclidean ball `|y| <= r`, corner balls sit at the lattice corner `0` and centre balls at the grid centre `3^level/2`. - The corner sweep runs every integer radius `r = 1 .. 3^level - 1`, the centre sweep every `r = 1 .. (3^level - 1)/2`, the inscribed radius about the grid centre. - All of it is integer arithmetic: a cell's three squared distances to the ball centre are doubled to stay integral, and `r` is read off by integer square root, so `N(r)`, the census columns and the defect `delta(r) = N(3r) - m N(r)` are exact. - A row prints `N` the count, `in` and `cut` the filled cells fully inside and crossing, `Nfull` the same three for the whole grid, the exact `delta`, the running maxima, and the error `E` against the self-similar main term with the certified band `Elow <= |E| <= Ehigh`. - The main term is `M(r) = lim N(3^j r) / m^j`; the row prints `E = N(r) - N(3^depth r) / m^depth` at the deepest radius the grid reaches and bands it by `cut(3^depth r) / m^depth`, which is the exact bound the crop page proves. - Centre rows print `err_num`, the exact integer `3^(dim * level) N - fill * Nfull`, so the density error is a printed rational, and `rel`, its size against the main term `rho_level Nfull`. - Every row is asserted before printing: `in <= N <= in + cut` both for the design and for the whole grid, `|delta(r)| <= cut(3r) + m cut(r)`, `|E| - band <= cut(r)`, and the crossing bound `Nfull_cut <= 3r + 5` in `dim 2`, `Nfull_cut <= pi sqrt(3) (r^2 + 1)` in `dim 3`. - The `E` band is live only where the grid reaches a deeper radius: the `totals` line prints `rows` and `banded`, `22028` rows of which `6802` carry a band of depth `1` or more, and a `bands` line prints the split per design. - The corner count does not depend on the level, so each level is asserted equal to its predecessor over the whole shared range before the deepest level prints its rows. - The oracle runs at every level, on fewer sample radii as the grid grows and on none at the largest level of each design, `41` radii in all: the exact `Frac` classifier of `mrlymath::shape::census` against the integer sweep, both the filled and the whole-grid columns. - `window` lines give the maximum per triadic window with its location, `trend` lines give the running maximum's growth over one triadic step as min, mean, max, a log-log fit and the endpoint slope, and `resonance` lines give the values at `r = 3^n` with the rank of `|delta(3^n)|` and of `cut(3^n)` inside the window `r = 3^n .. 3^(n+1) - 1`, the rank being the fraction of the window at most the value at `3^n`; past the deepest radius the defect reaches, the defect fields read `beyond_reach`. - `mean` lines give, for each triadic window `r = R .. 3R - 1` with `3R <= 3^level - 1`, the window sum of the crossing count, its mean over the `2R` radii, the exact sandwich `B(3R) - A(R)` to `2 (A(3R) - B(R))` in the two corner columns, the closed form `(m - 1) M(R) - C(3R) - C(R)` to `2 ((m - 1) M(R) + C(3R) + C(R))` with `M(R)` bracketed by its own columns at the deepest depth the grid reaches, and the multiplicity bracket `kappa = W(R) / ((m - 1) M(R))`; both the sandwich and the closed form are asserted. - `factor` lines give, over the same window, the minimum, mean and maximum of `Phi(r) = C(r) (3^dim/m)^level / C_full(r)` at `level` the least level with `r < 3^level`, the change in the minimum and the maximum from the window below, the window's band for `C_full(r) / r^(dim-1)`, and a `ground` for the mean built from the closed form's lower end and the proved cap on `C_full`, asserted against the printed mean; `C(r) <= C_full(r)` is asserted at every radius the windows cover, `r <= 6560` on the carpet and `r <= 242` on the sponge, and `C_full(r) >= (r / sqrt(dim - 1))^(dim-1)` at every radius of the corner sweep. - `C_full(r) = 2r + 1` is asserted