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.
• 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<l) (v_k - v_l - 1)/(l - k) < min_(l<k) (v_l - v_k + 1)/(k - l) with the upper end positive, asserted stable at search radius 12 and 15.
• alphabet lines assert that alphabet equal to the shell's truncated alphabet at r = 6560, 19682 and 12345 and print the one every-level state no line realises.
• frozen lines compare the derived pattern against the shell box by box at every level, once with the tangent at the box's left edge and once with the chord across it, splitting out the shallow half sigma <= 1 where the column parametrization is the right one.
• centre lines print the centre-crossed rate per level against the derived 1/3.
• line lines run the gain ladder on a straight line of slope num/den and offset off/den across 3^level columns at level 4..12, printing the cells, the survivors, the survival, the marginal product prod_j (1 - p_j), psi and its log, for the resonant slopes 1/3 and 1/7, the flat 1/2, sqrt(2) - 1, (sqrt(5) - 1)/2 and pi/4, and the near-resonant 1/3 + 3^-k.
• track lines run the resonance sweep: for every a/b in F_30, every fraction of [0, 1] with b <= 30, at r = 3^level - 1 for level 6, 7, 8, 9, the tracked run is the number of levels from the bottom of the tree at which some shell COLUMN BLOCK [3^j i, 3^j (i + 1)] has its whole slope span inside (a/b - 1/b^2, a/b + 1/b^2), read on the shallow half t <= 1 in exact integers, t(u) < p/q being u^2 (q^2 + p^2) < p^2 r^2. A block is not a box: a box's content spans w(X) <= 3^j columns and can be far flatter than its block.
• A track row prints the run, the proved cap #{j : 3^j b^2 < 2r}, the proved floor #{j : 8 * 3^(2j) b^4 < r^2} - zero where the sharpness hypothesis a >= 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.