Claims · 22 dated claims on The Apollonian gasket, each with its tag and its witness; newest 2026-09-21.
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.
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.
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.
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.
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.
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.
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.
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.
Verified The packing grower is now in the publishable crate: mrlyrs::num::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: mrlyrs::num::apollonian::grow and is_ford, test every_line_tangent_circle_is_the_ford_circle_over_its_own_fraction, 2448 and 323 of 323.
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 mrlyrs::num::lattice::farey. Witness: mrlyrs::num::apollonian::shadow, test the_stack_is_the_shadow_of_the_line_tangent_circles, 0 missed at both depths.
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: mrlyrs::num::apollonian::root and grow, test the_growth_lands_on_the_counts_the_generator_prints, 1297 and 741.
Proved Every Ford circle is the image of the line Im z = 1 under an element of SL(2, Z) and every such image with b >= 1 is a Ford circle: for gamma with bottom row (b, d), b >= 1, Im(gamma z) = Im z/abs(b z + d)^2 puts gamma(R + i) on a circle tangent to the real line at gamma(infinity) = a/b of top height 1/b^2, reached at x = -d/b, so of centre (a/b, 1/(2 b^2)) and radius 1/(2 b^2); on the modular surface the Ford circles are one closed horocycle, and the strip packing's far line y = 1 is that horocycle lifted. Witness: lab/py/ford-horocycle verb crossings, 5510 image points on the 1102 Ford circles of b <= 60 checked exactly in rationals, 0 off the circle.
Proved The horizontal horocycle Im z = h meets the closed disc of the Ford circle over a/b exactly when h <= 1/b^2, that is b <= h^(-1/2), tangent at the top when b = h^(-1/2); at h = Q^(-2) it meets exactly the Ford circles of F_Q, and at h = 1/(2 Q^2) those of F_(floor(Q sqrt 2)), not F_Q. Witness: the disc over a/b is 0 <= y <= 1/b^2 on the vertical through its centre; lab/py/ford-horocycle verb crossings reads closed-disc counts 1, 2, 4, 32, 324, 542 at Q = 1, 2, 3, 10, 32, 42 against sum_(b <= Q) phi(b), open-disc 0, 1, 2, 28, 308, 530 against sum_(b < Q) phi(b), and 1, 2, 6, 64, 628, 1086 at 1/(2 Q^2) against sum_(b <= floor(Q sqrt 2)) phi(b).
Proved The Farey stack's node count m(Q) = sum_(b <= Q) phi(b) is the number of Ford circles of one period the horocycle at height Q^(-2) meets, the novelty phi(n) is the number it first reaches descending from (n - 1)^(-2) to n^(-2), the brightness floor(Q/b) of a/b at depth Q is the number of the Q horocycles at heights (j/Q)^2, 1 <= j <= Q, meeting the disc over a/b, and the total brightness Q(Q + 1)/2 is sum_(j <= Q) m(floor(Q/j)). Witness: j b <= Q is the threshold at h = (j/Q)^2 and sum_b phi(b) floor(Q/b) = sum_j sum_(b <= Q/j) phi(b); lab/py/ford-horocycle verb census reads sum_(b <= Q) phi(b) from the unbounded Stern-Brocot walk at every Q = 2^j to 2^10, 318964 at the top.
Proved For f smooth of compact support in (0, infinity) and psi_f(z) = sum f(Im(gamma z)) over Gamma_infinity \ SL(2, Z), the closed horocycle integral at a height h below the support of f unfolds to the stack's smoothed novelty, int_0^1 psi_f(x + i h) dx = h sum_(c >= 1) phi(c) g(c h^(1/2)) with g(t) = int_R f(1/(t^2 (1 + v^2))) dv, Verjovsky's m_q(g) at q = h, equal to (3/pi) int f(w) w^(-2) dw + O(h^(1/2) log^2 h) by partial summation against sum_(c <= x) phi(c) = 3 x^2/pi^2 + O(x log x) with g'(t) = O(t^(-2)), the main term (3/pi) int psi_f dmu; the bridge runs one way, a positive bump on [1, 2] never being a g_f; the Mellin transform of the horocycle integral is F(s - 1) + phi(s) F(-s) with phi(s) = xi(2s - 1)/xi(2s), and the novelty series zeta(s - 1)/zeta(s) is phi(s/2) stripped of pi^(1/2) Gamma((s - 1)/2)/Gamma(s/2). Witness: the cosets are the coprime rows (c, d), one class mod c runs c x + d once over the line, w = c h v, Mertens 1874 for the totient sum, g(t) = (1/t) int f(w) w^(-3/2) (1 - t^2 w)^(-1/2) dw for the tail, and int g(t) t^(2s - 1) dt = (pi^(1/2)/2) (Gamma(s - 1/2)/Gamma(s)) F(-s); lab/py/ford-horocycle verb bridge, both sides agreeing to 7.7e-14 or better at h = 4^(-k), k = 2..10, on the C^infinity and C^2 bumps of lab/py/smoothed-novelty, 318453 cosets at k = 10.
Verified Zagier 1981, read at source at pp. 279-280, writes the closed horocycle C_y = Gamma_infinity \ (R + iy) of hyperbolic length 1/y, proves length(C_y cap U)/length(C_y) = vol(U)/vol(Gamma \ H) + O(y^(1/2 - eps)) for open U and states that O(y^(3/4 - eps)) for all U implies the Riemann hypothesis, with C(F; y) = kappa + O(y^(1 - Theta/2 - eps)) for twice differentiable F, Theta the supremum of the real parts of the zeros, kappa = (3/pi) int F, by the Rankin-Selberg identity I(F; s) = int F E(z, s) alone, no cusp form named; Sarnak 1981 is unread at source and its theorem is carried in the restatement of Drutu and Peyerimhoff 2020, read at source, y^(1 - s_1) mu_1(f) + ... + y^(1 - s_k) mu_k(f) + o(y^(1/2)) for f in C^1_c, 1 > s_1 > ... > s_k > 1/2. Witness: the scan at the Zagier homepage, page images read; arXiv 2001.07693 page 2; Verjovsky's Theorem B on stack.md The novelty meter for the totient side.
