The connection is flat. The field strength is zero everywhere. Every local statistical test of the CMB returns null. But the torus has holes—and around those holes, the monodromy is non-trivial. This is the Aharonov-Bohm effect applied to spacetime topology.
The present epoch sits at the topological phase transition. The Betti functional analysis of Planck data favors exactly this range. We are at the threshold of seeing the shape of the universe.
S + d² = 1 connects everything. Matched circles live in S. Power suppression lives in d². Their sum is always one.
The curvature 2-form vanishes identically on flat spatial sections: F = 0. This is why every statistical test of the CMB returns “random.” All structure lives in the monodromy around non-contractible loops.
The CMB two-point correlation decomposes into same-phase (matched circles, weighted by S) and different-phase (power suppression, weighted by d²). The same topology explains both the missing power and the missing circles.
The trichotomy parameter Γ(t) is strictly monotonically increasing. The universe does not expand into new space. It expands in fidelity: the same topology, seen more clearly.
Any compact flat topology with all cycle lengths exceeding 17.5 Hubble lengths will never be detected, regardless of observational technology.
For a 3-torus at L ≈ 2 Hubble lengths—the range favored by Planck Betti functional analysis—Γ = 0.98. The present epoch lies at the threshold of topological observability.
At holonomy phase Φ ≠ 0, matched circles should show anti-correlation, not correlation. No published search has tested this. A falsifiable prediction derived from S + d² = 1.
Local constraints propagate to determine global structure. Given the observed 30% quadrupole suppression, the chain extracts the topology uniquely. The Jacobian is nonsingular. The Sudoku has a unique solution.
The math does not change. We see more clearly.Download Paper (PDF)