Spin

• 2026-08-30 [Proved] The average of a picture over the q rotations by 2 pi / q keeps exactly the circular harmonics of order divisible by q, the average over all rotations keeps order zero only, and a design of rotation order g shows lcm(q, g) petals under a screen that turns it p/q of a turn per frame. Witness: mrlynum::spin the_harmonics_read_the_rotation_order, spin.
• 2026-08-30 [Proved] The rings of a spun square-lattice picture sit at sqrt(n) for n a sum of two squares with weight r2(n) = 4 (d1 - d3), silent exactly where a prime 3 (mod 4) divides n to an odd power, and their Dirichlet series is 4 zeta(s) L(s, chi_4); the hexagonal rings carry 6 zeta(s) L(s, chi_-3); the mass of a spun lattice is the Gauss circle count and Hardy's Bessel series for its error is the ring expansion. Witness: A001481, A004018, A003136, A004016, Hardy 1915, spin.
• 2026-08-30 [Proved] The exact ring profile of a raster integrates to its fill, int 2 pi r F(r) dr = fill, 512.0 at level 3 of the carpet; the carpet's profile is zero to side/6. Witness: mrlynum::spin the_mass_of_the_profile_is_the_fill, mrlydemo fixture.
• 2026-08-30 [Refuted] The coprime law survives the spin - flat layers at coprime odd scales are exactly uncorrelated, but their ring profiles over the inscribed disc correlate at +0.38 for (3, 5), -0.33 for (5, 7) and +0.38 for (9, 13), no better than gcd pairs; the cancellation is separable in x and y and the spin discards the angle. Witness: mrlylab test the_coprime_law_dies_under_the_spin.
• 2026-08-31 [Proved] The spin mass about the fixed point p_d = d/(base-1) of a filled digit d obeys M(r/base) = M(r)/fill exactly, since S(x) = (x+d)/base carries the design onto its d piece and divides the self-similar measure by the fill, so M(r) = r^dim G(log(r)/log(base)) with G of period exactly log base - the ripple's period is an identity and not a fit, valid for r/base below the distance from p_d to the other filled cells, that is r <= side at the corner digit and r <= side/2 at the centre. Witness: mrlynum::spin::mass_within, the_spin_mass_scales_by_the_fill_about_a_filled_corner.
• 2026-08-31 [Verified] The spin dimension read about the corner fixed point at level 6 over the window 27 <= r <= 729, three whole periods of log 3, gives slopes 1.465054, 1.649432, 1.783588, 1.761814, 1.897854, 1.879522, 2.000100 for codes 79, 95, 127, 239, 255, 495, 511 against the exact log(fill)/log 3, every gap at or below 1.9e-2 and the discretisation of the identity, max |M(3r)/(fill M(r)) - 1|, running 5.3e-3 to 1.8e-2; the exact integer shell histogram and the crate profile integral agree to 1.7e-6 on the total and 0.5% at partial radii. Witness: lab/rs/spin-census.
• 2026-08-31 [Verified] The corner ripple separates both equal-dimension pairs of the census where every density reading is identical: at level 7, 127 against 239 gives ripple gap 0.11984 on drift bar 0.04126 and 255 against the carpet 495 gives 0.12042 on bar 0.01461, with the solid square as the rippleless control at swing 0.00277 under its own bar 0.00578 and a code against its mirror at gap 0.00e0. Witness: lab/rs/spin-census.
• 2026-08-31 [Verified] The spin spectrum P_m, m = 0..12, read at levels 1 and 2 over all 511 nonempty base-3 plane codes, splits them into exactly 101 spectra, the number of nonempty orbits of the square group, with no pair outside one orbit agreeing to 1e-9: within the family it is a complete invariant of the dihedral class and no spin-isospectral witness exists. Witness: lab/rs/spin-census.
• 2026-08-31 [Verified] The ring-averaged powder of a design is not Porod: every sliding three-period window slope, over every fractal code at level 7 and at both pad 4096 and pad 8192, stays above -2.28 and so at least 0.72 from the -3 of a sharp interface, while the solid square control slides from -2.75781 to -2.35759, within 0.25 of -3 and never near its own -dim = -2. Witness: lab/rs/spin-census.
