# The spun stack

- 2026-09-09 [Proved] The dead-spin theorem: rotated layers `(m, alpha)` and `(n, beta)` of the line stack share a node off the origin iff `cos` and `sin` of `alpha - beta` are both rational, and the shared set is then `R_alpha (1/gcd(m,n)) Z^2`, of density `gcd(m,n)^2` per unit area; at whole degrees Niven's theorem makes the condition `alpha = beta mod 90`, exactly 4 of 360 degrees having both `cos` and `sin` rational by reduction of `zeta^d + zeta^-d` mod `Phi_360`; the rational rotations are exactly `w^2/N(w)` for nonzero Gaussian `w`, not `z/|z|`, `(1 + i)/sqrt 2` the counterexample, all 68 rational unit-circle points of denominator at most 60 reached from the box of side 12. Witness: lab/py/spun-stack `rational_angle_degrees`, `dead_spin_pairs`, `pythagorean_hits`.
- 2026-09-09 [Proved] The exact spun stack indexes layers by the nonzero associate classes of `Z[i]`, layer `z` the lattice `z^-1 Z[i]`; its lit nodes in the unit square are the Gaussian rationals `u/d` in lowest terms, the node with reduced denominator `d` is lit by exactly the layers `d` divides, and its brightness is `g(floor(N/N(d)))` with `g(t) = sum_j (floor(t/(4j+1)) - floor(t/(4j+3)))`, the Gauss circle count of nonzero classes of norm at most `t` and the Gaussian twin of `floor(N/b)`; the lit set has `sum_{[d], N(d) <= N} Phi(d)` nodes, the Gaussian totient sum; 672 nodes and 0 mismatches against exact literal stacking at norm bound 50, counts 672, 10608, 168088 at norm bounds 50, 200, 800. Witness: lab/py/spun-stack `literal_stack`, `closed_brightness`, `totient_sum`.
- 2026-09-09 [Proved] A fixed rotation with a fixed geometric scale per layer is multiplication by one complex `c`: the layers `c^-k Z[i]` overlap off the origin iff `c` is in `Q(i)` and nest iff `c` is in `Z[i]`, and then brightness is `depth + 1 - address`, a pure address with no moire, the base-`c` numeration tree (base `-1 + i` the twindragon); Verified at `c = 1 + i` depth 8 and `c = 2 + i` depth 4, 256 and 625 nodes, 0 mismatches, overlaps 440, 220 and 0 over the box of side 10 for `1 + i`, `3/2 + i/2` and `sqrt 2 e^i`. Witness: lab/py/spun-stack `base_depth_check`, `base_c_overlap`.
- 2026-09-09 [Verified] No Franel-Landau theorem for the Gaussian Farey set is found in the sources read (Sayous arXiv:2407.04380 proves equidistribution on `C/Z[i]` with no rate and a gap law, naming neither Franel nor Landau; Estala-Arias arXiv:1908.03658 states RH for `zeta_K` on measures over the positive reals; Huxley Acta Arith. 18 (1971) and Kanemitsu-Yoshimoto Acta Arith. 75 (1996) unread), so an RH-equivalent for `zeta_K` rendered by the spun stack is unstated, neither proved nor refuted; the named obstruction is that Franel-Landau needs a rank and `C/Z[i]` carries no canonical linear order. Witness: lab/py/spun-stack, REFS.md. Superseded: Huxley 1971 and Kanemitsu-Yoshimoto 1996 are read at source, the rank obstruction blocks only the rank functional, and the Fourier `L^2` equivalence is proved under Franel one field up.
- 2026-09-09 [Proved] The centre is a node of every spin schedule of the odd carpet stack: rotation about the centre preserves distance from it and layer `n`'s cell containing the centre has inradius `1/(2n)`, so a disc of radius `1/(2N)` lies inside one cell of all layers at every angle; at `N = 55` the centre reads ink `14/28`, the scales `n = 3 mod 4`, the unspun value, under every schedule and every raster. Witness: lab/py/spin-render `main`.
- 2026-09-09 [Proved] The unspun odd carpet stack at `N = 55` attains its global ink maximum `18/28` on exactly four square cells of side `1/159`, total area `4/25281 = 0.000158222`: ink at `(u, v)` is the size of the intersection of the two scale sets, so a maximum needs them equal and maximal; the one-dimensional maxima over 636 exact breakpoints are `[1/3, 18/53)` and `[35/53, 2/3)`, and the scale set is invariant under `x -> 1 - x` because `floor(n(1 - x)) = n - 1 - floor(nx)` with `n - 1` even, so both intervals carry the same 18 scales and all four products are maxima. Witness: lab/py/spin-render `diagonal_maximum`.
