Self-Referential Fibration Figure 1: Naturality Failure of Coarse-Graining—Explicit 2-Morphism Coordinates of the Commutation Obstruction

On a one-dimensional exactly solvable Ising ring, this paper constructs the com
posite difference of two coarse-graining paths in order to characterize explicit co
ordinates of the commutation obstruction. The two paths are, respectively, spatial
blocking followed by temperature stepping and temperature stepping followed by
spatial blocking. The effective energy difference is denoted by ∆(b), and the gauge
obstruction is defined as δ∗(b) = ∆(b)−lnfib(b), where fib(b) is the cardinality of the
fiber of block configurations b under the coarse-graining map. The gauge decompo
sition gives ∆(b) = lnfib(b) +δ∗(b), δ∗(b) = ln[Mb(βJ)2/Mb(2βJ)], where Mb(µ) is
the uniform fiber mean. When βJ ̸ = 0, δ∗(b) ≤ 0 follows from the Cauchy–Schwarz
inequality on the uniform distribution over the fiber, with equality if and only if
the bond variable is constant on the fiber; when βJ = 0, δ∗ degenerates to the zero
function and provides no diagnostic of fiber constancy. In the small-coupling ex
pansion, the first-order term cancels identically, the degeneracy term is eliminated
fiber by fiber in the gauge baseline, and the first surviving term is −λ2κ2(b), where
κ2(b) is the variance of the bond variable on the fiber; if C(R) = √︁ ∑︁ b κ2(b)2 > 0,
then ∥δ∗∥2 = C(R)λ2(1 + O(λ)). For the fully enumerated parameterized simula
tion of the N = 8 periodic Ising ring, the leading coefficients of the two rules are
C(D) = 35.7771 and C(S) = 17.6677, and the L2 ratio is about 2.025, showing that
the leading coefficient of the quadratic scaling is a functional of the coarse-graining
rule. In the componentwise vertical cell coordinate framework, the commutation ob
struction is carried by ∆ as the vertical coordinate; the conjugacy criterion uses the
diagonal-operator similarity modulo additive constants, the mean-centered compo
nent multisets of the eighteen cases are pairwise distinct, and neither relabeling-type
nor general linear-similarity-type 1-morphism corrections exist. Two-dimensional
scaling, all-order Laurent coefficients, and higher-level carriers of multi-rule cyclic
consistency are left as open problems.