EML Atlas

Current status · July 2026

The Atlas is a map of EML representations, depth/cost estimates, proof targets, and exploratory extensions. It is useful for education and review, but public promotion still depends on the current proof/evidence gate. Entries involving towers, special functions, zeta-style analogies, runtime advantage, or symbolic discovery should be read as bounded research notes unless a card says otherwise.

This page is rendered directly from src/data/atlas.json and src/data/pfaffian-towers.json at build time — the SuperBEST mini-table below is joined against src/data/superbest.json's canonical costs rather than keeping its own copy. The 295+-equation domain catalog linked below is not independently re-verified by this page.

Mathematical objects classified by EML depth and operator cost.

SuperBEST v5.3 headline: 14n positive / 16n general (joined from /superbest, not typed here) · 16 elementary operators (THEOREM) + 8 Pfaffian towers (OBSERVATION) = 24 total · 35/36 calculator coverage

Depth Strata

EML depth measures how many nested exponentiations a function requires. Depth 0 = no exp/ln. Depth 1 = one application. Depth ∞ = impossible as a finite real tree.

Depth 0 Arithmetic

Constants and polynomials. No exp or ln required.

Object Cost Ref
Integers, rationals 0 nodes —
Polynomials p(x) 0 nodes (constants only) —
Arithmetic: +, −, ×, ÷ see SuperBEST table T08
π (as constant terminal) 0 nodes —
Depth 1 Exponential

One application of exp or ln. The native depth of EML.

Object Cost Ref
exp(x) 1 node (EML) —
exp(−x) 1 node (DEML) T08
sin(x) over ℂ 1 node (ceml(ix, 1)) T_EULER_LEAN
cos(x) over ℂ 1 node (ceml(ix, 1)) T_EULER_LEAN
softplus(x) = ln(1+eˣ) 1 node (LEAd(x,1)) —
x · y (multiplication) 1 node (ELAd) T_SUPERBEST_UB
x / y (division) 2 nodes (EDL) T08
Fourier kernel exp(iωt) 1 complex node —
Laplace kernel exp(−st) 1 real node —
Depth 2 Logarithmic

Requires composing exp and ln once. Power functions, reciprocals.

Object Cost Ref
ln(x) 1 node (EXL) T08
neg(x) = −x 2 nodes (EXL∘DEML) T09
recip(x) = 1/x 1 node (ELSb) R16-C1
x^n (integer power), x>0 1 node (EPL/ELMl) T_SUPERBEST_UB
x^r (real power), x>0 1 node (EPL/ELMl) T_SUPERBEST_UB
sqrt(x), x>0 1 node (EPL, F13) T_SUPERBEST_UB
add(x,y), all reals 2 nodes (LEdiv+DEML) ADD-T1
sub(x,y) 2 nodes (EML) T33
mul(x,y), x,y>0 1 node (EPL/ELMl) — duplicate of the depth-1 entry above T_SUPERBEST_UB
abs(x) 2 nodes —
Gaussian exp(−x²/2) 3 nodes T08
log-sum-exp (N terms) N−1 LEAd nodes —
softmax denominator (N) N−1 nodes —
Depth 3 Oscillatory (ℂ path)

Requires the complex bypass — sin, cos, trig via Euler. Depth-∞ over ℝ, depth-1 to 3 over ℂ.

Object Cost Ref
sin(x) Taylor, 8 terms 101 nodes (real path) O-FOURIER
Fourier series, N terms N nodes (complex path) O-FOURIER
Heat equation mode eᵏᵗ 2 nodes CAL-4
tan(x) 3 complex nodes —
Depth ∞ Non-constructible (ℝ)

Provably unreachable as finite real EML trees. Require either infinite depth or complex domain.

Object Cost Ref
sin(x) over ℝ ∞ (Infinite Zeros Barrier) T01
cos(x) over ℝ ∞ T01
i = √−1 (strict branch) ∞ (nearest: 4.76×10⁻⁶ at depth 6) T17

SuperBEST Routing Table

Minimum node counts across the 16-operator family. All entries proved optimal by exhaustive search (T08). Total: 17 nodes vs 289 naive = 94% savings.

