SuperBEST Routing Table

Minimum F16-node constructions for every elementary arithmetic primitive. Proved optimal via derivative obstruction lower bounds and explicit constructions. Taxonomy definitively closed: PROVED — no 24th operator passes PGC+AIT filter on open domain.

23 elementary operators (PROVED closed via PGC+AIT filter) + 8 Pfaffian tower generators (OBSERVATION tier; empirically verified) = 31 operators mapping two tiers of mathematics.

Core Arithmetic Primitives (F16 Layer 1)

Extended Operators (23-op Beyond F16)

Category A: genuine F16 algebraic shortcut. Category B: genuine via EML sign-variant. Category C: notation-only (23-op counts 1n; F16 costs 3–4n; no F16 shortcut exists).

High-Impact Expressions

Layer 1 = F16 ground-truth; Layer 2 = 23-op extended (must be labeled)

ML / Smooth Functions

Quantum / Information

Physics

Special Functions (all require infinite F16 depth)

Pfaffian Tower Extensions OBSERVATION

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). Adding 8 tower generators to the EML grammar brings 33 previously-infinite-depth functions inside finite EML trees of depth ≤ 4. Tier label: OBSERVATION (empirical, not Lean-proved).

Verification: 2 towers exact (W, F); 4 towers to ~1e-31 — erf, Si, J, K (mpmath@30-digit precision at sample points); Γ to ~1e-18 (one tricky sample, others 0.0); Airy via the defining ODE Ai″(x) = x·Ai(x) to 1.5e-7 (the cubic-phase improper integral resists direct mpmath.quadosc convergence).

Hierarchy (3-rooted): Pairwise pass over all 28 tower pairs (28-pair probe in exploration/tower-independence-2026-04-27/): 5 SUBSUMES — T_F (₂F₁) covers T_erf, T_Si, T_J, T_Ai, T_K via DLMF identities (7.18.5, 6.11.2, 10.16.5, 9.4, 19.5.1). 1 PARTIAL — T_Ai ⇆ T_J via fractional-order Bessel (DLMF 9.6.1). 22 INDEPENDENT pairs. The 8 families form a 3-rooted hierarchy: in the F16 grammar all 8 are operationally needed; in the abstract Pfaffian-closure sense the minimum generating set is {T_F, T_Γ, T_W} = 3. Minimum generating set in the abstract Pfaffian-closure sense: {T_F, T_Γ, T_W} = 3 towers. All 8 remain operationally needed in the F16 EML grammar (which lacks parameter-substitution primitives).

Outside the closure: 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.

Honest Savings Summary

Verify it yourself

Reproduce the cost analysis on your own machine. `eml-cost 0.12.0` is live on PyPI; the 578-row corpus is bundled inside the wheel.

pip install eml-cost

from eml_cost import analyze
from sympy import symbols, exp, sin

x, t = symbols('x t')
a = analyze(exp(-x) * sin(t))
print(a.cost_class)        # 'p2-d5-w2-c1' — the headline cross-domain class
print(a.pfaffian_r, a.is_pne)   # 2  False

from eml_cost import analyze_dynamics
from sympy import symbols, exp, sin, cos

x, t = symbols('x t')
d = analyze_dynamics(exp(-x) * sin(t) * cos(2*t))
print(d.predicted_r)       # 5  (1 decay + 2 oscillation modes)
print(d.confidence)        # 'high'

from eml_cost import find_siblings
from sympy import symbols, sin, pi

x = symbols('x')
for s in find_siblings(sin(pi*x)/(pi*x), max_distance=0)[:3]:
    print(s.distance, s.expression, s.domain)

Code matches the published 0.12.0 API. `analyze_dynamics` and `find_siblings` ship in 0.10.0+ (rolled up into 0.11.0 / 0.12.0).