Constraint Network Topological Spectrum of Generalized Algebras F; Galois Arithmetic Locking and Categorical Closure– Unification in Constraint Network Theory for Generalized Algebras

Within the framework of Galois-equivariant constraint networks, this paper
completes the unification of three core problems in modular representation theory—
the Alperin–McKay equality, the Alperin weight equality, and Brauer’s k(B) up
per bound—at the level of information geometry. By strictly separating arithmetic
equivariant actions from geometric equivariant actions, and by establishing the
network–module representation correspondence law, the dimension of the abstract
equivariant module space is precisely identified with the number of irreducible
characters, the number of height-zero characters, and the number of irreducible
Brauer characters. Based on this, it is proved that the information volume defined
by the number of irreducible Brauer characters imposes a strict upper bound on
the capacity of the equivariant module space, thereby naturally deriving Brauer’s
k(B) ≤ |D| as a corollary of the rigid core boundary capacity. Abandoning the tra
ditional Dirichlet-to-Neumann continuous analogy, the boundary Cartan operator
with the Schur complement of the Cartan matrix at its core is constructed, thereby
establishing a boundary–capacity locking mechanism at the purely algebraic level.
Using the adjointness of restriction and induction functors, the synchronisation of
module space dimensions between the global network and the local network on
the normaliser of the defect group is revealed, and it is proved that the two are
strictly equal when the information potential is completely uniform. Ultimately, it
is clarified that the Alperin–McKay equality, the Alperin weight equality, and the
Brauer upper bound are not mutually independent combinatorial coincidences, but
rather triple projections of the same information-geometric structure in the limit of
vanishing information potential. The deviation of the uniformity degree from zero
to positive provides a unified order-parameter description for the transition from
equality to strict inequality among the three major conjectures.