Claims · 32 dated claims on Arc loops, each with its tag and its witness; newest 2026-10-03.
Proved Every curve of a dimension-two design level drawn in quarter-circle arcs, filled cells around the lower-left and upper-right corners and deleted cells around the other two, is a path or a cycle, and the level has exactly 2 side strands. Witness: arcs.md, section "Strands".
Proved The loop count of a design level in arcs is L = c(G) - 2 side - 1, where c(G) is the number of connected components, isolated lattice points included, of the mirror graph G that puts in each cell the diagonal its arcs do not cross; so L is the cycle rank of G, and it equals the number of components of that graph holding no boundary lattice point. Witness: arcs.md, section "Loops are cycles of the mirror graph".
Verified Union-find over edge midpoints, the mirror-graph cycle rank and the block recursion give the same loop count, and 2 side strands, on every code at base 2 to level 7 and at base 3 to level 4, 2688 levels. Witness: lab/rs/arc-loops, verb check.
Proved The half turn, the transpose and the anti-transpose of the mask fix the arc loop count at every level, while the quarter turn need not: base 2 code 9 has 2^n - 1 loops and its quarter turn code 6 has none. Witness: arcs.md, section "Loops are cycles of the mirror graph".
Proved The all-deleted block of side N drawn in arcs has strands B_t - R_(N-1-t) and L_t - T_(N-1-t) and no loop. Witness: arcs.md, section "The block recursion".
Proved The arc loop count satisfies L(n + 1) = k L(n) + J(n) with k the filled cells of the mask and J(n) the cycles of the glued strand matchings of the base^2 blocks. Witness: arcs.md, section "The block recursion".
Proved The carpet, bang dim 2, base 3, code 495, which is bang dim 2, code 7 at side number 3, has L(n) = (8^n - 1)/7 - 3^n + n + 1 arc loops at level n, and its gluing adds J(n) = 5 * 3^n - 7n - 5. Witness: arcs.md, section "The carpet law".
Proved The strand matching of carpet level n is the lower-left and upper-right corner families on A_n = {(3^j - 1)/2 : j <= n}, the lower-right and upper-left families on A_n minus its middle port, and on each side the same turn set P_n of (3^n - 2n - 1)/2 pairs, where P_0 is empty and P_(n+1) is P_n, 3^n + P_n, 2 * 3^n + P_n and the pairs (j 3^n - 1 - a, j 3^n + a) for j = 1, 2 and a = (3^i - 1)/2, i < n. Witness: arcs.md, section "The carpet law".
Verified The carpet arc loop law holds at levels 0 to 15, by union-find and cycle rank to level 7 and by the block recursion to level 15, and the carpet matching lemma equals the glued matching at levels 0 to 10. Witness: lab/rs/arc-loops, verb carpet.
Proved At base 2 codes 7 and 14 have 3^(n-1) - 2^n + 1 arc loops at level n >= 1, gaining 2^n - 2 at each gluing n >= 1 and none at gluing 0. Witness: arcs.md, section "Base 2, complete".
Proved At base 2 codes 11 and 13 have 3^(n-1) - 2^(n-1) arc loops at level n >= 1, gaining 2^(n-1) at each gluing n >= 1 and none at gluing 0. Witness: arcs.md, section "Base 2, complete".
Proved At base 2 code 9 has 2^n - 1 arc loops at level n, gaining one loop per gluing. Witness: arcs.md, section "Base 2, complete".
Proved The other 11 codes at base 2, namely 0, 1, 2, 3, 4, 5, 6, 8, 10, 12 and 15, have no arc loop at any level. Witness: arcs.md, section "Base 2, complete".
Verified The base 2 matching lemmas of codes 7, 11 and 9 equal the glued matchings at levels 1 to 14 and their gains hold at 13 gluings, and the base 2 census agrees to level 24. Witness: lab/rs/arc-loops, verbs two and census.
Proved When the mask keeps k > base cells, L(n) / k^n converges to sum_m J(m) / k^(m+1), positive unless L vanishes, and the carpet limit is 1/7. Witness: arcs.md, section "The census at base 3".
