Primes and Circle Duality K: From Constraint Networks to the Critical Line — A Strict Functorial Correspondence between Zeta Zeros and Three Operators

Within the framework of constraint-network theory and automorphic L-functions,
we construct a strict functorial correspondence from the sieve-tree category to the
truncated zero-configuration category. The key step is to introduce an arithmetic
marking category as an intermediary, separating the combinatorial topology of
sieve trees from the analytic spectral data of L-functions. On this intermediary
layer, the topological invariance of truncated L-functions is precisely formulated:
spectral information is determined solely by the set of primes, the character assign
ments, and the condensation partial order, independent of the tree-edge set carry
ing these data. Based on this decomposition, the three operators—condensation,
fission, and reconnection—are respectively mapped to the removal of prime local
factors with spectral coarse-graining, the introduction of prime local factors with
spectral refinement, and character phase transformations with spectral involution
rearrangement.
Strict results under finite truncation—functoriality, reconnection faith
fulness, and the fact that the fixed-point set of the reconnection involution is
the critical line—have been rigorously proved within a fixed truncation-parameter
framework, without relying on any unproved assumptions. Conditional results
involving infinite limits—in which the Riemann hypothesis is reconstructed as
a reconnection Pareto-optimal fixed-point condition for the trivial marking in the
arithmetic marking category, equivalent to the absence of infinite divergent causal
chains—explicitly depend on two working assumptions: the existence of direct lim
its and the preservation of Pareto optimality. This paper does not claim to provide
an analytic proof of the Riemann hypothesis. Throughout the paper, strict results
and conditional results are clearly distinguished; all conclusions involving infinite
limits are marked as conditional theorems depending on the working assumptions.