A Unified Reciprocal Cost-Functional Framework: Structural Approaches to Fluid Regularity, Gauge Theory, and Arithmetic Geometry

We develop a unified mathematical framework built on the reciprocal cost functional J(x) = 1/2 (x + x^-1)^-1, which we prove is the unique penalty satisfying five natural axioms (inversion symmetry, J(1) = 0, strict convexity, coercivity, and the multiplicative d’Alembert identity). The underlying composition law is itself forced by polynomial consistency. From this single functional we derive seven domain-agnostic foundation tools, covering log-domain cost geometry, topological frustration, periodic neutrality, Cayley–Schur phase analysis, algebraic pinch mechanisms, topological capacity obstructions, and an Iwasawa-theoretic bridge, each proved unconditionally from classical mathematics.

Applying these tools via problem-specific bridge theorems, we obtain new structural results in five areas:

1. A contradiction argument for 3D Navier–Stokes global regularity via ancient-element extraction, commutator depletion, the Topological Frustration Pinch, and an Alexander-duality linking veto.

2. A construction of Yang-Mills theory with mass gap via lattice reflection positivity, AF-free norm-resolvent convergence through a log-domain geometric embedding, and Osterwalder–Schrader reconstruction.

3. A prime-wise Iwasawa-theoretic approach to the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over Q, using a diagonalization engine, Lambda-adic Fredholm operator, Fitting–characteristic equality, and algebraic principal-ideal pinch.

4. A clean reduction of the Riemann Hypothesis to a three-part hypothesis H; the internal logic is proved unconditionally, but (H1) requires a non-circular global phase-margin estimate.

5. An exact affine encoding for 3-SAT with unconditional query lower bounds; the complexity consequence is conditional on a non-natural polynomial-time certifier.

This manuscript serves as an extended research announcement and architectural overview for a comprehensive series of companion papers. Every claim carries an explicit status label and a traceable dependency chain; full analytical details are developed in the companion papers cited herein.