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.