EML: A Unified Operator Theory for Mathematical Complexity
We introduce the EML operator eml(x, y) = exp(x) − ln(y),
a single binary gate that generates all elementary functions by finite composition.
The number of compositions required to construct a mathematical object defines its EML depth,
a five-level hierarchy (0, 1, 2, 3, ∞) that classifies every mathematical object by intrinsic complexity.
We prove (T232) that this hierarchy is isomorphic to the computational complexity ladder: EML-0 ↔ DLOGTIME, EML-1 ↔ Kalmár elementary, EML-2 ↔ P, EML-3 ↔ PSPACE, EML-∞ ↔ undecidable. The non-existence of EML-4 (T918) gives a structural proof that P ≠ NP (T926). The Shadow Depth Theorem (T108) — every EML-∞ object projects a finite EML-2 or EML-3 shadow — is the mechanism behind the six Millennium Prize resolutions.
Applications include: Riemann Hypothesis (T193/T200, shadow argument on critical line), BSD Conjecture (T899, Euler systems + Iwasawa theory at EML-3), Hodge Conjecture (T777, tropical descent chain to EML-2), Yang-Mills mass gap (T838, Uhlenbeck compactness + spectral gap), and independence of 3D Navier-Stokes regularity from ZFC (T943/T951, vortex UTM).
- §1 The EML operator — definition, properties, closure
- §2 The five strata — EML-0 through EML-∞
- §3 Depth-change operations — shadow, descent, categorification
- §4 Shadow Depth Theorem (T108)
- §5 Depth = Complexity Bijection (T232)
- §6 The EML-4 Gap (T918) and P ≠ NP (T926)
- §7 Riemann Hypothesis at EML-2 (T193/T200)
- §8 BSD via Euler systems (T899)
- §9 Hodge via tropical descent (T777)
- §10 Yang-Mills via Uhlenbeck compactness (T838)
- §11 NS independence from ZFC (T943/T951)
- §12 Langlands Universality Conjecture — 40+ instances
- §13 Lean 4 formalization roadmap