conduction-at-large-side

• Conduction on bang dim 2, code 7 at odd side N and level L: the side-N tile voids a cell iff both digits are odd, the render is its L-th Kronecker power in the unit square, unit conductivity on the filled cells, potential 0 on the left edge and 1 on the right, every other boundary insulated.
• The network: one node per cell of the render refined k times per axis, conductance 2ab/(a+b) between face neighbours of conductivities a, b, conductance 2a to an electrode, so a solid square reads exactly 1; at k = 1 it is the graph of the walk census. A symmetric render is solved on one quarter: the potential is odd about x = 1/2 and even about y = 1/2. Sparse LU (SuperLU, COLAMD) in dim 2, Jacobi-preconditioned CG to 1e-11 in dim 3.
• cell computes the level-1 conductivity at infinite side on the unit square minus its top right quarter, between the left edge and the lower half of the right edge, at mesh 2M per side, M = 64..512; Richardson with order 4/3; then inclusion conductivities z = 0.01, 1/9, 1/3, 3, 9, 100 by three-mesh Aitken against sqrt((1 + 3z)/(3 + z)), the products sigma(z) sigma(1/z), and z = 1e8 against sqrt(3). It asserts the insulating value within twice its bar of 1/sqrt(3), every contrast within 6e-8 of the formula and every Keller product within 7e-8 of 1.
• dual multiplies the conductance with insulating voids by the one with voids of conductivity 1e8 at (N, L) = (5, 1), (3, 2): Aitken on k = 16, 32, 64, the bar its shift from k = 8, 16, 32, and it asserts the product within its bar of 1.
• side prints N (sigma_k(N, 1) - sigma_k(infinity, 1)) at k = 4, 8, 16 for N = 11, 21, 41, 81, the infinite-side value at the matched mesh coming from cell, and sigma(N, 1)/sigma(N, 2) at k = 1, 2, 4 for N = 5..41.
• rate prints sigma(N, 1)/sigma(N, 2) at k = 2, N = 9..41, and N (sqrt(3) - rho(N, 1)), and fits it by r - a/N + b/N^(4/3) with r free and by sqrt(3) - a/N + b/N^q at q = 5/4, 4/3, 3/2.
• scale prints the per-level ratio rho(N, L) = sigma(N, L)/sigma(N, L + 1) at k = 1, 2 for N = 3, 5, 7, 9, 11 and every level whose render has side at most 3200 at k = 1 and 2700 at k = 2. It reads rho(N) as the deepest value at the mesh whose last level step is smaller, k = 1 at N = 3, 5, 7 and k = 2 at N = 9, 11, with bar the larger of that step and the mesh gap at the deepest common level; it also prints the cut and short bounds [beta_N, alpha_N], d_w = log(fill rho)/log N, the law 2 + log(3 sqrt(3)/4)/log N, (d_w - 2) log N with the ends of its trap, log(rho)/log(N^2/fill) and its gap to log 3/log(16/9), the drift from the first ratio to the deepest at k = 1, 2, and the table rows of the note. It asserts every ratio inside the bounds, which it prints as exact fractions.
• dim3 computes the infinite-side cell of bang dim 3, code 23, the unit cube minus the four octants with at least two upper coordinates, at M = 8, 16, 32, 64, Aitken on two triples and Richardson on the last pair, and prints log(1/sigma)/log 2.

RUN

• uv run python mrlyprod/research/lab/py/conduction-at-large-side/conduction.py runs every verb in about 32 s: cell 13 s, dual 0.5 s, side 3 s, rate 5 s, scale 6 s, dim3 5 s.
• One verb by name: uv run python mrlyprod/research/lab/py/conduction-at-large-side/conduction.py scale.
• Needs numpy and scipy; prints only, writes nothing.

WITNESSES

• walks.md, section "Conduction at large side": the cell value and its bar, the orders, the contrast and Keller readings, the duality products and bars, the level-1 boundary readings, the ratios sigma(N, 1)/sigma(N, 2) and the fits, the table, the drifts, the trap ends, the Archie readings and -0.1412, and the dim 3 value and exponent.