Reciprocal Internal Complementarity and the Conditional Emergence of Compact U(1) Gauge Geometry

Abstract

Reciprocal Internal Complementarity (RIC) denotes an internal paired relation whose complementary roles are mutually determining but not thereby identified with spacetime, gauge redundancy, or electromagnetic reciprocity. A preceding RIC analysis showed that, on a supplied real two-plane, reciprocal reversal admits elliptic, hyperbolic, nilpotent, and frozen generators, while a non-fixed two-sided bounded orbit selects the elliptic branch and a positive invariant quadratic capacity. This paper asks what further structure is sufficient for that selected phase plane to support compact Abelian gauge geometry. The positive result is a conditional emergence chain. The elliptic plane is one Hermitian complex line, and its compatible unit-frame group is $U(1)$. Smooth localization over a supplied base yields a Hermitian line bundle. Only after an independent physical equivalence under smooth vertical rephasings is imposed does the structure group acquire a gauge interpretation; only after a unitary path-comparison law is supplied do a connection, holonomy, curvature, and the Bianchi identity follow. Exact countermodels show that none of these transitions can be collapsed. An additionally admitted anti-linear RIC reversal conjugates the phase circle and pairs every nonzero representation weight $n$ with $-n$. We prove the same-bundle topology, strict-equivariance, and connection-preservation obstructions, and identify $L_n\oplus L_{-n}$ as the minimal reciprocal representation carrier. Under an explicit four-dimensional Lorentzian, local, first-derivative, constant, metric-only quadratic constitutive package, the construction imports the Maxwell kinetic term together with the constant $F\wedge F$ term and admits conserved electric sources. A compact Euclidean lattice realization preserves gauge invariance, holonomy, group-valued Bianchi closure, reciprocal $\pm n$ matter, principal-branch integer defects, and a manufactured $O(a^2)$ Maxwell continuum regression. Standard bundle, Maxwell, and lattice results are treated as imports. The contribution is the cross-release selector architecture linking the RIC bounded phase theorem to compact $U(1)$ gauge kinematics while preserving every remaining source problem.

Where it sits in the release

RIC-EM is one of the three Wave B preprints in Foundational Release III. It carries the bounded-phase result from RIC-P1 into a conditional gauge-geometry construction. Its reciprocal phase-frame type, electromagnetic gauge-frame freedom, and electric–magnetic duality rotation remain distinct unless explicit maps relate them.

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