• 2026-08-31 [Proved] The Menger sponge at level level blocks every lattice line down its space diagonal that meets its bounding cube: the shadow obeys S_(level+1) = union_d (3 S_level + proj d), and along (1,1,1) the 27 cube digits and the 20 sponge digits project onto the same 19 classes, so the induction gives equality at every level, the count 3^(2 level + 1) - 3^(level+1) + 1 = 19, 217, 2107, 19441, 176419. Witness: lab/rs/spin-census, A220978, A003215.
• 2026-08-31 [Proved] The sponge's axis shadow is exactly the Sierpinski carpet, 8^level against the cube's 9^level, dimension log 8 / log 3 = 1.892789: the 20 sponge digits project along an axis onto the 8 carpet digits, disjoint modulo 3. Witness: lab/rs/spin-census.
• 2026-08-31 [Verified] No direction other than the axis is deficient in the searched window - over the 13 directions with 0 <= a <= b <= c <= 3, read to level 4 against the cube, the axis is the only share that falls with the level, every other rising, (1,1,2) to 0.98568 and (0,1,2) to 0.97090 at level = 4. Witness: lab/rs/spin-census.
• 2026-08-31 [Proved] A radius sqrt(k)/n of the spun scale-n square lattice, read inside the disc of radius sqrt 2, is new at n exactly when no prime p | n has p^2 | k - the sum-of-two-squares condition at the smaller scale is automatic by a parity argument, so only integrality binds - and hence new(n) = sum_(d | rad n) mu(d) B(2n^2/d^2) with B the counting function of A001481, giving 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, 91, 64, 123, 97, 128, 126, 199, 136, 243, 180, the rule, the identity and a direct union agreeing at every n to 64. Witness: lab/rs/spin-census, A001481.
• 2026-08-31 [Proved] The Gaussian Farey's local factor is the Jordan totient J_2(n)/n^2 = prod_(p | n) (1 - 1/p^2), the square-lattice analogue of Farey's phi(n)/n, approached from below at rate 1/ln n because B(X) ~ K X / sqrt(ln X): the radical-6 family climbs 0.56250, 0.59813, 0.61126, 0.62594, 0.63276, 0.63801 at n = 6, 12, 24, 48, 96, 192 toward 2/3. Witness: lab/rs/spin-census, A064533.
• 2026-08-31 [Proved] The spin dimension about the raster centre is undefined for a design with an empty centre digit - the empty digit removes the open square of side side/3 about the centre and hence its inscribed disc, so M(r) = 0 for every r <= side/6 and the centre-spun mass carries neither power law nor ripple over a whole factor of base. The bound is attained, in exact integer arithmetic on doubled coordinates rather than cell centres, which would return hole + 1/2 whatever the hole: the squared distance to the nearest filled cell is (side/3)^2 = 6561 at level 5 for both bang dim 2, base 3, code 239 and the carpet 495, that is side/6 = 40.5 exactly, while 79 empties out to 56.572962. Witness: lab/rs/spin-census.
• 2026-08-31 [Conjecture] The near-degenerate corner ripples beyond the segment case: de-duplicated to transpose classes, the equal-fill class pairs sitting inside their own drift bar number 13 at level 6 and 6 at level 7, at gap-to-bar ratios 0.71 to 0.95, the tightest 287 against 315 at fill 6, gap 0.03574 on bar 0.04543, two designs that differ by moving one cell from (0,2) to (1,2); six survive both levels, 287-315, 63-123, 123-187, 31-59, 437-485, 37-261, and the count moves with the level and the estimator, so the list is a phenomenon and not a census. The ripple's Fourier coefficient at frequency 2 pi / log 3 should be a linear functional of the digit set whose kernel is what collides. Witness: lab/rs/spin-census.
• 2026-08-31 [Conjecture] The powder falls as k^-dim - at level 7 with pad 4096 the slopes -1.37986, -1.51762, -1.73012, -1.83071, -1.97886, -2.00433 sit within 0.24 of -dim over both pads, but the agreement is inside the instrument's own spread: sliding a three-period window a quarter period at a time moves the slope by 0.16 to 0.45, and doubling the pad to 8192 moves 127 from -1.73012 to -1.81607, 255 from -1.97886 to -2.03225 and the carpet from -2.00433 to -2.12289, with no monotone approach to -dim. The log-periodic ripple is not resolved either, the folded residual swinging 1.5 to 4.4 in ln power because the ring average of a lattice point set is spiked on the norms of A001481. Witness: lab/rs/spin-census.