* sin/cos column compares 1-node complex path vs 101-node real Taylor path

Operation Best family Nodes Naive
exp(x) EML 1 1
ln(x) EXL 1 3
neg(x) EXL+DEML 2 9
recip(x) ELSb 1 5
sqrt(x) EPL (F13) 1 7
sub(x,y) EML 2 5
add(x,y) EAL 2 11
mul(x,y) EPL/ELMl 1 13
pow(x,n) EPL/ELMl 1 15
div(x,y) ELSb 2 15
sin(x) ceml (ℂ) 1 101
cos(x) ceml (ℂ) 1 101
softplus LEAd 1 3
Total 17 289

16-Operator Census

Every binary combination of exp(±x) with ln(y) via arithmetic. Completeness class determined by the position of negation relative to exp (T26–T28).

■ Exactly complete (8) · ■ Approximately complete (1) · ■ Incomplete (7)

Name Formula Complete? Why
EML exp(x) − ln(y) YES exp(+x), no restriction
EAL exp(x) + ln(y) YES exp(+x), add
EXL exp(x) · ln(y) YES exp(+x), mul
EDL exp(x) / ln(y) YES exp(+x), div
EPL exp(x) ^ ln(y) YES exp(+x), pow
LEAd ln(exp(x) + y) YES exp(+x), composition
ELAd exp(x + ln(y)) YES exp(+x), composition
ELSb exp(x − ln(y)) YES exp(+x), composition
EMN ln(y) − exp(x) APPROX −exp(x) outside: unbounded ↓ but exp residual
DEML exp(−x) − ln(y) NO Slope barrier: self-comp slope = +1
DEMN ln(y) − exp(−x) NO Domain failure: output always < 0
DEAL exp(−x) + ln(y) NO Domain collapse: offset e⁻¹ irremovable
DEXL exp(−x) · ln(y) NO Dead constant: dexl(x,1) ≡ 0
DEDL exp(−x) / ln(y) NO Decay barrier: outputs → 0
DEPL exp(−x) ^ ln(y) NO Decay barrier: bounded above by 1
LEX ln(exp(x) − y) NO Domain barrier: domain → ∅ at depth 2

Structural rule: exp(+x) with no domain restriction ↔ exactly complete (T26, T27, T28). See Why exp(+x) Means Complete →

Pfaffian Tower Extensions OBSERVATION TIER

The 16 elementary operators reach every elementary function. But mathematics is larger than the elementary class. Functions like erf, Bessel J_n, and Gamma Γ have no finite composition of exp and ln (Liouville 1833 / Risch 1968). They live in Pfaffian families that need new primitives — one generator per family.

8 independent tower generators. 35 Pfaffian functions reachable as finite EML trees of depth ≤ 4 from {EML, T_X}. Source: monogate-research/exploration/E201_extended_atlas/ (extended_node_counts.csv + extended_node_counts_summary.json). Reconciled 2026-07-23: atlas.astro's own copy of this table and blog/src/data/superbest.json's `pfaffian_towers` key had each independently drifted from this source AND from each other (function_total 42 vs 33 vs the source's actual 35; several towers' `covers` lists included functions that were never in the verification run at all). This file is now the single copy both /atlas and /superbest read — see scripts/audit-citations.mjs for the check that would catch a future drift.

Tier note: the elementary 16 are THEOREM-tier (Lean-verified). The 8 towers are OBSERVATION-tier (Empirically verified at sample points, not Lean-proved. The 'no single operator spans two towers' theorem is open (CONJECTURE). The 8 tower families form a 3-rooted hierarchy under DLMF closed-form identities (see hierarchy_note below).)

