# The Apollonian gasket

- 2026-09-14 [Proved] The Descartes reflection needs no square root and acts on all three coordinates: in the coordinates `(k, k x, k y)`, a line being `k = 0` with `(k x, k y)` its outward normal, the fourth circle tangent to three given ones is `v' = 2(v_1 + v_2 + v_3) - v`, by Vieta, so an integral root quadruple grows an integral packing; six identities ride along under `B(u, v) = (sum u_i)(sum v_i) - 2 sum u_i v_i`, `B(k, k) = B(k, kx) = B(k, ky) = B(kx, ky) = 0` and `B(kx, kx) = B(ky, ky) = -4`, true on both roots and preserved by the reflection, which lies in the orthogonal group of `B`. Witness: lab/rs/apollonian, verbs strip and census, all six rechecked on 575969269 quadruples, 0 broken, an arithmetic check only.
- 2026-09-14 [Proved] The circles of the strip packing `(0, 0, 2, 2)` tangent to the line `y = 0` are exactly the Ford circles, one over every reduced `a/b`, no interval assumed, of curvature `2 b^2`: tangency is `k y = 1` at positive curvature; `(0, 2 b^2, 2 d^2, k)` has square discriminant `64 b^2 d^2` and roots `2(b + d)^2` and `2(b - d)^2`, the mediant and the Stern-Brocot parent; two are tangent exactly at `(a d - b c)^2 = 1`; the walk covers `(0, 1)`, the root pair `0/1` and `1/1` the ends, and the period-1 translation the rest; conversely Dirichlet forces an overlap at irrational `p` and nesting equality at rational `p`. Witness: lab/rs/apollonian, verb ford, 4863601 mediants to denominator 4000 against `sum_{b <= 4000} phi(b) - 1`, 0 broken, 0 misses.
- 2026-09-14 [Verified] The Ford identification holds in both directions on the grown packing, not only on the Stern-Brocot walk: one period of the strip packing grown to curvature `2097152` gives 20770674 circles of which 318963 carry `k y = 1`, every one passing the Ford test that `k/2` is a square `b^2` and `k x = 2 a b` with `gcd(a, b) = 1`, 0 off-Ford, and 318963 is `sum_{b <= 1024} phi(b) - 1`; the same count returns at `Q = 32` and `Q = 181` as 323 and 10059, the far line carries 318963 by the strip's reflection symmetry, and no circle leaves the open period, 0 outside `0 < k x < k`. Distinctness is controlled at `T = 2048`, 2448 circles and 2448 distinct. Witness: lab/rs/apollonian, verb strip.
- 2026-09-14 [Proved] The stack's brightness reads off the packing's curvature: the Farey stack lights the node `a/b` exactly `floor(Q/b)` times at depth `Q`, the one circle resting on that node has curvature `k = 2 b^2`, so the brightness is `floor(Q sqrt(2/k))`, and the nodes lit at depth `Q` are exactly the tangency points of the line-tangent circles of curvature at most `2 Q^2`; summing over the half-open period `[0, 1)` gives `sum_{b <= Q} phi(b) floor(Q/b) = sum_{n <= Q} sum over b dividing n of phi(b) = Q(Q + 1)/2`, the walk carrying `(0, 1)` and the node `0/1` adding its `Q`. Witness: lab/rs/apollonian, verb ford, brightness 1275, 20100, 500500, 8002000 at `Q = 50, 200, 1000, 4000` against `Q(Q + 1)/2`.
- 2026-09-14 [Verified] The curvature census grows like a power of `T` whose local exponent, read as the ratio `log(N(T_2)/N(T_1))/log(T_2/T_1)` and never as a fit, lands at `1.305`, the fourth place set by the grid: the bounded packing `(-1, 2, 2, 3)` gives `N(T) = 5, 165, 3325, 67163, 1359167, 27463391, 555198593`, ratios ending `1.3055, 1.3057`; the strip's one period gives `2, 48, 950, 19298, 390478, 7899138` on the decades, ratios ending `1.3061, 1.3060`, and `20770674` at `T = 2097152`, octave ratios ending `1.3050, 1.3056`. `N(T)` excludes the root quadruple, four circles bounded, one per strip period. Witness: lab/rs/apollonian, verbs census and strip, 67163 distinct against 67163 counted on the `T <= 10^4` control.
