node-stack

• Stacks the nodes of a design instead of its ink: the EDGE set of a layer, the cell sides that separate an inked cell from a paper cell or from the outside, and the CORNER set, the vertices of its inked cells.
• A layer at scale n is the design laid 1-periodically at period 1/n; the parity carpet's cell grid at odd n is n x n, the base-3 carpet's at level is 3^level n on a side. The edge stack over n = 1..N adds the layers' edge indicators as a measure on segments, the corner stack adds their corner indicators as a point measure, and a segment or point is lit by the number of layers containing it.
• Parity carpet, cell (i, j) inked iff i and j are both odd: the edge set is the interior grid lines restricted to the odd rows and columns, {k/n} x [j/n, (j+1)/n] for 1 <= k <= n-1 and j odd, and the border of the square carries no edge at odd n; the corner set is the full interior vertex set {(k/n, l/n) : 1 <= k, l <= n-1} (parity_edges_form, parity_corners_form).
• Base-3 carpet, cell paper iff some digit pair of (i, j) is (1, 1): a vertical grid line of layer n carries an edge iff its index is not a multiple of 3^level, so the period boundaries k/n are never edge lines and the removed blocks' walls are the whole edge set (edge_residues, carpet_line_lit).
• Brightness has a closed form in both designs, at line level and at segment level, and the lit line sets are exact Farey families (parity_line_brightness, carpet_line_brightness, carpet_lines_reach).

RUN

• uv run python research/lab/py/node-stack/node_stack.py
• From the repo root. One core, under two seconds.
• Domain is the parity carpet at N = 15 literal and N = 31 rendered, the base-3 carpet at N = 12, level 1 and N = 6, level 2 literal and N = 12, level 2 rendered, the restricted Farey sizes at level 1 and N = 1..30, and the Landau discrepancy at Q = 30, 60, 90.
• Every count is exact: cells and sides are integer pairs, points and segment ends are Fraction. The five PNGs beside this file are written; nothing else.

WHAT IT PRINTS

• check_parity_edges, check_parity_corners: the literal edge and corner sets of every odd layer against the closed forms, as a mismatch count.
• parity_corner_stack against parity_corner_predicted: the literal corner stack at N = 15 against the Farey closed form, with the count of points missed, invented and mis-valued, and both sides' sha256 digest of the sorted x,y:brightness list.
• check_line_brightness_parity, check_line_brightness_carpet: the number of lit lines and the number of form breaches, counting both wrong values on lit lines and nonzero predictions on unlit ones.
• parity_segment_check, carpet_segment_check: the segment forms tested at the midpoint of every elementary interval of the union of all layers' grids, on every lit line, with the breach count.
• edge_residues: the residue set J_level and, per residue, the rows in which the edge exists.
• carpet_lines_reach against literal union: the lit-line set at (N, level) from the reach form against the union of the layers' lines.
• top_segments: the top segment brightnesses with their exact x and [y0, y1].
• carpet_corner_stack against carpet_corner_form: lit corner count, breach count, digest and the top brightnesses at N = 12, level 1, beside the plain Farey pair count at Q = 36.
• farey_sizes, landau: the size of the restricted family beside the plain Farey size at Q = 3N, N = 1..30, and the Landau sum sum_i |f_i - i/m| of each at three Q, summed exactly as a Fraction and printed to six places.