- 2026-09-09 [Verified] Spinning by whole degrees destroys the unspun maximum and shrinks its cell: peak ink falls from `18/28` to `14/28` under a one-degree increment and `16/28` under the prime-degree schedule, the golden and Gaussian schedules, `17/28` under random angles, the peak cell area from `3.95523e-05` to `9.80453e-06` and `1.65596e-05`, at `R = 256, 512, 1024, 2048` and under a zoom at effective `R = 51200`; spinning leaves the fade law alone, `rms sqrt(L)` in the layer count `L` at `L = 28` running `0.401417` to `0.460395` over six schedules, the unspun raster matching the exact rational covariance sum `0.309477, 0.389754, 0.426869, 0.458411` at `L = 4, 8, 14, 28` to `0.4%`, the lane's `c = 0.522` being the limit constant and not the `L = 28` value. Witness: lab/py/spin-render `report`, `main`.
- 2026-09-09 [Proved] The eyes of the fixed increment: under the schedule that turns layer `k` by `k theta`, two layers with indices `j, k` share an exact lattice iff `(j - k) theta` is a multiple of 90 degrees (the odd carpet being invariant under a quarter turn), so at `theta = 90 p/q` in lowest terms the layers fall into exactly `q` angle classes and the sharing pairs number `sum_classes C(size, 2)`, while at an irrational `theta/90` no pair shares; on the 28 odd scales to 55 the count reads 378 at `theta/90 = 0`, 182 at `1/2`, 117 at `1/3` and `2/3`, 84 at `1/4` and `3/4`, 65 at `1/5` and `2/5`, 52 at `1/6`, 36 at `1/8`, 30 at `1/9` and 0 at `sqrt 2 - 1`, confirmed by the pairwise exact test and by the Niven-free rational-angle test at the whole-degree increments; the moire switches on exactly past the Farey fractions of a quarter turn. Witness: lab/py/spun-stack `increment_classes`.
- 2026-09-09 [Proved] The node-count constant of the spun stacks: for every imaginary quadratic field `K` with class number `h`, `w` units and discriminant `D_K`, `sum_{N(a) <= N} Phi(a) = (rho_K/(2 zeta_K(2))) N^2 + O(N^(3/2))` over nonzero ideals with `Phi = N * mu_K` and `rho_K = 2 pi h/(w sqrt |D_K|)`, by Dirichlet convolution and Abel summation on the ideal count `A(t) = rho_K t + O(sqrt t)`, itself derived one ideal class at a time from the lattice of covolume `N(a) sqrt |D_K|/2`; the Gaussian constant is `pi/(8 zeta(2) G) = 0.260634696495` and the Eisenstein constant `pi/(6 sqrt 3 zeta(2) L(2, chi_-3)) = 0.235217881630`, the nine published node counts recounted exactly and extended to norm bound 102400 where `count/(c N^2)` reads `0.999746` and `1.000049`, the deviation scaled by `N^(3/2)` never past `0.119` and the ratios oscillating about 1, with `D = -20` (`h = 2`, ratio `1.000001117`) and `D = -23` (`h = 3`, ratio `0.999886`) at 102400 as the class-number witnesses; the observed `N log N` size of the error stays Conjecture; the literature's complex Farey constant `pi/(sqrt |D_K| zeta_K(2))` counts element denominators and is `w` times this one, the sets being equal. For a general number field the same argument runs from any ideal count with error `O(t^theta)`, `0 < theta < 1`. Witness: lab/py/totient-constant `main`, `norm_totient_sum`, `farey_set`.
- 2026-09-09 [Conjecture] The Gaussian Farey stack's node count is asymptotically `pi N^2/(8 zeta(2) G) = 0.260635 N^2` with `G` Catalan's constant; the ratios read `0.268800, 0.265200, 0.262638` at norm bounds 50, 200, 800; the literature count of the complex Farey set is 4 times this, the order of the unit group, a convention difference unresolved. Witness: lab/py/spun-stack `totient_sum`. Superseded: the constant is Proved and the convention resolved, see the constant row under The spun stack in SETTLED.
- 2026-09-09 [Refuted] A whole-degree prime schedule (layer `k` at `p_k` degrees) is coincidence-free: 5 of the 435 layer pairs to `N = 30` share, all at relative angle exactly 90 (7 and 97, 11 and 101, 13 and 103, 17 and 107, 19 and 109 degrees), each sharing a lattice of density `gcd(m,n)^2` per unit area whose count in the open unit square with the origin excluded reads `1, 9, 49, 1, 9` and is angle-dependent (`2, 3, 4` at `g = 2` over degrees 1 to 89), not a formula; the other 430 pairs are dead with margin `0.003390`. Witness: lab/py/spun-stack `dead_spin_pairs`, `unit_square_shares`, `share_count_spread`.
- 2026-09-09 [Refuted] A resolution-stable off-centre maximum in a spun render is an exact coincidence: the stack is piecewise constant on cells of positive area, so any cell wider than a pixel holds its position at every resolution, and the prime-degree schedule shows one at `(0.19469, 0.15501)` drifting `0.29` px from `R = 1024` to `2048`; raster stability measures cell area, and the discriminators are the peak value and the cell area. Witness: lab/py/spin-render `drift`.