# 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.