WHAT IT FINDS

• The parity edge law is exact: the literal edge set equals the interior grid lines restricted to odd rows and columns at every odd n <= 15, 0 mismatches (check_parity_edges). A cell is inked iff both coordinates are odd, so crossing x = k/n inside an even row leaves paper on both sides, while inside an odd row it flips ink to paper for every interior k.
• The parity corner set is the full interior vertex grid, 0 mismatches at every odd n <= 15 (check_parity_corners): the two columns k-1, k contain exactly one odd index for every 1 <= k <= n-1, and none for k = 0 or k = n.
• The parity corner stack is the Farey field with an odd restriction. A point (a/b, c/d) is lit iff b and d are odd and lcm(b, d) <= N, with brightness (floor(N/lcm(b, d)) + 1)/2 rounded down, the count of odd multiples of lcm(b, d) up to N. At N = 15 the literal stack has 536 lit points and the closed form has 536, none missed, none invented, no value mismatch, both on digest 3dfbf194dcfc7dc08084acce38cfbf65b2ac63ef5a3de8f439411faeebcea9b6 (parity_corner_stack, parity_corner_predicted). The top brightness is 3, at the four points (1/3, 1/3), (1/3, 2/3), (2/3, 1/3), (2/3, 2/3).
• The parity edge stack factors: a whole line x = a/b is lit by (floor(N/b) + 1)/2 rounded down layers for odd b and by none for even b, 48 lit lines and 0 breaches at N = 15, and the brightness at a point (a/b, y) of that line is the count of odd n <= N with b | n and floor(n y) odd, 2352 midpoint tests and 0 breaches (check_line_brightness_parity, parity_segment_check). The edge stack is the line stack in one coordinate times the one-dimensional parity stack in the other.
• The base-3 edge residues are every nonzero residue: J_1 = {1, 2} with the edge in rows r = 1 mod 3, and J_2 = {1, ..., 8} with rows {1,4,7} at j = 1,2,7,8, {3,4,5} at j = 3,6 and {1,7} at j = 4,5 (edge_residues). The residue 0 is missing because the cells either side of a period boundary have all-2 and all-0 digits, which never collide, so x = k/n is never an edge line.
• The base-3 line criterion is exact: a reduced a/b is lit by layer n iff b | 3^level n and a (3^level n / b) mod 3^level lies in J_level, with 0 layer mismatches at N = 12, level 1 and at N = 6, level 2 (check_carpet_lines).
• The criterion reduces to a denominator condition, since J_level misses only 0: writing s for the exponent of 3 in b, the line is lit by some layer n <= N iff s >= 1 and b / 3^min(s, level) <= N, and its brightness is floor(N / (b / 3^min(s, level))) - floor(N/b), a telescoping count of the layers whose 3-adic valuation lands in [max(0, s-level), s-1]. Lit-line sets agree exactly, 106 lines at N = 12, level 1 and 100 at N = 6, level 2, with 0 brightness breaches on either (carpet_lines_reach, check_line_brightness_carpet).
• Segment brightness is smaller than line brightness and the study prints it exactly: at N = 12, level 1 the top is 4, on {1/3} x [1/3, 11/30] and three mirror segments, against line brightness 8 at x = 1/3; at N = 6, level 2 the top is 3, on {1/9} x [7/36, 2/9] and seven others. The segment form, edge at (a/b, y) iff the column residue lies in J_level and floor(3^level n y) mod 3^level lies in that residue's row set, passes 14628 midpoint tests at N = 12, level 1 and 10800 at N = 6, level 2 with 0 breaches (top_segments, carpet_segment_check).
• The base-3 corner set at level 1 is every vertex of the 3n grid, because the two columns k-1, k contain at most one index congruent to 1 mod 3, so at most one of a vertex's four cells is paper. The corner stack's brightness at (a/b, c/d) is therefore floor(N / (m / gcd(m, 3))) with m = lcm(b, d): at N = 12 the stack has 5029 lit corners with 0 breaches, digest 66676fdc633f950b85a70783f194aa98278dd7817ff7a5438063c2b4627e0ae5, top brightness 12 at the denominator-3 points (carpet_corner_stack, carpet_corner_form). Against it, the plain Farey pair count at Q = 3N = 36 is |F_36|^2 = 157609: the corner stack lights 5029 of those 157609 pairs, the rest being pairs whose two denominators are unreachable together.
• At level >= 2 the corner set is a proper subset of the vertex grid: of the 100 vertices of the level-2 tile, 96 are corners and the four points (4/9, 4/9), (4/9, 5/9), (5/9, 4/9), (5/9, 5/9) are not, each having all four of its cells carrying the digit pair (1, 1) at the first level, so they sit interior to the removed block (carpet_corners).
• The lit-line family is a denominator-restricted Farey family, not a numerator-restricted one. At level 1 it is exactly {a/b in F_Q : 3 | b} at Q = 3N; at general level it is {a/b : 3 | b, b / 3^min(v_3(b), level) <= N}, which at level 2 is the multiples of 9 up to Q together with the denominators of valuation 1 up to Q/3. Sizes in (0, 1) at level 1, N = 1..30, against |F_Q| in (0, 1]: 2, 4, 10, 14, 22, 28, 40, 48, 66, 74, 94, 106, 130, 142, 166, 182, 214, 232, 268, 284, 320, 340, 384, 408, 448, 472, 526, 550, 606, 630 against 4, 12, 28, 46, 72, 102, 140, 180, 230, 278, 344, 396, 474, 542, 628, 712, 806, 900, 1000, 1102, 1228, 1328, 1470, 1588, 1736, 1856, 2020, 2166, 2328, 2480 (farey_sizes).
• Its Landau sum sum_i |f_i - i/m| runs 0.784027, 1.557547, 1.962049 at Q = 30, 60, 90, against 1.298847, 1.710479, 2.153675 for the plain Farey sequence at the same Q (landau). Three points, no fit and no exponent is claimed.
• The tree's Farey door restricts the denominator by a cutoff, b <= Q; this family restricts it by a divisibility, 3 | b, and a level-dependent cutoff. They are different objects and neither is a numerator restriction.

FIGURES

• edges-parity.png: the parity edge stack over the odd scales n <= 31, accumulated as hairlines on a 1024 raster and block-maximum reduced to 512, peak raster count 10, 25733 bytes (edge_raster, save_png).
• corners-parity.png: the parity corner stack at N = 31, each lit point a square of half-width 1 + brightness pixels at 1024, peak brightness 5, 8085 bytes (corner_raster). The white cross on the centre lines is the odd restriction: every point with an even denominator is dark.
• edges-carpet.png, corners-carpet.png: the base-3 carpet at level 2 over scales 1..12, peaks 24 and 12, 45760 and 22936 bytes. The corner picture is a 9 x 9 lattice of Farey fields, one per level-2 cell.
• nodes-sheet.png: the four at 256, 63607 bytes (contact_sheet).
• Grey is monotone in brightness, the faintest lit value at grey 70 and the peak at black, so a picture shows which nodes are bright but reads no exact value. A raster cannot show a coincidence: every count on this page is exact rational arithmetic and the pictures are illustrations of it.

WITNESSES

• farey.md WHERE THE LINES LAND: the line stack's floor(N/b) is the shadow of these two laws. The parity corner stack replaces it by (floor(N/lcm(b,d)) + 1)/2 rounded down on pairs of odd denominators; the base-3 edge stack replaces it by a difference of two floors.
• farey.md THE STACK IS AN ADDRESS: brightness stays an address at the nodes as it is on the ink. Line brightness costs one gcd, segment brightness one pass over the layers dividing the denominator.
• core.md: the base-3 design here is the carpet, cell paper iff a digit pair equals (1, 1), at levels 1 and 2.