§ 3.5

Research

Open reformulation programs, illustrative diagnostic reports, and companion projects — alongside the latest research updates and findings.

Jump to Latest Research

Program	Latest milestone	Status	Notes
TNFR-Riemann	P12–P49 (ζ ↔ L attack surface)	OPEN — gap G4 (T-HP)	σ_c → 1/2 verified; paused at S(T)
REMESH-∞ closure	Branch A (N15)	CLOSED	13-operator catalog closed under τ_g → ∞ limit
TNFR-Navier–Stokes	Diffusive projection VALID	OPEN — Clay bound unresolved	Peak enstrophy debt bounded at fixed Re
Number theory	Cyclotomy law proved	ACTIVE	Primality as ΔNFR = 0; arithmetic pulse
Millennium reformulations	P vs NP, BSD, Hodge, Yang–Mills	EARLY	TNFR-internal structural reformulations

TNFR-internal structural reformulations

Interaction Sequences

Illustrative operator sequences. ./make.cmd report-interaction-sequences

Emergent Particles

Emergent-pattern detection report. ./make.cmd report-emergent-particles

Fundamental Particles Atlas

Particle-scale TNFR diagnostics. ./make.cmd report-fundamental-particles

Latest Research

Open Program

July 25, 2026

Unknown Author

APPLIED_STRUCTURAL_ANALYSIS

This document describes domain-specific applications of the nodal equation ∂ E P I / ∂ t = ν f Δ N F R ( t ) ∂EPI/∂t=ν f ΔNFR(t) that are implemented, tested, and computationally verified in this repository. The primary application is spectral factorization via Paley–Jacobi graphs, which has a dedicated implementation with 10 test modules, structural verification criteria, and reproducible certificates. Demonstration-level examples (particle collisions, elemental structure) are listed in §5 with their actual verification status.