Self-Referential Fibration Figure 4: Full-Order Combinatorial Enumeration Formula for Gauge Obstructions on Coarse-Graining Fibers

In real-space renormalization, the two operations of aggregating spin configura
tions by a block map and adjusting source parameters generally do not commute.
This paper splits this noncommutativity into two separable bookkeeping objects:
the fiber baseline depends only on the fiber cardinality and is a purely combi
natorial quantity of the block map; the gauge obstruction is defined by the fiber
moment-generating function and carries coupling-dependent energy-distribution in
formation. For an arbitrary surjective block map on a one-dimensional Ising ring,
the n-th order λ-coefficient of the gauge obstruction equals negative (2n − 2)/n!
times the n-th cumulant of the fiber energy distribution; the first-order cancella
tion follows from 2 − 2n vanishing at n = 1. The conversion between cumulants
and moments is accomplished by Bell polynomials, and the moments are written as
weighted sums of fiber Walsh coefficients over multisets of bond occupations. When
the block map is a GF(2)-linear map, the fiber Walsh coefficients degenerate into
the product of the indicator function of the row space and a sign factor; at general
rank this sign factor is independent of the choice of representative of the solution,
and full row rank is only the special case in which the solution is unique. Conse
quently, the computation of moments reduces to signed pure counting grouped by
the mod-2 sum of bond incidence vectors. For the dual-bit extraction rule, odd
order cumulants vanish, and even-order cumulants are the Rademacher even-order
cumulants multiplied by power sums of adjacent sums of block values; the explicit
expression for κ2 agrees pointwise with this closed form on all fibers for N = 8. The
non-positivity of the gauge obstruction is given by the Cauchy–Schwarz inequality,
with equality at nonzero source parameters if and only if the fiber energy is con
stant. This non-positivity does not constrain the signs of the coefficients order by
order; higher-order coefficients may change sign. At the algorithmic level, coeffi
cient enumeration for fixed order is polynomial in N, but the overall complexity for
an arbitrary block map involves 2N enumeration or a 2r row-space table, and does
not constitute a polynomial-time scheme in the general case.