Shortest paths

• 2026-10-02 [Proved] For bang dim 2, code 7 at odd side N and level L the closures of all voids are pairwise disjoint, so paths through corner contacts never arise, the closed set and its interior give the same shortest-path distance, and the visibility graph on void corners computes it exactly. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] The corner distance of bang dim 2, code 7 at odd side N and level L satisfies sqrt(2) <= D(N, L) <= 2 - (2 - sqrt(2)) (1 - 1/N)^L, so lim_(N -> infinity) D(N, L) = sqrt(2) at every level and limsup_(N -> infinity) D(N, cN) <= 2 - (2 - sqrt(2)) e^(-c). Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] For odd N >= 5 the limit set of bang dim 2, code 7 at side N holds no segment of non-axis slope, every rectifiable path in it is at least as long as the taxicab distance between its ends, and D(N, L) is nondecreasing in L with limit 2, so the side and level limits of the corner distance do not commute; no rate is proved. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] At side 3, D(3, L) = 2 sqrt(5)/3 at every level L >= 1 and in the limit, attained by (0,0) -> (2/3, 1/3) -> (1,1) of slopes 1/2 and 2. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] At level 1, D(N, 1) = sqrt(2) + (2 sqrt(5) - 3 sqrt(2))/N for every odd N. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] At infinite side the level-1 street grid of bang dim 2, code 7 has exactly the axes and the diagonals as free directions, and a line of slope s in (0, 1) that rises 3 rows passes a point whose disc of radius (1 - s)/(2 (1 + s)) lies inside a void, a sharp radius. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] The map Phi, the stable norm of the street grid carrying a norm, is monotone, satisfies Phi(nu) >= nu and fixes the free directions, and Phi^L(euclid) increases to the regular octagon gauge oct uniformly; every fixed point has as ball the octagon of its own free unit vectors, so oct is the only one equal to 1 on them. Witness: walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Proved] For bang dim 3, code 23 with paths in the closed filled cells, the map's free directions are the 6 axes and the 12 face diagonals, the body diagonals are blocked, and Phi^L(euclid) increases to the gauge of the hull of the 18 free unit vectors, which has 18 vertices, 48 edges and 32 triangular faces. Witness: walks.md, section "Shortest paths: two limits that do not commute"; lab/py/carpet-geodesics, verb hull, recounts.
• 2026-10-02 [Verified] Exact corner distances of bang dim 2, code 7: D(3, L) = 1.490711985000 at L = 1..4, D(5, L) = 1.460112615949, 1.485180310941, 1.504787873051 at L = 1, 2, 3, D(7, L) = 1.446998600642, 1.464884551783 at L = 1, 2, D(9, 2) = 1.458269875850, D(11, 2) = 1.449562910791. Witness: lab/py/carpet-geodesics, verb corner.
• 2026-10-02 [Verified] D(N, 1) = sqrt(2) + (2 sqrt(5) - 3 sqrt(2))/N at every odd N from 3 to 41, to 4.4e-16. Witness: lab/py/carpet-geodesics, verb level1.
• 2026-10-02 [Verified] Upper bounds from cycle points of the true ball, window 6 periods: Phi^L(euclid) at angle 22.5 degrees is at most 1.029173, 1.050508, 1.064891, 1.073404 at L = 1..4 against oct = 1.082392, so its gap to the octagon is at least 5.21e-2, 3.26e-2, 1.85e-2, 9.71e-3. Witness: lab/py/carpet-geodesics, verb map.
• 2026-10-02 [Verified] The exact level-1 distance from (0,0) to (1, (N-1)/(2N)) reads 1.122716, 1.135394, 1.136527, 1.139652 at N = 11, 21, 31, 41 against 1.128108, 1.135734, 1.138440, 1.139826 from the level-1 norm, within 0.06/N. Witness: lab/py/carpet-geodesics, verb bridge.
• 2026-10-02 [Verified] At level 2, N (D(N, 2) - sqrt(2)) reads 0.354834, 0.354697, 0.396507, 0.388843 at N = 5, 7, 9, 11 against 0.458991, twice the level-1 constant, and N (D(5, 3) - sqrt(2)) = 0.452872 against 0.688486. Witness: lab/py/carpet-geodesics, verb corner with 5,3 11,2.
• 2026-10-02 [Refuted] An excess of the corner distance over sqrt(2) of at least a L log N / N for some a > 0. Witness: the proved envelope D(N, L) - sqrt(2) <= (2 - sqrt(2)) L/N, walks.md, section "Shortest paths: two limits that do not commute".
• 2026-10-02 [Conjecture] Side first then level first, the render's distance tends to the octagon gauge, lim_L lim_N d_(N,L)(x, y) = oct(x - y); it follows from the map theorem once lim_N d_(N,L) = Phi^L(euclid) is proved; likewise the render of bang dim 3, code 23 tends to the gauge of the hull of the 18 free unit vectors. Witness: lab/py/carpet-geodesics, verb bridge, at dim 2 level 1 only; dim 3 untested.