The Coercive Projection Method: Axioms, Theorems, and Applications
Recognition Science, Recognition Physics Institute, Austin, Texas, USA
Summary
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, Goldbach, Riemann Hypothesis, and Navier–Stokes.
Abstract
The same projection/dispersion/aggregation pattern solves all four domains with structurally identical ingredients: a convex structured cone, a finite covering net, a rank‑one/Hermitian projection bound, and domain‑specific dispersion estimates. 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. RS predicts optimal parameter schedules (dyadic windows, $\varphi$‑scaling); conversely, proven classical results validate RS by converging to the unique zero‑parameter attractor.
Key Contributions
- CPM Axioms: abstract template with explicit coercivity/aggregator theorems
- Four case studies: Hodge (calibration–coercivity), Goldbach (medium‑arc), RH (boundary certificate), Navier–Stokes (critical vorticity route)
- Universality: structured cones, finite nets, rank‑one projection, and dispersion bounds recur across domains
- RS Bridge: CPM parameters and constants align with RS predictions (dyadic/$\varphi$‑tiers, eight‑tick structure)
Method Pattern
- Define structured set $\Struct$ and defect $\Defect$ (distance‑to‑structure)
- Coercivity: $\Energy(\alpha)-\Energy(\alpha_0)\ge c\,\Defect(\alpha)$ with explicit constant
- Finite $\varepsilon$‑net + rank‑one/Hermitian projection inequality
- Structured/dispersion split; bound dispersion with domain tools
- Aggregate to global positivity/existence