The unequal split

• 2026-10-03 [Proved] The patch, the attractor of x -> x/2 and x -> x/3 + t for t in {(2/3, 0), (2/3, 1/3), (2/3, 2/3), (0, 2/3), (1/3, 2/3)}, satisfies the open set condition with the open unit square and has dimension D = 1.778602507, the root of 2^(-s) + 5*3^(-s) = 1; it is nonlattice since log 2/log 3 is irrational. Witness: dimensions.md The unequal split, The patch; lab/py/unequal-split verb patch.
• 2026-10-03 [Proved] With the square of side 1/2 in a corner of the unit square at most five interior-disjoint squares of side 1/3 fit beside it, and with k >= 7 thirds no placement in the plane satisfies the open set condition, the squared ratios summing past 1. Witness: dimensions.md The unequal split, The patch.
• 2026-10-03 [Proved] The level render of the patch on side 6^level has fill (9 + 4k)^level, so the design reading log(fill)/log(base) gives log 29/log 6 = 1.879323585 and not D, and the level-2 render differs from the Kronecker square of its tile in 336 of its 1296 cells. Witness: dimensions.md The unequal split, The patch; lab/py/unequal-split verb patch.
• 2026-10-03 [Proved] Every zero of f(s) = 1 - 2^(-s) - k 3^(-s) lies in D_l <= Re s <= D with D_l the real root of k 3^(-s) = 1 + 2^(-s), D is the only zero on Re s = D, and a lattice string's zeros lie on finitely many vertical lines while the patch's do not. Witness: dimensions.md The unequal split, Its complex dimensions.
• 2026-10-03 [Verified] For every k = 1..27 the box Re [D_l - 1/4, D + 1/4], Im [-60, 60] holds exactly 21 zeros of 1 - 2^(-s) - k 3^(-s), certified by the winding number with a Lipschitz margin of at least 161; at k = 5 they have eleven distinct real parts and some imaginary part misses the multiples of 2 pi/ln 2, 2 pi/ln 3, 2 pi/ln 6 by at least 0.470, 0.058, 0.461 of a step. Witness: lab/py/unequal-split verb poles.
• 2026-10-03 [Verified] The root .7675115443 + 45.55415979 i printed for the 2-3 nonlattice equation by Lapidus and van Frankenhuijsen 2003, Section 3.1, lies 4.0e-8 from the root of its lattice approximant 1 - 2^(-s) - 2^(-485 s/306) and 7.6e-5 from the true complex dimension 0.7674996132 + 45.55423466 i. Witness: lab/py/unequal-split verb poles.
• 2026-10-03 [Proved] Near D + it with t ln 2 in 2 pi Z and theta = t ln 3 reduced mod 2 pi, the zero of 1 - 2^(-s) - k 3^(-s) is D + it - i Q theta/f'(D) - P Q (ln 2)^2 theta^2/(2 f'(D)^3) + O(theta^3) with P = 2^(-D), Q = k 3^(-D), so along the convergents of log2 3 the complex dimensions approach the line Re s = D; this is Lapidus and van Frankenhuijsen 2003 Theorem 4.3, equation (4-9), with multiplicities 1 and k, restated by the implicit function theorem. Witness: dimensions.md The unequal split, Its complex dimensions.
• 2026-10-03 [Verified] At k = 5 the zeros at the fifteen convergent denominators q = 2 to 53715833 of log2 3 meet the second-order law with ratio 0.988235 at q = 2 and 1.000000 from q = 665 on, the closest at D - Re w = 5.215918e-17 at height 4.869e8. Witness: lab/py/unequal-split verb poles.
• 2026-10-03 [Proved] The count N(r) of cells of side at least r satisfies N(r) = 1 + N(2r) + k N(3r), its Laplace transform in ln(1/r) is 1/(s f(s)), whose poles are the complex dimensions and s = 0, and the cells of side at most r with a larger parent number 1 + k L with L the count of cells larger than r. Witness: dimensions.md The unequal split, The count of cells.
• 2026-10-03 [Proved] N(r) r^D tends to 1/(D f'(D)), 0.573459971 at k = 5, the limit existing by Lalley 1989 Theorem 1 and its value forced by the Laplace transform, while the carpet's N(r) r^D is a fixed non-constant ln 3-periodic function of ln r less r^D/7 and never converges. Witness: dimensions.md The unequal split, The count of cells.
• 2026-10-03 [Verified] The count is exact to r = e^(-300) over 59448 sizes with the renewal identity at every one, and over the six printed windows of length 10 starting at U = 10, 20, 40, 80, 160, 290 the mean of N(r) r^D/C stays within 4.4e-4 of 1 while its swing falls from 0.470122 at [10, 20] to 0.089622 at [290, 300] and the carpet's stays between 2.06 and 2.10. Witness: lab/py/unequal-split verb count.
• 2026-10-03 [Conjecture] The swing of N(r) r^D/C over a window at U = ln(1/r) decays like 1.52 U^(-1/2), its product with sqrt(U) staying in [1.486, 1.546] on the six printed windows starting at U = 10, 20, 40, 80, 160, 290, read only at starts that are multiples of 10 and not at other starts. Witness: lab/py/unequal-split verb count.
• 2026-10-03 [Verified] On ln N(e^(-u)) - D u over u in [50, 300] the folded variance is 0.003, 0.005, 0.004 at ln 2, ln 3, ln 6 against the carpet's 0.999 at ln 3, and the ten highest periodogram peaks in (2, 600) lie within 0.001 of zeros of 1 - 2^(-s) - 5*3^(-s) with heights within 1.3% of their residue amplitudes, while the carpet's six highest sit at 1 to 6 times 2 pi/ln 3. Witness: lab/py/unequal-split verb count.
• 2026-10-03 [Proved] About the fixed point of one map of ratio rho with no other child within r_0, the natural measure's ball mass satisfies M(r) = rho^D M(r/rho) for r < r_0, so the spin ripple is exactly ln(1/rho)-periodic: ln 2 about (0, 0) with r_0 = 2/3, ln 3 about (1, 0) and (1, 1) with r_0 = 1/3. Witness: dimensions.md The unequal split, What the detectors see.
• 2026-10-03 [Verified] The enclosed ball masses meet M(r) = rho^D M(r/rho) at all 96 shifted radii about each of the three corners, the ripples swing 0.23906, 0.15388, 0.17843 against enclosures at most 8.2e-5, at least 2661-fold, and each folds 1.000 at its own period and at most 0.079 at the others. Witness: lab/py/unequal-split verb ripple.
• 2026-10-03 [Proved] Computer-assisted: a non-constant spin ripple cannot carry both periods ln 2 and ln 3, and the ripples about the three corners are non-constant by double-precision enclosures at least 2661 times narrower than their swings, so the ripples about (0, 0) and (1, 0) share no period, which no design can produce since every such identity on a design has period a multiple of ln base. Witness: dimensions.md The unequal split, What the detectors see.
• 2026-10-03 [Verified] Every self-similar measure on the patch with positive weights is Rajchman by Rapaport 2022 Corollary 1.6, the system being affinely irreducible with six non-collinear fixed points and 2^(n_2) = 3^(n_1) impossible. Witness: dimensions.md The unequal split, What the detectors see.
• 2026-10-03 [Verified] The patch is Minkowski measurable by Gatzouras 2000 Theorem 2.3(i) with Theorem 2.4, the open set condition holding and ln 2, ln 3 lying in no lambda Z. Witness: dimensions.md The unequal split, What the detectors see.