spin-render

• Renders the odd parity-carpet stack with each layer spun about the centre, and measures what survives a change of raster.
• Layer k is the carpet C_n(u, v) = chi_n(u) chi_n(v) at the k-th odd scale n = 2k - 1, n <= 55, rotated about (1/2, 1/2) by theta_k, sampled at pixel centres of an R x R raster.
• The stack is the paper coverage P in [0, 1], the mean over the 28 layers of the not-inked indicator; the ink field is D = 1 - P, and D takes values k/28.
• Peaks are read on D. In paper coverage the value 1 is attained on wide regions near the rim and carries no node structure; the isolated features are the ink concentrations, so every peak, plateau and drift number below is a statement about D.
• Measurement is restricted to the inscribed disc of radius 1/2 about the centre, so every layer covers every sampled pixel at every angle; the layers themselves are the periodic extension of the parity rule, so no pixel is ever undefined.
• The centre is the fixed point of every rotation and its cell has inradius 1/(2n) in each layer, so a disc of radius 1/110 about the centre lies inside one cell of all 28 layers whatever the schedule; the centre reads D = 14/28 in every schedule at every R, the 14 scales n = 3 mod 4 (main, dark_layers).
• Six schedules (schedules): unspun at 0; degrees at k degrees; golden at k * 137.507764 degrees; primes at 0, 2, 3, 5, 7, ..., layer 1 at 0 and layer k > 1 at p_(k-1), ending at 103; random at a seeded uniform angle, default_rng(1); gaussian at atan2(b, a) for a >= b >= 1 with a^2 + b^2 = n and 0 when no such pair exists, which rotates only the 9 scales 5, 13, 17, 25, 29, 37, 41, 45, 53.

RUN

• uv run python research/lab/py/spin-render/spin_render.py
• From the repo root. One core, about 10 seconds.
• Domain is R = 256, 512, 1024, 2048 for all six schedules, the fade table at R = 1024 and layer count L = 4, 8, 14, 28, and a 0.02-wide zoom at effective R = 51200 on each schedule's top peak.
• Writes the five PNGs beside this file and nothing else.

WHAT IT PRINTS

• report prints, per schedule and per R: the peak of D, the unit-square area of all disc pixels at that peak, the mean of P, the RMS contrast of D, the centre value, and the top three of the 20 local maxima with the pixel count of each maximum's connected plateau.
• Maxima are the pixels equal to their 3 x 3 maximum filter (maximum_filter3), taken greedily by value with an exclusion radius 0.01 in unit-square coordinates (peaks); each is reported at the centroid of its flood-filled plateau (plateau), not at its seed pixel.
• drift matches each of the 20 maxima at R to its nearest at 2R and prints the median and maximum displacement, in unit coordinates and in pixels at R, with the count of the 20 that land within one pixel.
• diagonal_maximum computes the unspun maximum and its whole locus exactly, with no raster. D(u, v) = |S(u) inter S(v)| with S(u) = {n : floor(nu) odd}, so a maximum needs S(u) = S(v) of maximal size, and the 1D maxima are read off the 636 breakpoints k/n in exact rationals; it returns every interval attaining the maximum and every product cell whose two factors carry the same S.
• exact_variance sums the paper's rational covariance formula over the L x L scale pairs, an expectation for the fade table computed without any render.
• dark_layers lists the scales inked at a peak's seed pixel, independently of the accumulator that measured it.

