# 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.