• 2026-08-31 [Conjecture] The axes are the sponge's only deficient shadow directions; the window checked is |v| <= 3. Witness: lab/rs/spin-census.
• 2026-08-31 [Conjecture] new(n) sqrt(ln n) / n^2 converges to sqrt 2 K prod_(p | n) (1 - 1/p^2) along each radical class, K the Landau-Ramanujan constant; the Mobius sum is proved but B has no closed form, so the Gaussian Farey carries a transcendental constant where the Farey carries none. Witness: lab/rs/spin-census, A064533.
• 2026-08-31 [Refuted] The corner ripple as a complete invariant of the transpose class - a design that is a solid segment has M(r) = c r exactly about a fixed point on it, so its ripple vanishes identically, and code 7, the solid row, and code 273, the solid diagonal, are two such designs of dimension exactly 1 in different transpose classes carrying the same zero ripple; the census reads them at swings 0.01114 and 0.01217 and mutual gap 0.01371, all discretisation, and no bar is needed for the conclusion. It does separate both named equal-mass pairs, at 2.9 and 8.2 times the drift bar. Witness: lab/rs/spin-census.
• 2026-08-31 [Refuted] The Gaussian Farey counted by primitive representations in Z[i] modulo units - the norms below 2n^2 with a primitive representation run 2, 3, 6, 9, 13, 17, 23, 29, 35, 44 against new(n) = 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, agreeing only at n = 1, 2; primitivity is the wrong condition, since (3,4) is primitive and 25 is a square, so sqrt(25)/5 = 1 is old at 5. The correct criterion is freedom from the squares of the primes of n. Witness: lab/rs/spin-census.
• 2026-08-31 [Refuted] The disc and the box read the same Gaussian Farey - restricting to 0 <= a, b <= n instead of the disc of radius sqrt 2 breaks the criterion at n = 3, witness the radius 4/3: 16 is free of 9 and 4/3 < sqrt 2, but 16 = 4^2 + 0^2 needs a coordinate above 3, and the box counts 2, 3, 7, 9, 17, 14, 31, 27, 41, 38 part from the disc counts from n = 3 on. Witness: lab/rs/spin-census.
• 2026-09-07 [Proved] The spin spectrum reads a pair census and nothing else: for a constant-valued 0/1 render on a raster of side base^level, every P_m is a quadratic form in the cell indicators whose Gram matrix is constant on the orbits of the raster's symmetry group acting on pairs, because turning a pair by theta multiplies both harmonic coefficients by e^(-i m theta) while a mirror at alpha sends c_m to e^(-2 i m alpha) conj(c_m) and the phase cancels in the real part; so P_m is a linear functional of the pair census Phi_level and equal censuses force equal P_m at every order, ring count and truncation, the base-3 plane carrying 11 pair classes at level 1 and 461 at level 2 and the level-1 coefficients solved from 11 independent censuses reproducing mrlynum::spin::harmonics at 1024 rings and m = 0..12 on all 511 codes at worst relative residual 1.14e-14. In dimension 3 the covariant object is the degree-l power summed over its orders, not a single (l, m). Witness: lab/rs/spin-census shape, spin.md the spin spectrum is a quadratic form.
• 2026-09-07 [Proved] The 101 spectra need level 2: Phi_1 takes exactly 97 values on the 101 nonempty base-3 plane orbits, four pairs lying in distinct square-group orbits with all 11 class counts equal, so each pair's level-1 spectrum coincides identically at every order and resolution; reading P_m at level 1 alone and bucketing greedily at 1e-9 returns 97 buckets, with 45-105 at gap 1.30e-16 and level-2 gap 0.151, 61-121 at 1.03e-17 and 0.0689, 78-102 at 6.51e-17 and 0.253, and 94-118 at 1.64e-16 and 0.105. Witness: lab/rs/spin-census shape, spin.md level 1 alone is not complete.
