The circle count

• 2026-09-06 [Proved] A design's corner disc count has a self-similar main term with a log-periodic multiplier: with F the base-3 digit-restricted set of a design, fill its digit count and N(r) the filled cells whose centre lies in the closed Euclidean ball of radius r about the lattice corner, N is independent of the level, M(r) = fill^level mu(B_(r 3^(-level))) = r^(log(fill)/log 3) G(log_3 r) with G positive and 1-periodic, and |N(r) - M(r)| <= C(r), the crossing count, which is O(r^(dim-1)) in every dim because a cell meeting the sphere lies in the shell | |y| - r | <= sqrt(dim), of orthant volume 2^(-dim) omega_dim ((r + sqrt dim)^dim - max(r - sqrt dim, 0)^dim), so N(r) = r^(log(fill)/log 3) G(log_3 r) + O(r^(dim-1)), an unconditional saving r^0.8927892607 on the carpet and r^0.7268330279 on the sponge; the density-times-volume main term fails outright at the grid centre, whose middle block is empty at every level and where the relative error is exactly 1. Witness: lab/rs/circle-crop (22028 asserted rows, 6802 with a live error band, 41 mrlymath::shape::census cross-checks, the crossing bounds C_full <= 3r + 5 at dim = 2 and C_full <= pi sqrt 3 (r^2 + 1) at dim = 3 asserted at every radius), crop.md THE CIRCLE COUNT.
• 2026-09-06 [Verified] The crossing exponent of the corner disc count reads min 0.871371 / mean 0.898741 / max 0.969141 per triadic step over r = 27..19682 on the carpet and 1.704391 / 1.733764 / 1.757218 over r = 27..728 on the sponge, both bands containing the dimension minus one; the defect exponent sits in [0.220478, 1.015046] (carpet) and [0.645285, 1.730726] (sponge); the powers of the base carry no resonance, ranking inside [0.1200, 0.1296] in the crossing profile at k = 3..8 on the carpet while the defect's eight ranks average 0.4788 and reach 0.8333; mu(B_1) is certified in [0.750767350, 0.751113415] on the carpet, excluding 3/4, and [0.475928750, 0.485478125] on the sponge. Witness: lab/rs/circle-crop.
• 2026-09-06 [Refuted] That the corner disc count resonates at r = 3^n, the heuristic the page's own transform identity suggests (hat mu(3t) = (P(t)/m) hat mu(t), equality on integer t): |delta(3^n)|, delta(r) = N(3r) - m N(r), ranks 0.5000, 0.8333, 0.1111, 0.0185, 0.6605, 0.5514, 0.7167, 0.4390 inside its own triadic window on the carpet (the fraction of the window with |delta(r)| <= |delta(3^n)|) with no trend and delta(27) = 0 exactly; what is periodic is the crossing profile, whose maximum sits at 2.9671 times each window's start and triples exactly from r = 721; and square-root cancellation over the crossing cells fails, every central defect estimate sitting above half the crossing exponent. Witness: lab/rs/circle-crop.
• 2026-09-07 [Proved] The crossing shell is a tree: the shell at level j is the whole grid's shell at real radius r/3^j, so it is a rooted tree of depth level with 2r+1 leaves and 2*floor(r/3^j)+1 boxes per level, and C(r) counts the leaves whose path never takes the centre seat; mean branching is 3 + (2k-2)/(2Q+1), exactly 3 at every level where floor(r/3^j) is 1 mod 3. Witness: crossing cells brute-forced from the cell definition at every level of every r <= 150 and at 18 large and boundary radii with no fault, the live leaves equal the corner ball's Cut column at every radius of every depth 1..5 for codes 7, 11 and 15, no box lacks a crossed parent at any r <= 242, and C(100) = 134 and C(242) = 296 read twice by paths sharing no code (lab/rs/circle-crop, crates/mrlydemo/tests/shell.rs, crop.md THE CIRCLE COUNT).
