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.