at every radius of every carpet level, the exact plane identity the crop page proves; the sponge keeps the measured band. - `digits` lines split the pointwise factor per triadic window into `ind = prod_j (1 - p_j) (3^dim/m)^level`, the independent digit model's survival times `(3^dim/m)^level`, and `Psi = (C/C_full) / prod_j (1 - p_j)`, the dependence correction, with min, mean and max of each, and print the union sum `sum_j p_j` as mean and maximum; `p_j(r)` is the fraction of the crossing cells of the whole grid whose base-3 digit vector at position `j` is one the design omits. - `digitrate` lines print, per window and per position, the min, mean and max of `p_j(r)` over the window's radii and the scaled drift `(mean p_j - null) 3^k/3^j`, the null being `1 - m/3^dim`. - `digitpair` lines print, per window, the consecutive and gap-two dependence ratios `p_(j,j') / (p_j p_(j'))` on the window's pooled counts, and `digittotal` lines the extremes over every window of the fine-position means, the scaled drift, both ratios, the union sum and both factors, so a global bracket is never assembled from the per-window rows; `p_(j,j')` is the fraction of crossing cells carrying an omitted digit vector at both positions. - The digit census enumerates the crossing shell directly from the column intervals rather than from the grid sweep, and asserts its own `C` and `C_full` against the sweep's Cut columns at every radius the windows cover, so the two generators of the same numbers are independent. - The cap `p_j(r) <= 2 * 3^j (2 floor(r/3^(j+1)) + 1) / (2r + 1)` is asserted in exact integers at every radius and position in `dim 2`; the Frechet-Hoeffding bound `C(r) >= C_full(r) - sum_j (cells omitted at j)` is asserted in exact integers at every radius of both designs. - `transform` `step` and `mass` lines price the digit route on the carpet: `step` prints the transfer step `h(u) = m^(-1) sum_e |P(u + e/3)|` at `u = 0`, asserted equal to `2`, and at `u = (155/243, 155/243)`, asserted below `2`; `mass` prints the `l^1` mass `Lambda_level = sum_(a != 0) |phi_level(a)|` at `level 1..6` by direct summation over `a mod 3^level`, its step, the step in `log_3`, and the cost and error exponents that step forces, with `Lambda_level + 1 >= 2^level` asserted at every level. - `transform` lines evaluate `hat mu(t) = prod_(j >= 1) P(t/3^j)/m` with the unnormalised `P(u) = sum_(d in S) e(-d . u)`, and assert both `hat mu(3t) = (P(t)/m) hat mu(t)` and `|P(t)/m| = 1` exactly on integer `t`. - `transfer` lines are the carpet's crossing shell read as a tree: the level-`j` boxes carrying a crossing cell are the whole grid's shell at the real radius `r/3^j`, `x -> floor(x/3)` maps one level into the next, and each level asserts its own size `2 floor(r/3^j) + 1` and that its cells are exactly the children counted by the level above. - `ladder` lines print, per radius, the marginal survival `1 - p_j`, the conditional survival `u_k = T_k / T_(k-1)` with `T_k` the crossing cells whose top `k` digits are all non-centre, the gain `g_k = u_(k+1) / (1 - p_(level-1-k))` indexed by the depth `k = level - 1 - j` from the top and its log in units of `10^-3`, and `psi = prod_k g_k` asserted equal to `(C/C_full) / prod_j (1 - p_j)` computed directly to `1e-12` relative; a `g0` line prints `g_0`, asserted `1` exactly, `g_1`, and the top two marginals that make `g_1` collapse to `1`. - `profile` lines sweep a triadic window at an even stride and pool the gains by the depth from the top `k = level - 1 - j`, printing the window mean of `log g_k` and of `|log g_k|`, the mean, deviation and extremes of `log psi`, the total of the mean `|log g_k|` over the window's ranks, the rank-to-rank decay of that mean, the worst decay from rank `2` on, and `rank1_flat`, the count of sampled radii at which `g_1 = 1` exactly. - The `totals` profile line prints the six window totals, their increments, the increments' decay, the geometric extrapolation of those increments and the `psi` band it gives, beside the envelope of the per-window `psi` extremes and `band_holds`, which reads false: the band is a fit on the window means and is not pointwise. - `stride` profile lines rebuild the same extrapolation with the three deepest windows resampled at strides `24`, `48` and `72`, printing each stride's totals, last increment, worst decay and extrapolated total, with `converges` false wherever the increments do not decay. - `operator` lines build the exact integer matrix of the transfer step: a state is the 9-bit pattern of a box's crossed children printed in octal with bit `4` the centre, `M[s][t]` counts parent-child pairs over the printed levels, and the mean offspring matrix is `M[s][t]` divided by the number of parents in state `s`. - The alphabet is collected at every radius and asserted equal across them, and a `wide` state pairs the pattern with the box's own seat in its parent, `30 * 9` states before the empty ones are dropped. - The operator reads levels `1..level-3` only: `every_level_states` lines rebuild the alphabet over levels `1..level` and print the extra state, `scan` lines count the radii of `3000..19682` whose truncated alphabet leaves the thirty, and pinned radii assert the state count at each of the four truncations `1..level-1` down to `1..level-4`. - `rowsum_is_popcount` lines assert the row sums of the mean offspring matrix state by state, then push the true state census down through it and print the model's level masses against the tree's own, one step and iterated. - `pair` lines rebuild the matrix from one level pair at a time and print the bracket's certified distance from `3` against the parent count, and a `sweep_radii` line brackets `rho` at eighteen radii the pass does not otherwise use and counts the signs. - Every Perron root is certified, never fitted: a positive integer test vector `v` gives the Collatz-Wielandt bracket `min_s (sum_t M[s][t] v_t) / (n_s v_s) <= rho <= max_s`, an exact ratio of integers, printed with the lower end truncated down and the upper end rounded up; the pruned matrix drops the states that cannot reach a surviving state before the bracket is taken. - `dobrushin` is the exact one-step ergodic coefficient of the row-stochastic pattern chain, and `doeblin` the certified minorisation `sum_t min_s P^n[s][t]` at `n = 1..6`, computed in fixed point with every rounding taken down so the floor is a bound, with `rate` the least `(1 - doeblin_n)^(1/n)`. - `index` lines carry the proved digit-rate bound and every identity beneath it, at every level of `r = 80, 242, 1000, 2186, 6560, 12345, 19682`, in exact integers: `leaves(X) = w(X) + h(X) - 1` box by box, `sum_X w = sum_X h = floor(r/3^j) + r + 1`, `sum_X leaves = 2r + 1`, the transpose equality of the two seat marginals, the seat identity `p_j(r) (2r + 1) = 2 sum_(seat X) w(X) - #{seat X}`, and both counts again from the floor formulas `sum_(floor(x/3^j) = 1 mod 3) [floor((y(x) - 3^j)/3^(j+1)) - floor((y(x+1) - 2 * 3^j)/3^(j+1))]` and its box-column twin, so an algebra slip in the proof shows up as a failed assertion and not as a shifted decimal. - The bound itself is asserted per level, `|p_j(r) - 1/9| <= 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1)` at `R = r/3^j`, with `(2r + 1) - seats >= 3^j` for the away-from-one lemma, and per radius the drift sum `sum_j |p_j - 1/9|` against `781` and `|log ind|` against `1191`; each line prints `live_levels`, the count of levels whose cap is under the trivial `8/9`, and it reads `0` at all seven radii, so those asserts check the identities and not the bound. - `index live` lines are the bound where it bites: at `j = 