0.99999524: The Near-Miss

How close can an EML tree get to i? A gap of 4.76×10⁻⁶ and the transcendental obstruction behind it.

One of the open problems on monogate.dev is constructing the imaginary unit i from the EML grammar with terminal {1}. We haven't solved it. But in exploring how close you can get, we found something more interesting than a near-miss — we found a proof of why the gap can't close.

The setup

The EML grammar over ℂ starts with 1 and applies eml(x, y) = exp(x) − Log(y) (principal branch complex log). At depth 1, the reachable set is a curve in ℂ. At depth 2, it's an arc. At depth 6, it's a dense-looking cloud of isolated points.

We want Im(z) = 1, Re(z) = 0. The question is: does anything in the depth-6 closure hit exactly i?

The near-miss

Enumerating depth-6 complex EML values near Im = 1, we found a point with Im = 0.99999524. The gap to Im = 1 is 4.76×10⁻⁶. For reference: depth-5 values have Im ≤ 0 (all negative imaginary part), so the jump to Im ≈ 1 at depth 6 is a structural phase transition, not a gradual convergence.

The near-miss requires Re(y) = π/tan(1) ≈ 2.0270 as the second argument in the outermost EML call. This is where the obstruction lives.

The obstruction: tan(1)

To reach Im = 1 exactly, you need Re(y) = π/tan(1) exactly. But π/tan(1) is transcendental — it's a ratio of π (transcendental by Hermite-Lindemann) and tan(1), which is transcendental by the Lindemann-Weierstrass theorem applied to e^{2i} ≠ 1.

The EML grammar with terminal {1} can only generate numbers in the EL field — the field of exponential-logarithmic numbers built from 1. π/tan(1) does not belong to this field at any finite depth.

This is the content of the real constructibility barrier. The gap at depth-6 (targeted search) is 6.67×10⁻⁷ for the real part. As depth increases, the gap decreases — but it can never reach zero by a constructibility argument, unless tan(1) has some unexpected algebraic relationship to EML-expressible constants.

Why depth-7 doesn't help

We found a structural reason the near-miss gap doesn't improve at depth 7:

All depth-5 complex EML values have Im = −π (exactly). That's because at depth 5, the values all pass through eml applied to negative real inputs, and sin(−π) = 0 makes the imaginary part of the resulting Log call collapse to −π regardless of the real part.

This means depth-6 inputs that come from depth-5 nodes contribute zero to Im(eml), and all depth-6 values arising from that route have positive imaginary part — positive arg. But reaching Im = 1, Re = 0 requires arg = π/2, while arg = −1 (pointing toward −i) is structurally inaccessible from positive-imaginary depth-6 values.

The 4.76×10⁻⁶ gap is not a computational limitation. It's structural.

What we have

We have: a near-miss, a transcendental obstruction, and a structural explanation for why depth-7 doesn't close the gap. What we don't have: a proof that i is unreachable at any finite depth. That remains the open problem.

The near-miss is written up as a draft paper: python/paper/near_miss_obstruction.tex. The main theorem there is the transcendental obstruction — that the exact gap is π/tan(1), which is provably outside the EL field at any finite depth under Schanuel's conjecture.

Challenge board: monogate.dev · Code: pip install monogate