Use the verified branch model
O_sing = Z_3 × R_loc, R_loc = {(a,b) in Z_3 × Z_3 : a ≡ b mod 3}.The arithmetic module is the regular module L_sing ≅ O_sing, so the module filtration is the ring filtration.
For the finite quotients modulo 3^k, write
O_k = (Z/3^k Z) × R_k, R_k = {(a,b) : a ≡ b mod 3}.Then
|O_k| = 3^(3k-1),
|3 O_k| = 3^(3k-4),
|J_k| = 3^(3k-3),
|J_k^r| = 3^(3k-3r) (1 <= r <= k),where J_k = 3(Z/3^k Z) × 3(Z/3^k Z × Z/3^k Z) is the Jacobson radical.
| modulus | dim(O_k / 3 O_k) | dim(O_k / J_k) | dim(J_k / 3 O_k) | dims of J_k^r / J_k^{r+1} | dims of 3^r O_k / 3^{r+1} O_k |
|---|---|---|---|---|---|
| 27 | 3 | 2 | 1 | [3, 3] | [3, 3] |
| 81 | 3 | 2 | 1 | [3, 3, 3] | [3, 3, 3] |
| 243 | 3 | 2 | 1 | [3, 3, 3, 3] | [3, 3, 3, 3] |
| 729 | 3 | 2 | 1 | [3, 3, 3, 3, 3] | [3, 3, 3, 3, 3] |
Three distinct structures coexist:
O_k / J_k has dimension 2: the semisimple two-branch skeleton.J_k / 3 O_k has dimension 1: the nilpotent shadow that collapses the two branches modulo 3.O_k / 3 O_k therefore has dimension 3: the intrinsic three-state residue core.Two distinct filtrations must be separated:
O_k / J_k of dimension 2, followed by repeated 3-dimensional layers J_k^r / J_k^{r+1};3-adic filtration has repeated 3-dimensional layers 3^r O_k / 3^{r+1} O_k from the start;1 is not a terminal Jacobson layer at all: it is the mixed quotient J_k / 3 O_k.So the correct arithmetic picture is:
2-state skeleton,1 nilpotent shadow,3-state residue machine,J-adic and 3-adic refinements repeating at width 3 rather than terminating after three layers.