0` and `r = 212957`, `531441` and `2000000`, an `O(r)` column loop counts the seats with `sum_(x = 1 mod 3) [floor((y(x) - 1)/3) - floor((y(x+1) - 2)/3)]` and asserts both `cap(r) < 8/9`, so the test can fail, and `|p_0(r) - 1/9| <= cap(r)`; `212957` is the least radius at which the cap clears `8/9` at all, and a cap under `1/9` needs `r >= 23157375`. - `line_alphabet` prints the frozen-slope alphabet: every integer quadruple `v_k = floor(u - k sigma)` realisable by some `(u, sigma)` with `sigma >= 0`, decided by the exact rational test `max_(k= 1` and `ab + 1 <= b^2` fails - the number of tracking boxes at the deepest tracked level and their columns censused mod `3`; both sides of the sandwich are asserted per row, and the down-set property is asserted level by level rather than assumed. - `budget` lines print, per level and per rank, the largest `F_30` denominator tracked there with `b^2 3^j` beside `2r`, the assert being the cap at that level. - `secant` lines print the deepest level at which some shell column block passes the three-point line test `abs(v(U) - 2 v(U + m) + v(U + 2m)) <= 1` at `m = (3^n - 1)/2`, with its witness column, and assert both `m^2 <= 3r` and `((3^(n+1) - 1)/2)^2 > 3r`, so the block cap is met and not merely satisfied. What is attained is the necessary condition, not a line. - `blind` lines compute exactly the measure of the offsets at which the thirty-mask state of slope `sigma` differs from that of `sigma + eps`, at `sigma = 1/3` and `eps = 3^(-k)` for `k = 2` to `7`, and assert it below the proved `6 eps`; the gap to the measured `4 eps` is the mask's clamp of its four floors to rows `0` to `2`. - `boxes` and `boxline` lines enumerate the level-`6` boxes at `r = 19682`, above the block cap, and test each box's content for a line by the exact rational feasibility bracket for `floor(s x + c) = v(x)`; they print how many boxes the arc crosses side to side, how many have both extents inside the theorem's reach, and every box whose content is a line, with its column range, its slope bracket, its span against the block floor and the finest `F_30` window it sits in. - The shallow-half stop truncates one row of the sweep: at `a/b = 1/1` the window `(0, 2)` reaches past `t = 1`, so the printed run `9` is short of the true run. No assert is endangered, the cap there being `10`. ## RUN - `bash scripts/cargo.sh cargo run --release -p circle-crop` - Under a minute, peak grid `3^9` squared and `3^6` cubed; prints only, writes nothing. ## WITNESSES - crop.md the level-free corner count: each level against the one below, `level 6` against `level 5` on `r = 1..242` up to `level 9` against `level 8` on `r = 1..6560` for the carpet, `level 4` against `level 3` on `r = 1..26` up to `level 6` against `level 5` on `r = 1..242` for the sponge. - crop.md the crossing exponent: carpet `min = 0.871371`, `mean = 0.898741`, `max = 0.969141`, fit `0.870673`, endpoints `0.898794` over `r = 27..19682`; sponge `1.704391`, `1.733764`, `1.757218`, `1.654302`, `1.720961` over `r = 27..728`. - crop.md the crossing shell: `C_full(r) / r^(dim-1)` in `[2.000152, 2.037038]` on the carpet over `r = 27..6560` and in `[2.298611, 2.380000]` on the sponge over `r = 9..242`. - crop.md the transform price: `h = 2.000000` at the lattice and `1.951261` at `(155/243, 155/243)`; `Lambda_level = 1.000000, 3.585973, 9.637999, 23.736907, 56.547512, 132.884543` at `level 1..6`, steps `3.585973, 2.687694, 2.462846, 2.382261, 2.349963`, `log_3` step `0.777708`, cost `1.277708`, error `1.170497`. - crop.md the window mean: carpet at `r = 2187..6560` sum `13758140`, sandwich `[11019880, 22055720]`, closed form `[11013332.750000, 22068746.000000]`, mean `3145.436671` against `1180.226337` at `r = 729..2186`; `kappa` in `[1.247746, 1.248322]` there and in `[1.331356, 1.331971]` at `r = 1..2`, sponge `kappa` in `[1.084118, 1.105871]` at `r = 1..2` and `[1.417534, 1.445977]` at `r = 81..242`. - crop.md the pointwise factor: carpet means `1.012500, 0.981078, 0.959982, 0.954469, 0.948442, 0.944582, 0.943419, 0.942790` over eight windows `r = 1..6560`, minima in `[0.588115, 0.900000]`, maxima in `[1.125000, 1.518945]` rising by `0.140625, 0.090402, 0.074920, 0.033718, 0.033360, 0.020111, 0.000808`; sponge means `1.080000, 1.042535, 0.980332, 0.968704, 0.965114` over five windows `r = 1..242`, minima in `[0.602555, 0.810000]`, maxima in `[1.350000, 1.673315]` rising by `0.176959, 0.086036, 0.054979, 0.005341`. - crop.md the exact plane shell: `C_full(r) = 2r + 1` asserted at every radius of every carpet level, `r = 1..19682` at `level 9`, and the level-`j` box bound `2 floor(r/3^j) + 1` behind the digit cap. - crop.md the digit census: 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`, per-radius bands `[0.102466, 0.121271]` finest and `[0.000000, 0.447092]` coarsest, fine positions inside `[0.108363, 0.111141]` across the five windows deep enough to have one and scaled drift inside `[-0.111806, 0.063806]` over all eight; sponge means `0.259211, 0.259237, 0.259663, 0.256864, 0.286061` at `r = 81..242` against `7/27`, fine positions inside `[0.259103, 0.259237]` on the three readings its two deep-enough windows supply, wholly below `7/27`, scaled drift inside `[-0.108311, 0.026803]`. - crop.md the pair dependence: carpet 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` at `r = 2187..6560`, over all windows `[0.939130, 1.714286]` and `[0.988460, 1.126957]`; sponge `0.999847, 1.000927, 1.003308, 0.954574` and `0.999497, 0.999197, 0.997414`, over all windows `[0.954573, 1.151415]` and `[0.997413, 1.019127]`. - crop.md the two factors: carpet `ind` means `1.012500, 0.973637, 0.954993, 0.953180, 0.945805, 0.943289, 0.942499, 0.942104` and `Psi` means `1.000000, 1.010516, 1.009082, 1.006311, 1.007708, 1.006371, 1.005977, 1.005714`, the `digittotal` brackets over all eight windows `[0.542697, 1.515753]` and `[0.793296, 1.374208]`, the sponge's over five `[0.532793, 1.624661]` and `[0.795518, 1.233429]`; the union sum `sum_j p_j` mean `0.941002` maximum `1.349974` on the carpet at `level 8`, mean `1.321036` maximum `1.627693` on the sponge at `level 5`. - crop.md the defect exponent: carpet `min = 0.220478`, `mean = 0.527490`, `max = 1.015046`, fit `0.544749`, endpoints `0.648815` over `r = 27..6560`; sponge `0.645285`, `1.056561`, `1.730726`, `1.001255`, `1.273634` over `r = 27..242`. - crop.md the measure brackets: carpet `mu(B_1)` in `[0.750767350, 0.751113415]` at `r = 6561`, sponge in `[0.475928750, 0.485478125]` at `r = 243`. - crop.md the powers of three: `delta` and its window maximum and rank at `r = 1, 3, ..., 2187` on the carpet and `r = 1, 3, ..., 81` on the sponge, the crossing rank out to `r = 6561` and `r = 243`. - crop.md the tombstone off the lattice: carpet `|hat mu(t)| = 0.59332804` and `|hat mu(3t)| = 0.29666402` at `t = (1/2, 0)`, both `0.10072687` at `t = (1, 0)`. - crop.md the crossing maximum's place: `at_over_start = 2.9671` for the carpet at window rank `k = 5, 6, 7, 8`, the argument tripling exactly from `r = 721`; the defect's is `2.3333, 2.7407, 2.8724, 2.8628, 2.7979` at `k = 3..7`. - crop.md the centre hole: first filled cell at `r = 122` and `r = 365` on the carpet at `level 6, 7`, at `r = 20` and `r = 58` on the sponge at `level 4, 5`, against the middle block's inradius `121, 364, 13, 40`. - crop.md the centre relative error: `1` through the hole at every level, falling only to `0.059636` (carpet `level 7`) and `0.301915` (sponge `level 5`) at the inscribed radius. - crop.md the shell tree: `2 floor(r/3^j) + 1` asserted at every level of every radius the transfer pass runs, and the children of one