# 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`.