Canonical Directed Laplacian on F_2^4, Symmetry-Induced Two-Scale Coupling, and the Structural Isomorphism of a 64-Copy Boolean Graph Assembly
Year: 2026
Type: preprint
Abstract: We study the canonical construction of a directed Laplacian O on the Hasse diagram G16 of the Boolean lattice 2^|4| over F_2^4 = (Z/2Z)^4, and its extension to an assembly of 64 copies. We prove that among all linear operators on R^16 satisfying locality, conservation, fixed binomial spectrum {4, 3, 2, 1, 0} with multiplicities (4 over k), and S4-invariance (canonicity), the operator O = D_out - A is the unique solution. Assembling 64 copies under the organisation 16 x 4, we show that any Sym(16 x 4)-invariant coupling operator decomposes uniquely into intra-group and inter-group terms. We establish a structural isomorphism between this operatorial decomposition and the combinatorial two-scale relational structure (TRS) of the assembly. Among all uniform partitions mk = 64, we prove that the organisation 16 x 4 is the unique one simultaneously satisfying the spectral separation criterion (which favours minimal k) and the S4-symmetry requirement (which imposes k >= 4), with indices m = 16 = |F_2^4| and k = 4 = dim_F2(F_2^4) emerging from independent arguments. The unique Sym(16 x 4)-invariant null mode of the effective operator is identified as a distinguished collective invariant. The derivational chain F_2^4 -> G16 -> O -> (A_intra, A_inter) -> O_ef -> v0 is closed and contains no additional postulates beyond those inherited from L1.
Author: Avelino Carlos Diaz
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166474521