level asserted to exhaust the level below. - crop.md the pattern alphabet: `30` states over levels `1..level-3`, `001,003,007,011,013,017,036,070,074,111,113,132,136,170,174,222,226,264,322,326,360,364,400,440,444,600,640,644,700,740`, asserted equal at `r = 6560`, `19682` and `12345`. - crop.md the thirty-first state: `31` states over levels `1..level`, the extra being the root's own pattern `744`, `744`, `032` at those three radii, and `228` of the `16683` radii `r = 3000..19682` leaving the thirty over levels `1..level-3`, `1966` over levels `1..level-1`, none reading fewer, pinned at `r = 15122`, `3182`, `1395`, `1739`, `6570`. - crop.md the memory-one masses: row sums exactly `popcount(s)`, one-step masses `161, 485, 1457, 4373, 13121` equal to the tree's at `r = 6560` and the four-step iterate within `8.694464` of `13121`, against the pooled branching `3.037736, 3.012422, 3.004124, 3.001373, 3.000457`. - crop.md the finite-size gap: certified distance from `3` reading `0.323840, 0.002623, 0.002385, 0.000086` as the parent count rises `53, 161, 485, 1457`, and over eighteen unused radii `3001..18301` every bracket excluding `3`, nine above and nine below, at sizes `0.000336` to `0.005951`. - crop.md the branching mean: `meanb = 3.003711, 3.001532, 3.000976` and the centre rate `0.326994, 0.328381, 0.335367` at those three radii, three times the rate reading `0.980983, 0.985143, 1.006102`. - crop.md the certified Perron brackets: `[3.000861,3.000862]`, `[3.000948,3.000949]`, `[2.997616,2.997617]` for the whole matrix against `3`, `[2.657742,2.657743]`, `[2.672261,2.672262]`, `[2.668083,2.668084]` for the pruned matrix against `8/3`, ratios `[0.885659,0.885661]`, `[0.890471,0.890473]`, `[0.890068,0.890069]` against `8/9` and per-level drifts `[-0.003314,-0.003311]`, `[0.001618,0.001621]`, `[0.001206,0.001208]`. - crop.md the wide state: `263`, `270`, `270` states and brackets `[2.964909,2.964910]`, `[3.000989,3.000990]`, `[2.998702,2.998703]` against `3`, ratios `[0.893588,0.893590]`, `[0.891339,0.891341]`, `[0.890199,0.890201]`. - crop.md the operator's gap: `dobrushin = 1.000000` at every radius, `doeblin = 0.000000, 0.073937, 0.217716, 0.347398, 0.458035, 0.552167` at `r = 6560` and `rate = 0.874687`, `0.874812`, `0.872736` at the three radii. - crop.md the gain profile: window means of `|log g_k|` in units of `10^-3` reading `0.000000, 36.111664, 34.280021, 24.412289, 14.937825, 11.086863, 6.332857, 3.731719, 2.371680` at `r = 6561..19682`, worst decay from rank `2` of `0.752924, 0.704680, 0.772659, 0.785194, 0.784415, 0.742201` over the six windows, the rank-one to rank-two step being `0.949278` at the deepest window and outside that statistic, and `g_1 = 1` at `18, 54, 35, 35, 35, 32` of the `54, 162, 98, 98, 98, 97` sampled radii. - crop.md the gain totals: `89.734824, 105.695558, 118.801020, 125.674943, 130.335999, 133.264920` in units of `10^-3`, increments `15.960734, 13.105462, 6.873923, 4.661056, 2.928921` decaying by `0.821106, 0.524508, 0.678078, 0.628381`, extrapolated total `146.708428` and the band `psi` in `[0.863545, 1.158017]`, which the per-window extreme envelope `[0.824797, 1.321206]` falls outside; resampled at strides `24`, `48`, `72` the last increment reads `1.993070`, `2.186657`, `4.336745` and the extrapolated total `120.468507`, `141.506415`, `151.873506`, the first not converging at all. - crop.md the psi windows: `log psi` deviation `0.070520, 0.074837, 0.074889, 0.076942, 0.078042, 0.078954` and per-window `psi` extremes `[0.881494,1.276712]`, `[0.824797,1.321206]`, `[0.842485,1.281241]`, `[0.836172,1.283999]`, `[0.837527,1.290536]`, `[0.838220,1.314879]` on the sampled radii. - crop.md the index bound where it bites: `j = 0` at `r = 212957`, `531441` and `2000000`, caps `0.888888350`, `0.586410834`, `0.323672329`, gaps `0.000590`, `0.000447`, `0.000192`, ratios `0.000664`, `0.000761`, `0.000592`; at `r = 80, 242, 1000, 2186, 6560, 12345, 19682` the cap exceeds the trivial `8/9` at every level, `live_levels = 0`, drift sums `0.223603, 0.245132, 0.328170, 0.256778, 0.260804, 0.446659, 0.267260` and `|log ind|` `0.069975, 0.105310, 0.341219, 0.140802, 0.144928, 0.281717, 0.158483`. - crop.md the staircase identities: `leaves(X) = w + h - 1`, `sum_X w = sum_X h = floor(r/3^j) + r + 1`, `sum_X leaves = 2r + 1`, the transpose equality of the seat marginals, `p_j (2r+1) = 2 sum_(seat) w - #seats`, both floor formulas and `(2r+1) - seats >= 3^j`, all asserted in exact integers at every level of those seven radii. - crop.md the frozen-slope alphabet: `30` masks from the feasibility test, stable at search radius `12` and `15`, equal to the shell's thirty at `r = 6560`, `19682` and `12345`, the every-level extra `744`, `744`, `032` not realisable by any line. - crop.md the pattern law's faults: chord rates `0.011434, 0.020590, 0.035052, 0.068323, 0.113208, 0.235294, 0.600000` at levels `0..6` for `r = 6560` and `0.006631, 0.009604, 0.018531, 0.051546, 0.062112, 0.188679, 0.235294, 0.600000` at levels `0..7` for `r = 19682`, tangent rates `0.023325, 0.040494, 0.101031, 0.161491, 0.415094, 0.411765, 0.600000` at the first, and on the shallow half `2` faults of `2188` at level `0` and none at levels `1..5`. - crop.md the centre rates: `0.342099, 0.334248, 0.309278, 0.310559, 0.339623, 0.294118, 0.400000` at `r = 6560` and `0.340980, 0.329522, 0.330130, 0.317526, 0.310559, 0.339623, 0.294118, 0.400000` at `r = 19682`, against the derived `1/3`. - crop.md the frozen refutation: at slope `1/3`, `log psi = 0.039821, 0.063195, 0.087646, 0.111808, 0.136048, 0.160267, 0.184491, 0.208714, 0.232938` at `level 4..12`, increment `0.024224` a level at the offsets `0`, `0.411523` and `0.906094`, survival rate `0.910342`; at `1/7` the increment is `-0.009633`; at `1/2`, `sqrt(2) - 1`, `(sqrt(5) - 1)/2` and `pi/4` the log settles at `-0.008195`, `0.002837`, `0.021450`, `-0.009142`; at `1/3 + 3^-9` the resonant profile is tracked through depth `k - 2`, splitting one depth later at `0.132569` against `0.136048`. - crop.md the tracking sandwich: `279` slopes at each of `r = 728, 2186, 6560, 19682`, the cap live at `276` of them, the run equal to the cap at `102, 91, 93, 98` and to the floor at `26, 72, 66, 63`, cap less floor never above `2` and attaining it over the `277` slopes with `a >= 1` and `a b + 1 <= b^2`, `1108` rows, the other `8` rows being `0/1` and `1/1` where the floor is `0` by that guard; at `r = 19682` the run is `8` at `1/3` (cap `8`, floor `7`), `7` at `1/5` (`7`, `6`), `4` at `1/30` (`4`, `2`) and `2` at `29/30` (`4`, `2`). - crop.md the Farey budget: largest tracked `F_30` denominator `30, 30, 30, 30, 19, 11, 6, 3, 1` at levels `0` to `8` for `r = 19682`, against the theorem's `22, 12, 7, 4, 2` at the last five, the first four levels held by the sweep's own `b <= 30`. - crop.md the half-tree line bound: line depth `4, 4, 5, 5` at `r = 728, 2186, 6560, 19682` with witness column `0` at each, and `14641 <= 59046 < 132496` at `r = 19682`. - crop.md the alphabet's blindness: the differing offset measure is exactly `4 eps` at `sigma = 1/3` and `eps = 1/9, 1/27, 1/81, 1/243, 1/729, 1/2187`, against the proved `6 eps`. - crop.md the box against the block: at `r = 19682` and level `6`, above the block cap `5.633`, `12` of the `53` boxes are crossed side to side, `19` have both extents inside the theorem's reach and `3` have their whole content equal to a line's staircase, `(16, 20)` on `12369..12392`, `(19, 19)` on `13851..13983` and `(20, 16)` on `15290..15308`; `(16, 20)`'s span is `0.002487440`, `6.7157%` of the block floor `0.037038919`, inside the Dirichlet window of `17/21`.