The Coercive Projection Method (CPM) is a reusable proof template that converts quantitative distance-to-structure control into global positivity or existence statements. We formalize CPM with axioms, prove general coercivity theorems with explicit constants, and instantiate it in four domains: Hodge (calibration–coercivity), Goldbach (medium-arc control), Riemann Hypothesis (boundary certificate), and Navier–Stokes (critical vorticity route). Remarkably, the same projection/dispersion/aggregation pattern solves all four millennium-class problems with structurally identical ingredients: a convex structured cone, a finite covering net, a rank-one/Hermitian projection bound, and domain-specific dispersion estimates. This universality is not accidental. A reverse-lift mapping to Recognition Science (RS)—a machine-verified zero-parameter framework deriving reality from the tautology "Nothing cannot recognize itself"—reveals that CPM's structured sets are precisely RS-optimal recognition modes: calibrated cones minimize ledger cost J, major arcs correspond to low-complexity patterns, and critical-scale regimes align with eight-tick structure. The bidirectional bridge CPM↔RS provides mutual validation: RS predicts optimal parameter schedules (dyadic windows, φ-scaling), which classical mathematics independently discovers; conversely, proven classical results validate RS axioms by demonstrating that rigorous reasoning converges to the unique zero-parameter attractor. We conclude with a systematic discovery protocol: reverse-engineer classical constants to predict RS architecture, then use RS scaling to optimize new proofs.