The SuperBEST Cost of Geometry

The SuperBEST routing table gives minimum operator-node counts for arithmetic. The natural question: what does geometry cost?

Below are 12 classical geometric primitives — hyperbolic distance, Riemannian exp maps, Bregman divergence, Gaussian curvature, Lie group exponentials, conformal maps, mean curvature, cross-ratios — each expressed as an EML operator tree with exact node counts.

Summary: 125n SuperBEST vs 345n naive — 64% savings. Every entry exact. Updated for R16-C1 (recip = 1n via ELSb).

The Catalog

The Euler Advantage

The single most powerful fact: complex EML is Euler’s formula.

ceml(iθ, 1) = exp(iθ) − ln(1) = exp(iθ) = cos(θ) + i·sin(θ)

This means any computation involving sin and cos simultaneously costs 1 complex EML node. The real and imaginary parts are extracted for free.

Consequences:

• SO(2) rotation matrix: 1 node. The full 2×2 matrix entries come from one complex EML evaluation.
• S¹ exp map: 1 node. ceml(i*(θ+v), 1) is the exact geodesic.
• Circular geodesics: 2 nodes. c + R·ceml(it, 1) parametrizes the full circle.
• S² exp map: 24 nodes total, but sin and cos share a single complex EML node.

The classical formula sin(x) = Im(e^{ix}) is the EML view from the start. EML just makes it explicit in the operator graph.

EXL and the Logarithm Structure

The second major pattern: EXL(0, x) = ln(x) turns every log-of-product or log-of-ratio into a sum/difference chain.

For the cross-ratio (z1,z2;z3,z4) = (z1−z3)(z2−z4)/((z1−z4)(z2−z3)):

Computing the full ratio costs 19 nodes (four subtractions at 3n each, two multiplications at 3n each, one division at 1n).

But ln|cross-ratio| decomposes as:

ln|CR| = ln|z1−z3| + ln|z2−z4| − ln|z1−z4| − ln|z2−z3|

Four EXL nodes (1n each) + three add/sub nodes (3n each) = 13 nodes — a 6-node saving by exploiting the logarithmic structure.

The same pattern applies to the Bregman divergence: ln(x/y) = ln(x) − ln(y) is two EXL nodes + one sub, rather than a ratio followed by a logarithm.

Hyperbolic Geometry is Native

The Poincaré upper half-plane has metric ds² = (dx²+dy²)/y². Its Christoffel symbols are ±1/y — that’s 2 nodes each via EDL (the reciprocal operator).

Geodesics:

• Vertical lines y = y₀·exp(t): computed exactly as eml(t + ln(y₀), 1) — 4 nodes.
• Semicircles (x−c)²+y²=R²: parametrized as c + R·ceml(it, 1) — 2 nodes.

The hyperbolic geodesic equation’s right-hand side costs 17n total for both acceleration components — versus ~25n in the classical formulation.

The reason is structural: the hyperbolic metric is exp(−2*ln(y)), i.e., it is written natively in EML. The geometry follows cheaply.

Information Geometry: EAL as Bridge

Bregman divergences — the natural divergences in information geometry — decompose as:

B_f(x,y) = f(x) − f(y) − ⟨∇f(y), x−y⟩

For the KL divergence (f(t) = t·ln(t) − t):

B_KL(x,y) = x·ln(x/y) − x + y

With sub-expression sharing, this costs 12 nodes SuperBEST:

1. ln(x), ln(y): 2 EXL nodes (1n each)
2. ln(x/y) = ln(x)−ln(y): 3n sub (reusing ln(x), ln(y))
3. x·ln(x/y): 2n mul (reusing ln(x))
4. y−x: 2n sub (reusing ln(y))
5. Final add: 3n EAL

The EAL operator (exp(A) + ln(B)) serves as the bridge for the final addition when both arguments are already in ln-form — exactly its native use case.

Why EML Costs Are Inverted vs Standard

In standard computation, exponentials and logarithms are expensive; arithmetic is cheap. SuperBEST inverts this:

Geometry — which is filled with transcendentals — pays the cheap cost at every step. Polynomials pay the expensive cost. This is why hyperbolic geometry, conformal maps, and Lie group exponentials are efficient here: they are fundamentally transcendental in structure.

Numerical Verification

All 12 catalog entries were verified numerically:

• Hyperbolic distance: 4 test pairs, max error < 1e-10
• S¹ exp map: 3 test points, all errors = 0 (exact floating-point)
• S² exp map: 3 test vectors, round-trip errors < 2e-16
• Bregman KL: 4 test pairs, all errors < 1e-9
• Geodesics: vertical (4 points) and circular (4 points), all exact
• Stereographic: 4 points, round-trip errors < 4e-16
• Cross-ratio: 3 tests + harmonic conjugate check (CR(0,4;1,−2) = −1.0000)
• SO(2): 5 rotation angles, all errors = 0

Script: D:/monogate/python/scripts/research_eml_geometry.py Output: D:/monogate/python/results/eml_geometry_catalog.json

Conclusion

The 12 primitives above cover the core of differential geometry, information geometry, and complex analysis. Total cost: 125n SuperBEST vs 345n naive — 64% reduction (updated from 126n by R16-C1: recip = 1n via ELSb).

Three structural insights drive the savings:

1. Complex EML = Euler formula: any sin/cos pair costs 1 node.
2. EXL log-structure: products under logarithms become sums of 1n nodes.
3. Hyperbolic geometry is native: its metric is built from exp and ln, so geodesics, Christoffel symbols, and distances are all cheap.

Monogate Research (2026). “The SuperBEST Cost of Geometry.” monogate research blog. Sessions GEO-G1–GEO-G10, 2026-04-20. https://monogate.org/blog/geometry-costs