Arc loops

• 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".
• 2026-10-03 [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".
• 2026-10-03 [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.
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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.
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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".
• 2026-10-03 [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.
• 2026-10-03 [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".
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.
• 2026-10-03 [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.