• 2026-09-07 [Proved] The crossing ladder is an exact ratio of integer counts: Psi(r) = prod_k g_k(r) with g_k = u_(k+1)/(1 - p_(level-1-k)), u_k = T_k/T_(k-1) and k = level-1-j the depth from the top; g_0 = 1 identically, and g_1 = 1 exactly whenever the level-(level-1) or level-(level-2) centre box is uncrossed, so the profile carries at most level-1 informative ranks and often level-2. Witness: lab/rs/circle-crop ladder lines, the product asserted against Psi computed directly to 1e-12 at r = 80, 242, 1000, 6560.
• 2026-09-07 [Proved] The crossing shell's transfer operator is the tripling map on the offset: with R_j = r/3^j and y_j(x) = sqrt(R_j^2 - x^2), the offset a_j(i) = frac(y_j(i)) satisfies a_(j-1)(3i) = frac(3 a_j(i)) at every level and column, because y_(j-1)(3x) = 3 y_j(x) is an identity of reals and needs no hypothesis; the 9-bit box pattern is mask(floor(u - k sigma)) for k = 0..3 clipped to [0,2], with sigma the scale-free slope. Witness: lab/rs/circle-crop derivation pass, the derived pattern law reproducing the shell with 2 faults of 2188 boxes at level 0 and none at levels 1..5 on the shallow arc at r = 6560, the steep half following by the shell's own symmetry.
• 2026-09-07 [Proved] A 9-bit box pattern is realisable by a straight line exactly when 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; the test returns exactly thirty masks, which are character for character the crossing shell's thirty, and it forbids the thirty-first, a zero step beside a step of two, so the extra pattern seen only at the top of the tree is a curvature state and not a line state. Witness: lab/rs/circle-crop derivation pass, stable at search radius 12 and 15 and asserted at r = 6560, 19682, 12345.
• 2026-09-07 [Verified] The Perron root of the memory-one pattern matrix differs from 3 at r = 6560, 19682, 12345, brackets [3.000861, 3.000862], [3.000948, 3.000949] and [2.997616, 2.997617], but this does not test the Markov property: the matrix row sums are exactly popcount(s), so the model returns each level mass exactly (161, 485, 1457, 4373, 13121 at r = 6560), the tree's own pooled branching is 3.037736, 3.012422, 3.004124, 3.001373, 3.000457 and never the exact 3, the certified distance from 3 falls monotonically 0.323840, 0.002623, 0.002385, 0.000086 as the parent count rises, and over 18 unused radii the sign of rho - 3 is positive 9 times and negative 9 times at sizes 0.000336 to 0.005951. Witness: lab/rs/circle-crop operator, pair and sweep_radii lines.
• 2026-09-07 [Verified] The thirty-pattern alphabet of the crossing shell's transfer operator belongs to the truncation levels 1..level-3 and not to the tree: over levels 1..level the alphabet is 31 at r = 6560, 19682 and 12345, the extra being the root's own pattern, and 228 of the 16683 radii r = 3000..19682 leave the thirty over levels 1..level-3 and 1966 over levels 1..level-1, none reading fewer. Witness: lab/rs/circle-crop every_level_states and scan lines, with r = 1395, 1739, 6570, 15122, 3182 pinned at four truncations.
• 2026-09-07 [Verified] Folding N(r)/r^(log(fill)/log 3) at 16 offsets of log_3 r mod 1 and comparing consecutive triadic windows measures the collapse instead of assuming it: the carpet's largest gaps run 0.353553, 0.222183, 0.110138, 0.042663, 0.015114 from R = 1 to R = 243 with shares of the window level 0.534078, 0.305035, 0.145250, 0.055448, 0.019546, so R = 81 and R = 243 agree to two percent and the ladder falls like 1/r; the deepest window opens at N(243)/243^(log(fill)/log 3) = 0.751038, inside the certified mu(B_1) bracket [0.750767350, 0.751113415], while the sponge reads 0.229755 at R = 9 against R = 27 and the grid centre does not collapse at all, its share reading 3.081886. Witness: mrlydemo crop_collapse, crates/mrlydemo/tests/crop.rs, site/check.ts.
