Conduction at large side

• 2026-10-03 [Proved] For bang dim 2, code 7 at odd side N with insulated voids, lim_(N -> infinity) sigma(N, L) = 3^(-L/2) at every level L, so lim_N rho(N, L) = sqrt(3) and at infinite side level L conducts its solid fraction (3/4)^L to the power log 3/log(16/9) = 1.9094; the proof uses G-convergence as stated in Allaire 2010 and the div-curl lemma of Murat 1978. Witness: walks.md, section "Conduction at large side".
• 2026-10-03 [Proved] The infinite-side level-1 cell of bang dim 2, code 7 reduces by reflections to the unit square minus its top right quarter, and abs(sigma(N, 1) - 1/sqrt(3)) <= C/N. Witness: walks.md, section "Conduction at large side".
• 2026-10-03 [Proved] In the continuum, for bang dim 2, code 7 at every odd side and level, the conductance with insulated voids times the conductance with floating perfectly conducting voids is exactly 1, and sigma(N, L; z) sigma(N, L; 1/z) = 1 for voids of conductivity z. Witness: walks.md, section "Conduction at large side".
• 2026-10-03 [Proved] For bang dim 2, code 7, (3N^2 + 1)/(2N(N + 1)) <= sigma(N, L)/sigma(N, L + 1) <= 2N/(N + 1) at every level, so d_w(N) = log(fill rho(N))/log N > 2 at every odd N, and (d_w(N) - 2) log N has every limit point in [log(9/8), log(3/2)]. Witness: walks.md, section "Conduction at large side".
• 2026-10-03 [Verified] The square array of square inclusions of area fraction 1/4 has effective conductivity sqrt((1 + 3z)/(3 + z)), 1/sqrt(3) at z -> 0; the network on the L gives 0.5773502680 with bar 6.4e-9, meets the formula at six contrasts within 6e-8, and gives Keller products within 7e-8 of 1. Witness: Moulinec, Suquet and Milton 2018, eq. (58), crediting Obnosov 1999; lab/py/conduction-at-large-side, verb cell.
• 2026-10-03 [Verified] Network duality products 1.0000152 with bar 5.0e-5 at N = 5, L = 1 and 1.0000145 with bar 2.5e-5 at N = 3, L = 2; N (sigma_k(N, 1) - sigma_k(infinity, 1)) = 0.65535, 0.65895, 0.66063, 0.66144 at N = 11, 21, 41, 81, k = 16; rho(N, 1) = 1.50940, 1.54066, 1.58189, 1.61788, 1.64991, 1.66765 at N = 9, 11, 15, 21, 31, 41, k = 2. Witness: lab/py/conduction-at-large-side, verbs dual, side, rate.
• 2026-10-03 [Verified] Resistance scale of bang dim 2, code 7, deepest per-level ratio: 1.25149 +- 1e-5, 1.38642 +- 2e-6, 1.46028 +- 6e-5 at N = 3, 5, 7 (k = 1) and 1.50712 +- 3e-3, 1.53971 +- 1e-3 at N = 9, 11 (k = 2), giving d_w = 2.0970, 2.0947, 2.0903, 2.0865, 2.0835; side 3 meets the reported rho ~ 1.251 and the spectral 2.097 of code 495. Witness: lab/py/conduction-at-large-side, verb scale.
• 2026-10-03 [Conjecture] sqrt(3) - sigma(N, 1)/sigma(N, 2) is of order 1/N, with coefficient between 3.1 and 4.1 for the fitted models; N (sqrt(3) - rho(N, 1)) still rises at N = 41, where it reads 2.6404, and a free fit r - a/N + b/N^(4/3) returns r = 1.7316. Witness: lab/py/conduction-at-large-side, verb rate.
• 2026-10-03 [Conjecture] For bang dim 2, code 7 the side and level limits of the resistance scale commute, lim_(N -> infinity) rho(N) = sqrt(3); equivalently d_w(N) = 2 + log(3 sqrt(3)/4)/log N + o(1/log N), and the level-first Archie exponent tends to 1.9094. Witness: lab/py/conduction-at-large-side, verb scale, a drift from rho(N, 1) falling from 2.96e-2 to 4.99e-3 over N = 3..11 at k = 1.
• 2026-10-03 [Verified] The infinite-side cell of bang dim 3, code 23, the unit cube minus the four octants with at least two upper coordinates, conducts 0.31588 with bar 1e-5. Witness: lab/py/conduction-at-large-side, verb dim3.
• 2026-10-03 [Conjecture] Side first, level L of bang dim 3, code 23 conducts 0.31588^L at solid fraction (1/2)^L, an Archie exponent 1.6626. Witness: lab/py/conduction-at-large-side, verb dim3, cell only.