Tower Family Generator Covers Chain Depth Verified
T_erf Error Function ∫₀ˣ exp(−y·t²) dt erf, erfc, erfi, dawson, fresnels, fresnelc 2 4 1.2×10⁻³¹
T_Si Sine/Cosine Integral ∫₀ˣ sin(y·t)/t dt Si, Ci, Shi, Chi, Ei, li 3 3 2.1×10⁻³⁰
T_J Bessel (1/π)∫₀^π cos(x sin(t) − y·t) dt J_0, J_1, J_n (general), Y_0, I_0, K_0 3 3 1.2×10⁻³¹
T_Ai Airy (1/π)∫₀^∞ cos(t³/3 + x·t + y) dt Ai, Bi, Ai′, Bi′ 3 2 1.5×10⁻⁷ (via ODE)
T_Γ Gamma ∫₀^∞ t^(x−1) exp(−y·t) dt Γ, lnΓ, ψ (digamma), β, ψₙ (polygamma), n! = Γ(n+1) 2 4 8.0×10⁻¹⁸
T_W Lambert W solve w·exp(w) = x + y W₀, W₋₁ (real branches) 2 1 0
T_K Elliptic Integrals ∫₀^(π/2) 1/√(1 − m sin²(t) + y) dt K(m), E(m), F(φ,m) 3 2 4.3×10⁻³¹
T_F Hypergeometric (series) ∑ (a)ₙ(b)ₙ/(c)ₙ · zⁿ/n! ₂F₁, ₁F₁ 2–4 1 0

What the towers add

What's still outside

Even with all 24 operators, some functions remain outside: Riemann zeta ζ(s) (infinite Dirichlet series + functional equation), modular forms, non-computable functions (Chaitin's Ω), and discrete operations like mod (no continuous Pfaffian representation). Functions still outside the 23-op + 8-tower closure: zeta-family beyond polylog, modular forms, modulo, set/Boolean ops, generic non-computable. See the atlas page for the full out-of-closure list.

Special Functions — EML Classification

Exact depth and cost over ℝ and ℂ for standard special functions.

Function Depth ℝ Depth ℂ Notes
exp(x) 1 1 Native. eml(x,1).
exp(−x) 1 1 deml(x,1).
ln(x) 2 2 1 EXL node.
x^r 2 2 3 nodes via EXL.
neg(x) 2 2 2 nodes. T09.
x + y 2 2 3 nodes (EAL, x>0). 11n (general).
x · y 2 2 1 node (ELAd) or 3n (EXL).
softplus(x) 2 2 1 LEAd node (definitional: LEAd(x,1) = ln(exp(x)+1)).
sin(x) ∞ 1 T01 (ℝ barrier). T_EULER_LEAN (ℂ bypass).
cos(x) ∞ 1 T01 (ℝ barrier). T_EULER_LEAN (ℂ bypass).
tan(x) ∞ 3 sin/cos ratio.
sinh(x) 3 3 (eˣ − e⁻ˣ)/2: 3 real nodes.
cosh(x) 3 3 (eˣ + e⁻ˣ)/2: 3 real nodes.
sigmoid σ(x) 3 3 1/(1+e⁻ˣ): 2–3 nodes.
i = √−1 ∞ ∞* T17. *Nearest depth-6: gap 4.76×10⁻⁶.

Calculus Costs

Node counts for standard calculus operations under SuperBEST routing (T08, T25).

Operation Cost Notes
Taylor series, N terms 9N − 3 nodes T25. sin(x) 8 terms = 69n.
Fourier series, N harmonics N complex nodes O-FOURIER. Each harmonic 1 ceml.
Derivative d/dx [exp(f)] + 0 nodes Native: d/dx eml(f,1) = exp(f)·f′.
Gaussian integral ∫e^(−x²)dx 3 nodes (no elementary antiderivative) Depth-2 approx.
Heat equation mode exp(−kt)sin(nx) 4 nodes 2 real + 2 complex.
U(t) = exp(−iHt) (quantum) 1 matrix EML node Q-3. Exact.
Partition function Z = Tr[exp(−βH)] 2 matrix nodes Q-1. Tr + matrix exp.
Softmax over N logits N − 1 LEAd + N EML nodes N-1 LEAd nodes for the log-sum-exp denominator (not independently Lean- or paper-cited; see /theorems T_SUPERBEST_UB for the LEAd operator itself).

Domain Catalog — 295+ Equations

Selected SuperBEST node counts across technology, sports, neural networks, physics, biology, quantum, and finance. Per-row values reflect the basis state at the time each row was added; the current routing-table headline is the v5.2 figure shown in the SuperBEST page. Full catalog: 295+ Equations →

Expanded from 157 (Monster Sprint) → 214 (COMP-ALL) → 295+ (domain-2: FIN, INFO, QM, THERMO, CHEM, BIO, ECON sessions).