• 2026-09-07 [Proved] At dim = 2, with level the least level with r < 3^level, the crossing shell's digit rate is pinned to 1/9 with an unconditional power saving and the index ind(r) = prod_(j<level) (1 - p_j(r)) (9/8)^level is bounded: the shell is a monotone lattice path, so a level-j box carries leaves(X) = w + h - 1 cells with sum_X w = sum_X h = floor(r/3^j) + r + 1 and sum_X leaves = 2r + 1, the coordinate swap makes the seat class's two marginals equal, and p_j(r) (2r+1) = 2 sum_(seat X) w(X) - #seats exactly; that turns both counts into sums of floor of one circle arc over 3^j residue classes mod 3^(j+1), where van der Corput's Satz 5 on closed subintervals [a, b] with 3^(j+1) b + c < r and a residue split of the monotone increments y(x) - y(x+1) give |p_j(r) - 1/9| <= 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1) at R = r/3^j, hence sum_(j<level) |p_j - 1/9| <= 781, and with 1 - p_j >= 3^j/(2r+1) from box column 0, |log ind(r)| <= 1191 for every r >= 1; Psi is untouched, so Phi does not follow, and the staircase is a plane fact so the sponge has none of it. Witness: lab/rs/circle-crop index lines, every identity asserted in exact integers at every level of r = 80, 242, 1000, 2186, 6560, 12345, 19682 and the bound asserted live at j = 0 for r = 212957, 531441, 2000000 where the cap clears the trivial 8/9; crop.md THE CIRCLE COUNT.
• 2026-09-07 [Proved] At dim = 2, with level the least level with r < 3^level, no level-j column block [3^j i, 3^j (i+1)] of the crossing shell of radius r tracks a slope window of width eps unless 3^j < 2 eps r, so a leaf's tracked run is at most floor(log_3(2 eps r)) + 1 levels, and no level-n column block is a line's staircase once n > log_3(1 + 2 sqrt(3r)): a block's slope span t(3^j (i+1)) - t(3^j i) is at least 3^j/r because t(u) = u/sqrt(r^2 - u^2) has t' = r^2 (r^2 - u^2)^(-3/2) >= 1/r, and tracking is inherited downwards so the tracked levels are a run from the bottom and never a gap; at the Dirichlet width eps = 1/b^2 no level-j block tracks a denominator b >= sqrt(2 r/3^j), so no block at rank k from the top of the Psi ladder tracks a rational with b >= sqrt(6) 3^(k/2) and ranks 0 to 4 are held to b <= 2, 4, 7, 12, 22; the block cap is sharp, since a >= 1, ab + 1 <= b^2 and 8 * 3^(2j) b^4 < r^2 force a level-j block to track; and if the staircase floor(sqrt(r^2 - X^2)) agrees with a line's at three columns U, U + m, U + 2m in [0, r) then the second difference reads at least -1 from the line and at most 2 - m^2/r from the arc, so m^2 <= 3r, which gives (3^n - 1)^2 <= 12 r for a block but only w(X), h(X) <= 2 + 2 sqrt(3r) for a box and so excludes no box the arc enters and leaves through one side; hence every rank k < level - 1 - log_3(1 + 2 sqrt(3r)), which is level/2 - 2.14 ranks to leading order and never half of them, carries no frozen-slope resonance across a block, halving the exponent the refuted frozen route produced without bounding Psi, which stays Conjecture. Witness: lab/rs/circle-crop track, budget, secant, blind, boxes and boxline lines, the run asserted between the two exact integer counts on all 1116 slope rows of F_30 at r = 3^level - 1 for level = 6, 7, 8, 9, run equal to the cap at 102, 91, 93, 98 of 279 slopes and to the floor at 26, 72, 66, 63, the three-point condition attained at exactly the largest level the block cap allows at each radius, and at r = 19682 level 6 exactly 3 of the 53 boxes with their whole content equal to a line's staircase, (16, 20), (19, 19) and (20, 16), the witness that the box statement fails; crop.md THE CIRCLE COUNT.