Verified At base 3 the 512 codes fall in 168 classes under the half turn and the two diagonal reflections; 48 classes and 149 codes have no arc loop to level 14 and the other 120 classes give 74 distinct nonzero loop sequences. Witness: lab/rs/arc-loops, verb census.
Conjecture 61 of the 74 base 3 arc loop sequences satisfy a linear recurrence with integer roots only, of order at most 6, whose largest root is the number of filled cells. Witness: lab/rs/arc-loops, verb census.
Conjecture 8 base 3 arc loop sequences, 66 codes, have the factor x^2 - 3x + 1 with roots phi^2 and phi^-2 in their minimal recurrence. Witness: lab/rs/arc-loops, verb census.
Verified Base 3 code 13 gains the odd-indexed Fibonacci numbers J(n) = F_(2n-3) at the gluings n = 1 to 13. Witness: lab/rs/arc-loops, verb gains.
Conjecture Base 3 code 13 gains J(n) = F_(2n-3) at every gluing n >= 1, so its arc loop sequence is A104487 shifted by two. Witness: lab/rs/arc-loops, verb gains.
Conjecture Base 3 code 287 has an arc loop recurrence with the factor x^2 + x + 1, fitted to level 17 with two terms of margin. Witness: lab/rs/arc-loops, verb census.
Verified The arc loop sequences of base 3 codes 43, 171, 175 and 181 fit no linear recurrence of order at most 8 with two terms of margin through level 17. Witness: lab/rs/arc-loops, verb census.
Verified Counting block strands with void-block strands included, base 3 code 43 closes a loop of 6032 strands at the gluing from level 11 to 12, while the carpet's longest new loop over gluings 0 to 11 has 8 strands. Witness: lab/rs/arc-loops, verb lengths.
Conjecture The parity design bang dim 2, code 14 at side number 3 gains 2 * 3^(n-1) at each gluing n >= 1 and 1 at gluing 0, and has 2 * 5^(n-1) - 3^(n-1) arc loops at level n >= 1. Witness: lab/rs/arc-loops, verb gains.
Conjecture The parity design bang dim 2, code 9 at side number 3 gains 2 * 3^n per gluing and has 5^n - 3^n arc loops. Witness: lab/rs/arc-loops, verb gains.
Conjecture The parity design bang dim 2, code 6 at side number 3 gains 2^(n+1) per gluing and has 4^n - 2^n arc loops. Witness: lab/rs/arc-loops, verb gains.
Conjecture The parity designs bang dim 2, code 11 and code 13 at side number 3 gain 2^(n+1) - 2 per gluing and have (7^n - 6 * 2^n + 5)/15 arc loops. Witness: lab/rs/arc-loops, verb gains.
Conjecture The carpet's parity code 7 has arc loop recurrences with roots 12, 4, 1 at side number 4 and 21, 5, 1, 1 at side number 5. Witness: lab/rs/arc-loops, verb bases.
Conjecture Every one-cell deletion at base 4 has arc loop roots 15, 4, 2, 1 except code 57343 with 15, 4, 3, 2, 1, and at base 5 the corners, the centre and the inner ring have roots 24, 5, 3, 1. Witness: lab/rs/arc-loops, verb bases.
Verified The base 5 one-cell deletions 25165823, 29360127 and 31457279 fit no arc loop recurrence of order at most 4 with integer coefficients and two terms of margin on levels 0 to 10, and no fit at bases 4 and 5 among the parity codes and one-cell deletions has a root outside the integers. Witness: lab/rs/arc-loops, verb bases.
Verified The carpet arc loop sequence 0, 3, 50, 509, 4444 and 53 other base 3 arc loop sequences are absent from the OEIS by their first seven terms from the first nonzero one, while 20 of the 74 match an entry. Witness: lab/rs/arc-loops, verb census.
Verified The parity codes and one-cell deletions drawn in arcs give 10 distinct nonzero loop sequences at base 4 to level 12 and 15 at base 5 to level 10, of which 3 and 3 match an OEIS entry by their first seven terms from the first nonzero one. Witness: lab/rs/arc-loops, verb bases.