# 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](../notes/spin.md).
- 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](../notes/spin.md).
- 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.