• 2026-09-07 [Refuted] That a spectral gap of the frozen-slope transfer operator bounds Psi: at slope 1/3, where the pattern law is exact, the straight-line ladder gives log Psi rising by 0.024224 a level over level = 4..12 identically at three line offsets, so Psi ~ 1.024520^level with survival rate 0.910342 above 8/9, while at slope 1/7 it falls by 0.009633 a level; the hole is a full triadic cylinder only at sigma = 0, where the digits are independent and Psi = 1 identically. Witness: lab/rs/circle-crop derivation pass; the circle escapes the frozen model only because it tracks a resonance for k-2 levels at width 3^-k.
• 2026-09-19 [Proved] At dim = 2, with R = r/3^j, the three constants of the crossing shell's digit-rate bound are explicit: over the 3^j classes each column sawtooth sum is at most 4.5310 r R^(-1/3) + 101.0364 r R^(-1/2) + 2 r R^(-1) + 1 and each box sawtooth sum at most 4.5310 R^(2/3) + 101.0364 R^(1/2) + 3 from 3 pi 3^(-2/3) and 175 * 3^(-1/2) rounded up, the residue split costs (7/9) sqrt(2r) <= 1.1000 r^(1/2) and (10/3) sqrt R + (4/3) sqrt(2R) <= 5.2190 R^(1/2), the counting terms are exact to 2 * 3^j and 2/3, the main terms cancel to (2r + 1)/9 plus a residue below 1/9, and the errors add to 27.1860 r R^(-1/3) + 610.1581 r R^(-1/2) + 9.3334 r R^(-1) + 10.3334, which 2r + 1 >= 2r and R <= r fold into 13.60 R^(-1/3) + 305.08 R^(-1/2) + 9.84 R^(-1). Witness: van der Corput 1923 Satz 5 as restated in Laugesen and Liu Theorem 18, with lab/rs/circle-crop asserting the bound per level at r = 80, 242, 1000, 2186, 6560, 12345, 19682 and live at j = 0 for r = 212957, 531441, 2000000.
• 2026-09-19 [Proved] The geometric sum behind the drift cap is explicit: sum_(j < level) R_j^(-delta) <= 3^delta/(3^delta - 1) reads 3.2612, 2.3661 and 1.5 at delta = 1/3, 1/2, 1, which is what carries sum_(j < level) abs(p_j(r) - 1/9) <= 781, and a Huxley-type delta = 77/208 in place of van der Corput's 1/3 moves only the first of the three, to 2.9927. Witness: arithmetic on the closed form, with the drift sums 0.223603, 0.245132, 0.328170, 0.256778, 0.260804, 0.446659, 0.267260 printed against 781 by lab/rs/circle-crop.
• 2026-09-19 [Proved] The certificate abs(log ind(r)) <= 1191 splits into two explicit blocks: where R_j clears 23157375, at which the digit-rate bound first falls under 1/9 with the crossing at 23157374.055, 1 - p_j >= 7/9 and the mean value theorem gives abs(log(1 - p_j) - log(8/9)) <= (9/7) abs(p_j - 1/9), so that block costs at most (9/7) 781 <= 1004.15, while at most 16 levels fall below it, since R_j < 23157375 asks j > log_3 r - 15.44 and log_3 r >= level - 1, each costing log(3 R_j) < (i + 1) log 3 at the i-th from the top, a tail of at most log 3 * n(n + 3)/2 over the top n levels and so at most 167.0 at n = 16. Witness: lab/rs/circle-crop index lines, abs(log ind) asserted against 1191 at every level of r = 80, 242, 1000, 2186, 6560, 12345, 19682.
• 2026-09-19 [Verified] Over the 277 slopes of F_30 meeting the sharpness hypothesis a >= 1 and a b + 1 <= b^2, 1108 rows at r = 3^level - 1 for level = 6, 7, 8, 9, the proved cap #{j : 3^j b^2 < 2r} less the proved floor #{j : 8 * 3^(2j) b^4 < r^2} is never above 2 and attains it, so the two exact integer counts pin those tracked runs to within two levels at every radius swept; the remaining 8 rows are 0/1 and 1/1, where the floor reads 0 by that hypothesis. Witness: lab/rs/circle-crop track lines.