- 2026-09-14 [Verified] The residues are the arithmetic the census can see: the bounded packing `(-1, 2, 2, 3)` uses exactly the eight classes `2, 3, 6, 11, 14, 15, 18, 23` mod 24 over all 555198593 circles of curvature at most `10^7`, at counts 83211520, 55422929, 55455852, 83378348, 83354132, 55617906, 55576284 and 83181622, while the imprimitive strip packing uses exactly the four classes `0, 2, 8, 18` at 4144636, 6223160, 6241134 and 4161744 of its 20770674 circles; which integers inside those classes occur is closed by others and this tree makes no claim on it. Witness: lab/rs/apollonian, verbs census and strip.
- 2026-09-14 [Verified] The residual dimension is `1.3056867280498771846...`, rigorous to 128 places by an effective Ruelle-Bowen computation on a Chebyshev-Lagrange approximation of the transfer operator, Theorem 1.1 of Vytnova and Wormell 2024 reading `1.3056867280 4987718464 5986206851 0408911060 ... +- 10^(-129)`; the counting asymptotic `c T^alpha` is Kontorovich and Oh 2011 with `alpha ~ 1.30568(8)`, McMullen 1998 reads `1.305688`. The census's bounded `1.3057` is `alpha` correctly rounded to four places, off `1.3e-5`; the strip's `1.3056` and `1.3060` agree to three, off `8.7e-5` and `3.1e-4`. Witness: REFS.md, read at source; lab/rs/apollonian, verbs census and strip.
- 2026-09-14 [Verified] No design carries the gasket's dimension inside the window the tree can see: a design is the attractor of similarities of one ratio `1/base` under the open set condition so its dimension is `log N/log base` for an integer cell count `N`, equal to `alpha` only if `base^alpha` is an integer, and over `2 <= base <= 100` the nearest approach is `52^alpha = 174.005426001` at gap `0.005426001`, then `68`, `89`, `49`, `23`, `20` at gaps `0.008182, 0.011684, 0.015094, 0.022279, 0.026750`, worst `0.488110` at `base = 47`; the table refutes equality and nothing weaker, the nearest design dimension being `log 351/log 89 = 1.305694144`, off `alpha` by `7.4e-6`. Witness: lab/rs/apollonian, verb design.
- 2026-09-14 [Verified] The packing grower is now in the publishable crate: `mrlynum::apollonian` takes a named integral root, grows it by the square-root-free reflection in exact `i64` triples `(k, k x, k y)` and rechecks all six invariants of `B` on every quadruple, reproducing the generator's numbers from the crate: 2448 circles on one period of the strip to curvature 2048 and 950 to curvature 1000, the root excluded, 0 broken and 0 circles outside the open period, and 323 circles carrying `k y = 1` below 2048, every one passing the Ford test `k = 2 b^2`, `k x = 2 a b`, `gcd(a, b) = 1`. Witness: `mrlynum::apollonian::grow` and `is_ford`, test `every_line_tangent_circle_is_the_ford_circle_over_its_own_fraction`, 2448 and 323 of 323.
- 2026-09-14 [Verified] The stack and the packing's tangency points agree fraction by fraction and not only in count: at depth 32 the 323 nodes the Farey stack lights inside the open period and the 323 tangency points of the line-tangent circles of curvature at most `2 Q^2 = 2048` are the same set of reduced fractions with 0 missed either way and 0 off-Ford, the brightness of the period summing to 528 against `Q(Q + 1)/2`, and at depth 16 the same reading gives 79 against 79 with 0 missed and brightness 136; the agreement is checked at every depth from 2 to 64 against `mrlynum::lattice::farey`. Witness: `mrlynum::apollonian::shadow`, test `the_stack_is_the_shadow_of_the_line_tangent_circles`, 0 missed at both depths.
- 2026-09-14 [Verified] Two further bounded roots sit in the integer coordinates with all six invariants exact and hand back new censuses: `(-2, 3, 6, 7)` placed as `(-2, -1, 0), (3, 1, 0), (6, 5, 0), (7, 5, 2)` and `(-3, 4, 12, 13)` placed as `(-3, -1, 0), (4, 1, 0), (12, 7, 0), (13, 7, 2)`, each carrying a double Descartes root because `k_1 k_2 + k_2 k_3 + k_3 k_1 = 0`, giving `N(1000) = 1297` and `N(1000) = 741` beside 3325 for `(-1, 2, 2, 3)` and 950 for the strip period, the root quadruple excluded throughout. Witness: `mrlynum::apollonian::root` and `grow`, test `the_growth_lands_on_the_counts_the_generator_prints`, 1297 and 741.