# 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.