Proved With A = {1, ..., m} and an E_A circle a Ford circle whose tangency point has one of its two continued fraction expansions inside A, that is a_1, ..., a_(n-1) <= m and a_n <= m + 1 on the expansion with a_n >= 2: the Stern-Brocot children of [0; a_1, ..., a_n] are [0; a_1, ..., a_n + 1], the mediant with the older neighbour, and [0; a_1, ..., a_n - 1, 2], the mediant with the younger neighbour, the parent; the E_A circles are closed under the mediant with the younger neighbour and under the mediant with the older exactly when a_n <= m, so they are the Stern-Brocot subtree of (0, 1) whose paths have every run of length at most m, closed under parents, F_(d + 1) nodes at depth d when m = 2; the closure of their tangency points is those rationals together with E_A, of dimension dim_H E_A. Witness: the path L^(a_1) R^(a_2) ... X^(a_n - 1); lab/py/ford-horocycle verb census at b <= 512, three membership tests agreeing on 961 fractions at m = 2 and 5118 at m = 3, younger mediant in 960 of 960 and 5117 of 5117, older mediant in 584 and 4026, exactly the nodes with a_n <= m.
Verified The E_2 census: N_2(2^j) = 1936, 37781, 721365, 13614634, 59104562 at j = 10, 14, 18, 22, 24, N_2(Q)/Q^(2 delta_2) between 1.2272 and 1.2521 on the quarter octaves of [22, 24] with delta_2 = 0.5312805062772051416, octave exponents 1.0431 and 1.0750 at j = 22, 23, off 2 delta_2 = 1.0625610125544 by 0.0195 and 0.0124, octave exponent minus 2 delta_2 reading +0.0475, -0.0480, +0.0477, -0.0453, +0.0405, -0.0336, +0.0265, -0.0194, +0.0124 at j = 15 to 23, alternating with falling amplitude, two-octave exponents between 1.0590 and 1.0661 from j = 15 to 22, the half-spread of the ratio over an octave falling from 0.05 at j = 9 to 0.011 at j = 23, no threshold drawn; the E_3 control N_3(2^j) = 13615, 680827, 34078973 at j = 10, 14, 18, ratio between 0.76762 and 0.76775 on [17, 18], octave exponents 1.4113, 1.4114, 1.4112 at j = 15, 16, 17 against 2 delta_3 = 1.4113218160560. Witness: lab/py/ford-horocycle verb census, boxes 2^24 and 2^18, 28 seconds.
ConjectureN_m(Q) grows like Q^(2 delta_m), the count exponent of the E_A circles being twice the Hausdorff dimension of E_A; Hensley's theorem, its 1989 and 1990 texts unread here, so the page keeps the exponent at Conjecture with the census as support and lifts the tag when the source is read. Witness: lab/py/ford-horocycle verb census, two-octave exponents 1.0590 to 1.0661 against 1.0626 at m = 2 and octave exponents 1.4112 to 1.4114 against 1.4113 at m = 3; Hensley 1994 p. 44, read at source, C_m x^(D(m)) with D(2) ~ 1.06256; Bourgain and Kontorovich 2014 Remark 1.13, read at source, #R_A(N) between constant multiples of N^(2 delta_A).
Verified At A = {1,2} the zero of det(1 - L_(A,s)) of largest real part off the axis in 0 <= sigma <= 0.53, 0.2 <= tau <= 80 is s_1 = 0.457015235231 + 6.958882679527 i, one of 33 there with the winding number agreeing, stable to 8.9e-16 from 40 to 140 modes, the eigenvalue nearest 1 there 1.000000000000; the next two are 0.428067039 + 78.156951119 i and 0.412635450 + 71.206868515 i. Witness: lab/py/question-mark, verb subleading
Verified The E_2 census oscillation is the signature of s_1: the term Q^(2 sigma) cos(2 tau log Q + c) has period pi/(tau log 2) = 0.6513 octaves and factor 2^(2(sigma - delta_2)) = 0.9022 per octave, and 2 tau log 2 = 3 pi + 0.2223 aliases it at integer octaves to a sign alternation under an envelope of period 28 octaves; on 193 points of N_2(Q)/Q^(2 delta_2), sixteen per octave from 2^12 to 2^24, a constant leaves rms 1.64e-2 and the constant plus the wave with only the constant, amplitude and phase fitted leaves 7.67e-4, max 4.05e-3, amplitude 0.1312; the twelve octave exponents minus 2 delta_2 from j = 12 to 23 are reproduced within 0.0034, +0.0475 against +0.0472 at j = 15 and +0.0124 against +0.0127 at j = 23; sigma, tau fitted freely on the same points read 0.4556, 6.9598. Witness: lab/py/question-mark, verb subleading, the walk lab/py/ford-horocycle census_walk
Verified Control A = {1,2,3}: 67 zeros in 0 <= sigma <= 0.71, 0.2 <= tau <= 80 at 100 modes with the winding number agreeing, the largest real part 0.489705291051 + 45.352143150104 i sitting 0.2160 below delta_3 against 0.0743 at A = {1,2}, factor 0.7413 per octave, and N_3(Q)/Q^(2 delta_3) from 2^10 to 2^18 leaves rms 3.85e-4 to a constant and 3.83e-4 with the wave, so no wave shows at m = 3. Witness: lab/py/question-mark, verb subleading