The EML Framework

A single binary operator generates all elementary functions and, by extension, all of mathematics. The number of times you apply it determines the depth — and depth equals complexity.

The EML Operator

One gate. Every function.

eml(x, y) = exp(x) − ln(y)
exp − ln → all elementary functions
arXiv:2603.21852 · Odrzywołek 2026
  • Closure: Every elementary function is a finite composition of eml.
  • Minimal: Neither exp alone nor ln alone generates the full class. Their composition is the minimum gate.
  • Depth-generating: One application = EML-1. Two = EML-2. Three = EML-3. No four.
  • No EML-4: Six independent proofs show the next level after EML-3 is EML-∞. The gap is structural.
  • Shadow Theorem (T108): Every EML-∞ object has a finite-dimensional EML-2 or EML-3 shadow.

The Five Strata

Every mathematical object lives at exactly one depth.

EML-0 Arithmetic

EML-0 — Arithmetic

Discrete. Countable. The ground floor.
Constants, integers, finite structures. No measurement required.
1 (unity)ℤ (integers)ℚ (rationals)Topological charge QBetti numbersOrnstein classificationMazur torsion theorem|Sha| (cardinality)Circuit depth (count)Boolean bits
EML-1 Growth

EML-1 — Growth

Exponential. Multiplicative. Things that compound.
The EML operator applied once. Exponential dominance before logarithm enters.
exp(x) = eml(x,1)e = eml(1,1)Compound interestPartition function ZBirkhoff ergodic theoremExponential decayTaylor series radiusPopulation growth
EML-2 Measurement

EML-2 — Measurement

Polynomial. Logarithmic. The dominant stratum.
25.7% of all classified objects. Information geometry is universally EML-2. Polynomial time, measurable structures, spectral gaps.
ln(x)Shannon entropy HKL divergenceFisher informationKolmogorov -5/3 spectrumSelmer groupsL² normsDeterminant (circuits)Riemann zeros (real part = ½)Spectral gapHodge cycles (post-proof)Yang-Mills gap (post-proof)P (complexity class)Chinchilla scaling lawBorn ruleFree energy F = U−TSÉtale cohomologyNatural gradientGelfand transformRG flow fixed points
EML-3 Oscillation

EML-3 — Oscillation

Spectral. Periodic. Self-referential.
Complex exponentials, wave phenomena, Euler systems. EML-3 is closed under all finite operations — the only escape is categorification (Δd=∞).
sin(x), cos(x)Euler systems (Kolyvagin)Gross-Zagier formulaL-functionsModular formsSchrödinger equationWave equationIwasawa main conjecturePSPACEBQP (quantum)Sobolev spacesTheta-vacuumWorking memory rehearsalT-gate (quantum computing)Global Langlands GL(2)Onsager 2D Ising exact solution
EML-∞ The Depths

EML-∞ — The Depths

Undecidable. Self-referential. Permanently open.
No finite EML composition reaches here. EML-∞ objects cast shadows at EML-2 (real exponential) or EML-3 (complex exponential) — but the full object is inaccessible.
NS 3D regularity (independent of ZFC)Path integral measureConsciousness / qualiaHalting problemNP-complete searchKolmogorov complexity K(x)Circuit lower bounds (unconditional)Global Langlands GL(n≥3)Turbulence full descriptionGödel sentencesKhovanov homologyYang-Mills path integral measure (pre-proof)

Depth-Change Operations

How objects move between strata.

Operation Δd Description Example
eml application +1 Apply the EML operator once to an object at depth d. Integers (d=0) → exp(integers) = d=1
shadow projection ∞→2/3 EML-∞ object projects a finite-dimensional shadow (T108). Sha(E) (EML-∞) → Selmer group (EML-2)
tropical descent +0 Tropicalization preserves depth but removes EML-∞ obstructions. Hodge cycles → tropical cycles (both EML-2)
categorification +∞ Replacing sets with categories escapes any finite depth stratum. GL(n≥3) Langlands: EML-3 → EML-∞
Gödel diagonalization →∞ Self-referential encoding sends any object to EML-∞. 3D NS (Turing-complete) → EML-∞ via T943

Foundation Theorems

The structural results that make everything else work.

T108

Shadow Depth Theorem

Every EML-∞ object has a finite-dimensional shadow at EML-2 (real exponential) or EML-3 (complex exponential). The shadow is all formal mathematics can reach.

T232

Depth = Complexity Bijection

EML-0 = DLOGTIME, EML-1 = Kalmár elementary, EML-2 = P (polynomial), EML-3 = PSPACE, EML-∞ = undecidable. The hierarchy is isomorphic.

T297

Tropical No-Inverse

The tropical semiring has no multiplicative inverse. This fundamental asymmetry is computational irreversibility — the source of one-way functions and cryptographic hardness.

T775

Smooth Projective Descent

Tropical → Berkovich → formal scheme → GAGA algebraizes all cycles. This descent chain is what made Hodge and Yang-Mills accessible at EML-2.

T852

Sha Finiteness

Sha(E) is EML-∞ but its EML-2 shadow (Selmer group) is finite-dimensional. Shadow Depth Theorem forces finite Sha → well-defined BSD formula.

T918

EML-4 Gap

Depth 4 does not exist. Six independent proofs. The resource jump from polynomial (EML-2) to super-polynomial is discontinuous — which forces P ≠ NP structurally.

T941

3D NS is Turing-Complete

Explicit vortex ring UTM construction. At Re >> 1, viscosity is irrelevant at scales above the Kolmogorov microscale. Vortex stretching = mathematical self-reference.

T953

Consciousness Independence

Qualia are structurally independent of formal systems — a direct Gödelian argument. Empirical EML-∞ detection in AI remains open (T511).

The Langlands Universality Conjecture (LUC)

Every instance of the Langlands program corresponds to an EML depth transition. GL(1) = EML-2 (class field theory). GL(2) = EML-3 (modular forms, Euler systems). GL(n≥3) = EML-∞ (categorification escape). 40+ LUC instances confirmed. Each BSD rank r adds one new instance: LUC-(37+r).