Technology
PageRank (per page, N pages) 5N+4 nodes web ranking
TF-IDF 7n document relevance
Haversine distance 28n GPS/navigation
Sports
NFL passer rating 33n most complex sports formula
ELO rating update 26n chess/competitive ratings
Pythagorean expectation 11n same structure as Hill equation
Neural Networks
Softplus ln(1+eˣ) 1n via LEAd(x,1) cheapest smooth activation
Sigmoid σ(x) 7n standard activation
Cross-entropy loss (N classes) 8N−3n training loss
Physics
Ohm's law V=IR 2n product of two scalars
Kinetic energy ½mv² 7n v² = mul chain
Gravitational force GMm/r² 9n power-law denominator
Biology / Epidemiology
Basic reproduction number R₀=β/γ 2n cheapest important metric
SIR model per-step 20n full epidemic simulation
Quantum
Fermi-Dirac distribution 9n same structure as sigmoid
Partition function Z (N terms) 6N−3n same as softmax denominator
Finance
Sharpe ratio 4n mean/std ratio
Black-Scholes full DAG (shared) 83n most expensive EML computation

Cost Theory

Structural decomposition of SuperBEST node counts. T38–T43.

Cost(E) = NaiveCost(E) − SharingDiscount(E) − PatternBonus(E)

Structural Classes

Class Signature Description Mean cost Example
A (1, 0) Pure exponential 10.4n Boltzmann, Arrhenius
B (0, 0) Rational / polynomial 12.2n Newton gravity, Coulomb
C (0, 1) Log-ratio (cheapest) 9.5n Shannon entropy, pH
D (1,1)+ Mixed exp+log 20.1n Black-Scholes (47n)

Key Cost Facts

Theorem Statement Detail
T34 (Naive UB) Cost(E) ≤ NaiveCost(E) Always achievable; no sharing or pattern required.
T38 (Decomp) Three-term formula Cost = NC − SD − PB. Unique decomposition for each expression.
T38-NNP No Nesting Penalty Cost(O₁(O₂(A,B),C)) = c(O₁)+c(O₂)+Cost(A)+Cost(B)+Cost(C).
T40 (Add Law) Cost(op(E₁,E₂)) = Cost(E₁)+Cost(E₂)+1 For independent sub-expressions. Explains Lorentz (57n).
T40 (Linear) N-term sum: (α₀+3)N − 3 Exact formula. Softmax: 4N−3. Shannon: 6N−3. Taylor: 8N−3.
Domain routing (2026-07-22 correction) Positive routing still saves on mul/sqrt/pow/div, no longer on add add is now 2n for ALL reals (ADD-T1) — no positive/general gap left to route around. mul (1n vs 3n), sqrt (1n vs 2n), and pow (1n vs 3n) still have a real gap; div (2n vs 3n) is Lean-proved for the exp-outer case only (see /superbest). The old "~1750n saved across 295-eq catalog" figure was computed from add's pre-ADD-T1 numbers and is retracted — recomputing it needs the underlying equation dataset, not attempted here.
O-ISO 8 cross-domain isomorphism families Radioactive decay = compound interest = population growth (same tree).
QCC Cost ≤ O(N²) conjecture No standard scientific equation exceeds quadratic growth. Open.
T39 Cost ≤ O(N) conjecture Linear Ceiling: no standard equation exceeds O(N). Open.

Pattern Bonus Table (top 6)

Pattern Bonus Formula
DEML −5n exp(−x) − ln(y)
EAL −4n exp(x) + ln(y)
LEAd −4n ln(exp(x) + y)
ELAd −4n exp(x + ln(y))
EML −3n exp(x) − ln(y)
EXL −3n exp(x) · ln(y)

Blind test validation: 27/30 exact (MAE = 0.20) on held-out fluid dynamics, optics, acoustics, materials, epidemiology equations. See Cost Theory → and 295+ Equations →

Theorem catalog → · Research blog → · Full catalog on monogate.dev ↗