• 2026-09-07 [Verified] The 13 orders see 9 of the 11 level-1 census directions, and the odd cap of 3 is exact: the half turn rho is itself in the square group and acts trivially on classes, but half-turning one member, tau: {j, k} -> {rho j, k}, is well defined on classes because rho is central in D4, and g_(m, rho j) = (-1)^m g_(m, j) gives Q_m . tau = (-1)^m Q_m; tau fixes 5 of the 11 classes, so the antisymmetric part has dimension (11 - 5)/2 = 3 and no number of odd orders can exceed rank 3, which the six odd orders reach exactly while the seven even orders reach the proved cap of 6. The level-1 spectrum is strictly coarser than the census it factors through and splits it into the same 97 classes anyway. Witness: lab/rs/spin-census shape, spin.md the 13 orders see 9 of the 11 census directions.
• 2026-09-07 [Verified] The completeness is not about base 3, as a statement about the census: over all 2^25 base-5 plane codes the level-1 pair census takes 3993511 values on the 4211743 nonempty square-group orbits with 204856 ties over 423088 orbits and largest tie 8, and over all 2^27 base-3 dim = 3 codes it takes 1461693 values on the 2852287 nonempty orbits of the order-48 cube group with 757066 ties over 2147660 orbits and largest tie 32, and every tie breaks at level 2, the budget-capped weight window failing to bind and covering every group, all 204856 out to weight 21 and all 757066 out to weight 24 against level-2 censuses of 24805 and 6325 classes, with the canonical counts matching the Burnside averages 4211744 and 2852288 computed from the cycle index in the same pass. Witness: lab/rs/spin-census shape, spin.md the completeness is not about base 3.
• 2026-09-14 [Proved] The level-1 rank cap is 9 overall, 6 even and 3 odd; the odd cap of 3 is the known tau argument, the even cap of 6 and the total of 9 are new and supersede the earlier even bound of 6 of a possible 8. The common kernel of the coefficient vectors Q_m on the 11 base-3 pair classes holds the corner-centre basis vector, since the centre cell's farthest point and a corner cell's nearest point are both at r = sqrt2/6, so the two radial supports meet only in that null set and the integral vanishes at every order; and it holds (4, 16, 8, 8, 16, 4, 4, 8, 8, 4, 1) less 9 in the centre-centre slot, since the raster is similar to its own centre cell at ratio 1/3 and P_m(S/3) = P_m(S)/9. Both are tau-symmetric, so odd caps at 3 and even at 6. Witness: paper spin-harmonics Theorem 5.4, scripts/verify.py blocks 6 and 7.
• 2026-09-14 [Verified] The three caps are attained, on closed-form cell arcs and tanh-sinh quadrature rather than the 1024-ring discrete transform of the tree: 5 and 19 exactly located radial segments at 113 nodes give rank 9, even 6, odd 3 at m = 0..12, stable from 1e-9 to 1e-13, smallest pivot 1.878e-4 against largest coefficient 9.806e-2; the Gram matrix is constant on the 11 classes to 1.44e-17 and on the 461 level-2 classes to 1.52e-18 under both generators of the square group, a quarter turn and a reflection; the same pass returns 97 level-1 and 101 level-2 spectra over all 511 codes, every level-2 bucket one orbit. Witness: paper spin-harmonics scripts/verify.py blocks 5, 7, 9 and 10.
• 2026-09-14 [Proved] The four base-3 homometric pairs are two pairs and their centre augmentations, and homometry alone forces it: classes 0, 6 and 10 of the level-1 census are the self-pairs of a corner, an edge cell and the centre, so homometric designs already share their corner count, their edge count and their centre occupancy, and adjoining the centre to two that avoid it adds the corner count to the corner-centre class, the edge count to the edge-centre class and 1 to the centre-centre class: 61 = 45 + centre, 121 = 105 + centre, 94 = 78 + centre, 118 = 102 + centre, the augmented censuses reading 2, 2, 1 there against 0, 0, 0 before. Witness: paper spin-harmonics Lemma 4.4, scripts/verify.py block 3.
• 2026-09-14 [Verified] The 13 orders are a genuine truncation, and the solid square is the design the truncation flatters most: at level 1 the orders m = 0..12 hold 0.977541 of the solid square's angular energy against the exact Parseval total 1, the largest share over all 511 codes, tied only by the lone centre cell code 16 where P_m(S/3) = P_m(S)/9 forces it, while the smallest is 0.899436 at the four one-corner codes 1, 4, 64, 256, so every design spills at least 2.2 percent into the unread orders and some spill 10. Witness: paper spin-harmonics Fact 6.3, scripts/verify.py block 11.