WHAT IT FINDS

• The unspun global maximum is 18/28, the 18 scales n = 3, 5 mod 6, attained on two diagonal intervals [1/3, 18/53) and [35/53, 2/3), each of width 1/159, and so on four square cells: S(1 - x) = S(x) exactly for odd scales, since floor(n(1 - x)) = n - 1 - floor(nx) with n - 1 even, so both intervals carry the same scale set and all four products are maxima. The locus is 4 cells of total area 4/25281 = 0.000158222, one cell 1/25281 = 3.95554e-05 (diagonal_maximum).
• The raster finds all four: peak 18/28 at (0.33667, 0.33667) and its three mirrors at every R, plateaus of 4, 9, 49, 169 pixels at R = 256, 512, 1024, 2048, measured locus area 0.000244141, 0.000137329, 0.000186920, 0.000161171 against the exact 0.000158222, and the zoom measures the single cell at the top peak as 3.95523e-05 against the exact 3.95554e-05 (report, main).
• Median drift of the unspun top 20 is 0.71, 0.35, 0.35 pixels over 256 -> 512 -> 1024 -> 2048, under one pixel at every step, with 20/20 and 19/20 inside one pixel on the last two steps (drift).
• Every whole-degree schedule loses the unspun maximum: degrees peaks at 14/28, exactly the centre's value, at all four resolutions; primes peaks at 16/28; neither reaches 18/28, and the zoom at effective R = 51200 finds nothing above the raster peak in any schedule (report).
• The spun peaks sit on far smaller cells. The area of the single cell at the top peak, from the zoom, is 3.95523e-05 unspun, 1.65596e-05 primes, 9.80453e-06 degrees, 2.27585e-06 random, 1.30005e-06 gaussian, 1.0128e-06 golden; the golden peak is one pixel wide even at R = 2048 (report).
• The discriminator is the single cell, never the whole locus: a schedule whose maximum is lower spreads it over more cells, so the total locus area at the peak runs to 0.000415802 under degrees at R = 2048, larger than the unspun 0.000158222 at a value four layers higher (report).
• Off-centre maxima do stay put for whole-degree schedules: the primes peak 16/28 at (0.19469, 0.15501) drifts 0.29 pixels over 1024 -> 2048. A raster cannot read this as an exact coincidence. The stack is piecewise constant on cells of positive area, so any cell wider than a pixel is found at every resolution; plateau stability measures cell area, not lattice coincidence, and the only honest statements a raster supports are resolution-drift statements.
• Top-20 lists reshuffle under refinement wherever the peak value is degenerate: 8/20 to 15/20 of the spun maxima land within one pixel of a maximum at 2R, against 19/20 unspun, and the maximum displacement runs to 470 pixels where a small cell drops out of the top 20 (drift).
• The centre is never the strict global maximum: 18/28 beats it unspun, 16/28 or 17/28 beats it in five schedules, and only degrees leaves it tied at the top with 14/28 (report).
• The fade constant c = rms * sqrt(L) at R = 1024 over the full square runs 0.309907, 0.390837, 0.427756, 0.460395 at L = 4, 8, 14, 28 unspun, against the exact 0.309477, 0.389754, 0.426869, 0.458411 from exact_variance; the tree's c = 0.522 is the L -> infinity limit, not the value at L = 28, and 0.4604 against 0.4584 is the raster's 0.4% discretisation bias (main).
• Spinning does not change the fade law: c at L = 28 reads 0.401417 degrees, 0.427986 golden, 0.425189 primes, 0.423811 random, 0.440940 gaussian, against 0.460395 unspun, all within 13% and all falling as 1/sqrt(L) (main).
• The mean of P over the disc is 0.7682 in every schedule at every resolution, 0.7689 unspun at R = 2048; rotation moves no ink (report).
• float32 against float64 at R = 512 on the golden schedule: 8 pixels of 262144 differ, by one layer (main).

FIGURES

• stack-unspun.png, stack-degrees.png, stack-primes.png, stack-gaussian.png: the R = 1024 render of each schedule, block-averaged to 512 and quantised to the 29 stack levels, 45820, 61142, 63814, 55405 bytes; stack-sheet.png is the four at 256, 69865 bytes (save_png, contact_sheet).
• Unspun is a lattice of straight moire rays through the diagonals; the whole-degree fan pleats them into curved bands; the prime and golden schedules wind them into a vortex of arms about the centre, with the centre node visible as a single dark speck.

WITNESSES

• farey.md THE STACK IS AN ADDRESS: the unspun render this study spins is the same field lab/py/carpet-stack-address renders at 512 x 512, and its global maximum 18/28, on four cells of side 1/159 against the two intervals [1/3, 18/53) and [35/53, 2/3), is the closed form read at a denominator-3 node and its mirror.
• The stack's L^2 fade: c = 0.522 is the limit of sqrt(L Var(G_L)), whose exact values 0.309477 .. 0.458411 at L = 4 .. 28 this study reproduces from the raster to 0.4%.