The node stack

• 2026-09-09 [Proved] The corner stack of the odd parity carpet is the Farey field, odd-restricted: with the edge set of a layer the cell sides separating ink from paper and the corner set the vertices of inked cells, layer n's edges are the interior grid lines restricted to the odd rows and columns and its corners the full interior vertex grid, so the corner stack lights exactly the pairs (a/b, c/d) with b and d odd and lcm(b,d) <= N, at brightness the number of odd multiples of lcm(b,d) up to N, and the edge stack is the line stack in one coordinate times the 1D parity stack in the other; at N = 15 literal stacking gives 536 lit points against 536 predicted, none missed and none invented, and 2352 segment tests with no breach. Witness: lab/py/node-stack parity_corner_predicted, parity_segment_check.
• 2026-09-09 [Proved] The base-3 carpet's edge stack is a denominator-restricted Farey family: a period boundary k/n is never an edge line, every other residue mod 3^level is (J_1 = {1, 2}, J_2 = {1, ..., 8}), so the lit lines are the reduced a/b with 3 | b and b/3^min(v_3(b), level) <= N, at brightness floor(N/(b/3^min(v_3(b), level))) - floor(N/b); at level = 1 the lit set is exactly the multiples of 3 in F_3N, 106 lines at N = 12 and 100 at N = 6, level = 2, with no breach on 14628 and 10800 segment midpoints; the level-1 corner stack at N = 12 lights 5029 corners at brightness floor(N/(m/gcd(m, 3))) against the plain Farey pair bound 157609. Witness: lab/py/node-stack carpet_lines_reach, check_line_brightness_carpet, carpet_corner_stack.
• 2026-09-09 [Refuted] The Sierpinski carpet's edge family is a numerator-restricted Farey sequence: the edge condition a (3^level n/b) mod 3^level in J_level is vacuous on the numerator because J_level is every nonzero residue, so the family is restricted by a divisibility of the denominator and must not be conflated with the cutoff-restricted Farey sequence of the Farey page; sibling designs removing another digit vector have proper residue sets and are the open door. Witness: lab/py/node-stack edge_residues.