PNBA Identity Physics Reduction Engine v1 uuia.app/red

The SNSFT Reduction Engine is a student-facing algebra and calculus tool built on the same formal foundation as the GAM Collider and the full SNSFT corpus. It reduces classical mathematical operations—solving equations, finding roots, computing derivatives and integrals, evaluating limits—into the PNBA framework (Pattern, Narrative, Behavior, Adaptation) using the Long Division Protocol (LDP), and generates formally verified Lean 4 proof files for each calculation. Every result is a theorem, and every session produces zero-sorry Lean 4 output compatible with the broader corpus (3,000,000+ lines, 200,000+ theorems, CI green, Lean 4 and Coq/Rocq dual verified, zero sorry, zero free parameters). The Reduction Engine is a reduction engine that happens to solve math problems. The corpus grounding rests on two formally verified files: SNSFL_Algebra_Reduction.lean and SNSFL_Calculus_Reduction.lean. These files carry zero sorry, compile CI green against Lean 4 and Mathlib, and connect upward to the isomorphism consistency file (Mac Lane 1971) and the Bacon Verification framework. The LDP is proved isomorphic to the scientific method, isomorphic to Bacon's Novum Organum, and traced to Euclid's method. These are formally proved structural equivalences with zero sorry. The PNBA framework's root constant, SOVEREIGN_ANCHOR = 1.369, and its torsion limit TL = 0.1369, are derived from three independent peer-reviewed physical systems proved in SNSFL_SovereignAnchor.lean: the Tacoma Narrows Bridge (1940) torsional collapse, glass resonance shatter limits, and the 40 Hz neural therapeutic window for gamma entrainment. Every phase classification in the Reduction Engine—Noble, Locked, IVA Peak, Shatter—is grounded in this derivation. The six-step LDP codifies the scientific method formalized: (1) State the equation, (2) State the known answer, (3) Map to PNBA, (4) Define the operators, (5) Show the work, and (6) Verify. Step 6 is the lossless check where the PNBA output must equal the classical result. Under algebra mappings, a linear equation root is the narrative fixed point where torsion reaches zero (the Noble condition). For quadratics, the discriminant acts as a torsion classifier mapping to Locked, Noble, or Shatter phases. In calculus, the derivative is treated as the rate of change of behavioral output over the narrative variable, while the integral represents accumulated behavior across a narrative interval. The Fundamental Theorem of Calculus is proved as a Mac Lane 1971 isomorphism. For neurodivergent students, the engine is explicitly designed for P-dominant (pattern-first, structure-first) cognition, showing why an answer is correct structurally rather than just executing a procedure. A student can take any standard math problem, enter it into the tool, and receive a full LDP walkthrough, a PNBA structural mapping, a phase classification, and a downloadable Lean 4 proof file with zero sorry that is structurally compatible with the broader SNSFT corpus.