EML Atlas
Mathematical objects classified by EML depth and operator cost.
SuperBEST v5.3 headline: 15n positive / 18n general (joined from /superbest, not typed here) · 16 elementary operators (completeness classes: CONJECTURE) + 8 Pfaffian towers (OBSERVATION) = 24 total · 35/36 calculator coverage
Depth Strata
Atlas depth measures how many nested exponentiations a function requires, with arithmetic free. Depth 0 = no exp/ln. Depth 1 = one application. Depth ∞ = impossible as a finite real tree. It is not EML tree depth, which counts every eml node on the longest root-to-leaf path: there ln x needs exactly 3 and x + 1 exactly 4 on (0, ∞). Node costs below are F16 nodes (the sixteen operators on /framework); a row that also names EXL or ELSb, census operators outside F16, gives that count separately.
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 | — |
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 for x, y > 0 (F16: exp(ln x + ln y)) | T_SUPERBEST_UB |
| x / y (division) | 2 nodes for x, y > 0 (F16(x, F13(−1, y))) | T08 |
| Fourier kernel exp(iωt) | 1 complex node | — |
| Laplace kernel exp(−st) | 1 real node | — |
Requires composing exp and ln once. Power functions, reciprocals.
| Object | Cost | Ref |
|---|---|---|
| ln(x) | 2 nodes (LEdiv(0, F13(−1, x))); 1 with the census operator EXL (EXL(0, x)) | T08 |
| neg(x) = −x | 2 nodes (LEdiv(0, EML(x, 1)); EXL∘DEML with the census EXL) | T09 |
| recip(x) = 1/x | 1 node for x > 0 (F13(−1, x); ELSb(0, x) in the census); 7 for all x ≠ 0 | R16-C1 |
| x^n (integer power), x>0 | 1 node (F13; EPL/ELMl in the census) | T_SUPERBEST_UB |
| x^r (real power), x>0 | 1 node (F13; EPL/ELMl in the census) | T_SUPERBEST_UB |
| sqrt(x), x>0 | 1 node (F13; EPL in the census) | T_SUPERBEST_UB |
| add(x,y), all reals | 2 nodes (LEdiv+DEML) | ADD-T1 |
| sub(x,y) | 2 nodes (LEdiv∘EML) | T33 |
| mul(x,y), x,y>0 | 1 node (F16: exp(ln x + ln y)) — duplicate of the depth-1 entry above | T_SUPERBEST_UB |
| abs(x) | no real F16 tree over all reals (|x| is not analytic at 0) | — |
| Gaussian exp(−x²/2) | 3 nodes | T08 |
| log-sum-exp (N terms) | at most 2N−2 nodes (LEAd and EML alternating) | — |
| softmax denominator (N) | at most 2N−1 nodes | — |
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 | — |
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
Node counts from the canonical SuperBEST table, in F16 nodes. Only some entries have a proved lower bound; /superbest gives each entry's status. Total: 18 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) | LEdiv+F13 | 2 | 3 |
| neg(x) | LEdiv+EML | 2 | 9 |
| recip(x) | F13 | 1 | 5 |
| sqrt(x) | F13 | 1 | 7 |
| sub(x,y) | LEdiv+EML | 2 | 5 |
| add(x,y) | LEdiv+DEML | 2 | 11 |
| mul(x,y) | F16fn | 1 | 13 |
| pow(x,n) | F13 | 1 | 15 |
| div(x,y) | F16fn+F13 | 2 | 15 |
| sin(x) | ceml (ℂ) | 1 | 101 |
| cos(x) | ceml (ℂ) | 1 | 101 |
| softplus | LEAd | 1 | 3 |
| Total | 18 | 289 | |
16-Operator Census
The census list of Sixteen_Operator_Census.tex: exp(±x) combined with ln(y) by arithmetic, with LEAd, ELAd, ELSb and LEX. It is a different sixteen from the F16 on /framework. Completeness class conjectured from the position of negation relative to exp (T26–T28, all conjectures; over ℝ the rule fails for EAL, whose real trees are all nondecreasing).
■ 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; conjectured over ℂ; over ℝ not complete, since every real EAL tree is nondecreasing (T26) |
| 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 | Conjectured; the domain argument is refuted: lex(x, lex(lex(1,1),1)) is defined on all of ℝ (T28) |
Conjectured structural rule: exp(+x) with no domain restriction ↔ exactly complete (T26, T27, T28; none has a proof). See Does exp(+x) Mean 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 completeness classes of the elementary 16 are CONJECTURE-tier (argued on paper, with gaps, and searched computationally; not formalized in Lean). 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
- Calculator coverage 34/36 → 35/36.
factorial(n) = Γ(n+1)becomes a finite Pfaffian tree at depth 2 via T_Γ. Modulo stays at 1/36 (discrete; no continuous representation). - 35 named special functions reachable across the 8 families. Max depth 4 from any tower; node range 1–11 in BEST routing.
- 8 / 8 towers independent. No single fixed integral kernel spans more than one tower at finite depth (C-198).
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 | 2 F16 nodes (1 with the census operator EXL). |
| x^r | 2 | 2 | 1 node for x > 0: F13(r, x). |
| neg(x) | 2 | 2 | 2 nodes. T09. |
| x + y | 2 | 2 | 2 nodes for all real x, y: LEdiv(x, DEML(y,1)) (ADD-T1). The April figures were 3 nodes (EAL, x>0) and 11n (general). |
| x · y | 2 | 2 | 1 node for x, y > 0 (F16fn); 3 for all reals (LEdiv(0, F13(y, DEML(x, 1)))). |
| 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).
| PageRank (per page, N pages) | 5N+4 nodes | web ranking |
| TF-IDF | 7n | document relevance |
| Haversine distance | 28n | GPS/navigation |
| NFL passer rating | 33n | most complex sports formula |
| ELO rating update | 26n | chess/competitive ratings |
| Pythagorean expectation | 11n | same structure as Hill equation |
| 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 |
| 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 |
| Basic reproduction number R₀=β/γ | 2n | cheapest important metric |
| SIR model per-step | 20n | full epidemic simulation |
| Fermi-Dirac distribution | 9n | same structure as sigmoid |
| Partition function Z (N terms) | 6N−3n | same as softmax denominator |
| 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) ≤ Σ c_op·n_op(E) | Holds when every unit cost is a gadget valid on the domain used; the old table's div = 1 has no such gadget over all reals. |
| T38 (Decomp) | Cost = NC − SD − PB (a definition) | PatternBonus is the remainder after sharing, so this is not a formula for Cost; with a fixed pattern catalogue it fails (e^{x₁} + e^{x₂}). |
| T38-NNP | No Nesting Penalty (upper bound) | Cost(O₁(O₂(A,B),C)) ≤ c(O₁)+c(O₂)+Cost(A)+Cost(B)+Cost(C). Equality fails: LEdiv(LEdiv(x,1),1) = x costs 0. |
| T40 (Add Law) | Cost(op(E₁,E₂)) ≤ Cost(E₁)+Cost(E₂)+1 | For independent sub-expressions. Equality fails: LEdiv(EML(x,2),1) costs 1. The Lorentz 57n is an upper bound. |
| T40 (Linear) | N-term sum: naive (α₀+c_add)N − c_add | Cost ≤ (α₀+2)N − 2 with the 2-node add (ADD-T1). The old exact (α₀+3)N − 3 is false: Σ e^{xᵢ} costs at most 2N − 1, not 4N − 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. In F16, mul (1n vs 3n), recip (1n vs 7n) and div (2n vs 8n) still have a real gap, and sqrt, pow and ln have no all-reals entry; the general div lower bound 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 →