The Dual Membrane and Korselt Colour-Blindness in Eisenstein Carmichael Ideals

This is the closing part (Part XXXVIII) of the series Arithmetic Geodynamics on the 6N Skeleton. It lifts the one-dimensional 6N construction -- the integers with the primes 2 and 3 stripped, every surviving prime on a wing 6N+-1 -- to its two-dimensional source, the Eisenstein integers Z[omega] (omega = exp(2*pi*i/3)) with their exact six-fold symmetry, and studies Carmichael ideals there under the generalised Korselt criterion N(p)-1 | N(I)-1. Two results of opposite character are obtained. First, a DUAL MEMBRANE THEOREM (a rigid exterior boundary): no Eisenstein Carmichael ideal is divisible by the prime above 3 (the ramified (1-omega), norm 3) or by the prime above 2 (the inert 2, norm 4). The two exclusions rest on independent valuations -- a 3-adic congruence and a 2-adic parity -- so that the stripping of 2 and 3 that DEFINES the 6N skeleton in one dimension is, in two dimensions, a THEOREM: the Carmichael structure expels the primes above 2 and 3 on its own. Second, KORSELT COLOUR-BLINDNESS (a free interior, a measured negative result): above the membrane the surviving factors are of two unit-invariant types -- split (norm p = 1 mod 3) and higher inert (norm q^2, q = 2 mod 3, q >= 5). The type-composition spectrum of Eisenstein Carmichael ideals is statistically indistinguishable from the null model induced by the factor pool alone (pure-split fraction 0.663 for the Carmichael population versus 0.642 under the norm constraint with the Korselt condition removed). Within the resolution measured, the Korselt criterion constrains only the multiplicative congruence of norms and expresses no preference among factor types; the one-dimensional monochromatic collapse has no type-level analogue in Z[omega]. In keeping with the series' honest ledger: Carmichael ideals over rings of integers are a studied object (Steele 2008; Bae-Hu-Sha 2023), and the dual membrane is most likely an elementary corollary of that abstract theory rather than a new algebraic theorem. The contribution claimed is its placement -- the recognition that the 6N skeleton's defining exclusion of 2 and 3 is exactly the 2-adic and 3-adic geometric blockage in the six-fold phase space of Z[omega] -- together with the first large-scale empirical map of the Carmichael type spectrum above the membrane. A standard-library reproducer and the enumerated dataset are openly available at https://github.com/Ruqing1963/6N-eisenstein-dual-membrane