The Depth-6 Phase Transition

The EML closure of {1} stays real through depth 4. At depth 5, complex values appear — all with Im = −π. At depth 6, everything changes.

The experiment

Starting from the terminal set {1} and the EML operator under extended complex semantics (where ln of a negative real returns a complex value via the principal branch), we enumerated all EML tree values at each depth level.

Depths 1–4: all real

At depths 1 through 4, every value produced by the EML closure of {1} is real. The grammar generates e, e^e, various compositions of exp and ln — all positive reals. No imaginary parts.

Depth 5: first complex values

At depth 5, complex values appear for the first time. But they are remarkably constrained: every complex value at depth 5 has Im = −π exactly.

This is not a coincidence. The principal-branch logarithm gives ln(−r) = ln(r) + iπ for r > 0. The first time a depth-5 subtraction produces a negative real argument to ln, the imaginary part locks at −π. All other depth-5 paths through the grammar land on the same structural value.

Depth 6: the phase transition

At depth 6, the imaginary parts diversify dramatically. Dozens of distinct imaginary values appear. The structural bound Im = −π that held for depth ≤ 5 breaks.

The most notable new value: Im = 0.99999524 — within 4.76 × 10⁻⁶ of 1. This is the nearest approach to Im = 1 (i.e., to constructing i) found in the depth-6 closure. The gap does not close at depth 7 or 8; see the near-miss post for the structural obstruction.

A second obstruction route approaches π/2 from below, with a gap of 9.75 × 10⁻⁶. Two independent transcendental obstructions are active simultaneously.

Why this matters

The depth-6 phase transition shows that the EML grammar has discrete structural changes at specific depths — not a smooth progression. The jump from "all Im = −π" to "Im values everywhere" is not gradual. It is a combinatorial transition triggered by new tree shapes becoming available at depth 6.

This is evidence for the more general claim that EML complexity is genuinely stratified: each depth level has qualitatively different expressive power, not just quantitatively more nodes.

Reproduce

python experiments/complex_closure_depth6.py
# Enumerates EML closure from 1 through depth 6
# Prints depth-by-depth summary of imaginary values

Cite this work

Monogate Research (2026). "The Depth-6 Phase Transition." monogate research blog. https://monogate.org/blog/depth-6-phase-transition

License

CC BY 4.0 — free to share and adapt with attribution. · Code: pip install monogate · Paper: arXiv:2603.21852