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.
Correction (2026-09-13): the script this post named, experiments/complex_closure_depth6.py, was never in this repository, and no other was kept. The enumeration was rebuilt and is printed below. It confirms depths 1–5 and the 0.99999524 near miss. It does not confirm three things this post said: that depths 1–4 give only positive reals (0 and negative values appear), that "dozens" of imaginary parts appear at depth 6 (there are 13,600), or a second near miss approaching π/2 with a gap of 9.75 × 10⁻⁶ (the closest is 4.1 × 10⁻⁴ away).
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. Depth is tree depth: eml(a, b) is one level
deeper than the deeper of a and b.
Depths 1–4: all real
At depths 1 through 4, every value produced by the EML closure of {1} is real: 397 distinct
values in all. The grammar generates e, ee and various compositions of
exp and ln. Not all of them are positive: eml(1, eml(eml(1,1),1)) = e − ln(ee) = 0
at depth 3, and eml(0, ee) = 1 − e at depth 4, with 0 that depth-3 value.
Depth 5: first complex values
At depth 5, complex values appear for the first time: 13,598 of the 77,528 distinct values. But they are remarkably constrained: every complex value at depth 5 has Im = −π exactly.
This is not a coincidence. A depth-5 value is ea − ln b with a and b of depth at most 4,
so both are real. The principal-branch logarithm gives ln(−r) = ln(r) + iπ for r > 0,
so whenever b is negative the imaginary part is exactly −π, and otherwise the value is real.
Depth 6: the phase transition
At depth 6, the imaginary parts diversify dramatically: 13,600 distinct values. Every depth-5 value a has imaginary part 0 or −π, so ea is real, and the imaginary part of eml(a, b) is −arg b: one value for each complex b, plus 0 and −π. 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 near-miss post argued that depth 7 cannot improve on it; its re-run search finds a depth-7 imaginary part within 7.9 × 10⁻¹² of 1, on a value 1.32 from i.
This post also reported a second route approaching π/2 from below with a gap of 9.75 × 10⁻⁶, and called the two routes independent transcendental obstructions. The enumeration does not reproduce it: the depth-6 imaginary part closest to π/2 is 1.5712099, above π/2 by 4.1 × 10⁻⁴.
Counting nodes instead
Counting eml nodes rather than depth gives a smaller picture, which /theorems (T25) also records: trees with at most 4 nodes are real, trees with 5 nodes have imaginary parts 0 and −π, and trees with 6 nodes add one more, 1.50795.
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
The script enumerates every value by depth with the principal complex logarithm, in mpmath at 50 digits (80 digits gives the same counts). A cancellation smaller than 10⁻³⁰ of its operands is taken as an exact 0. Pairs whose exponential ea has Re a > 10⁵ are skipped; they give real values, or complex values with imaginary part −π, so the depth-5 statements hold for them too and the depth-6 count is a lower bound. It runs in a few seconds.
import itertools, mpmath as mp # pip install mpmath
mp.mp.dps = 50
REL = mp.mpf(10) ** -30 # a cancellation this small next to its operands is an exact 0
def eml(a, b): # principal log; None where ln 0 or e^a is out of reach
if b == 0 or mp.re(a) > 1e5:
return None
ea, lb = mp.e**a, mp.log(b)
if abs(mp.im(ea)) <= REL * abs(ea): # e^(r - i*pi) is real: drop the rounding in sin(pi)
ea = mp.re(ea)
z = ea - lb
return mp.mpf(0) if abs(z) <= REL * max(abs(ea), abs(lb)) else z
S = {'1': mp.mpf(1)} # by tree depth: every value of depth <= d, deduplicated
for d in range(1, 6):
for a, b in itertools.product(list(S.values()), repeat=2):
z = eml(a, b)
if z is not None:
S.setdefault(mp.nstr(z, 30), z)
nonreal = [z for z in S.values() if mp.im(z) != 0]
print(f"depth <= {d}: {len(S)} values, {len(nonreal)} non-real, Im in {sorted({mp.nstr(mp.im(z), 12) for z in nonreal})}")
assert all(mp.im(z) in (0, -mp.pi) for z in S.values()) # so e^a is real and Im eml(a, b) = -arg(b)
ims = {mp.nstr(-mp.arg(b), 20) for b in S.values() if b != 0}
print("depth 6:", len(ims), "distinct imaginary parts")
for t, name in ((1, "1"), (mp.pi / 2, "pi/2")):
im = min((mp.mpf(s) for s in ims), key=lambda v: abs(v - t))
print(f" closest to {name}: {mp.nstr(im, 10)} (gap {mp.nstr(abs(im - t), 3)})")
N = {0: [mp.mpf(1)]} # by number of eml nodes
for n in range(1, 7):
N[n] = [z for i in range(n) for a in N[i] for b in N[n - 1 - i] for z in [eml(a, b)] if z is not None]
print(f"{n} nodes: Im in", sorted({mp.nstr(mp.im(z), 12) for z in N[n]}))Output:
depth <= 1: 2 values, 0 non-real, Im in []
depth <= 2: 5 values, 0 non-real, Im in []
depth <= 3: 26 values, 0 non-real, Im in []
depth <= 4: 397 values, 0 non-real, Im in []
depth <= 5: 77528 values, 13598 non-real, Im in ['-3.14159265359']
depth 6: 13600 distinct imaginary parts
closest to 1: 0.9999952372 (gap 4.76e-6)
closest to pi/2: 1.571209895 (gap 0.000414)
1 nodes: Im in ['0.0']
2 nodes: Im in ['0.0']
3 nodes: Im in ['0.0']
4 nodes: Im in ['0.0']
5 nodes: Im in ['-3.14159265359', '0.0']
6 nodes: Im in ['-3.14159265359', '0.0', '1.50795316919']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