Two Sides of the Same Coin: The EML Operator, Continued-Fraction Rules of Unity, and an Interference Law on the Logarithmic Branch Cut

Odrzywołek's operator eml(x,y) = eˣ − ln y generates, together with the single constant 1, all elementary functions and the constants e, π, i. The Geometry of Unity generates φ, e and π from the unit by three continued-fraction rules. We show that both constructions are two readings of one mechanism and make the common core explicit. (i) A single parameter t = 1 + 4 lim aₖ/(bₖbₖ₋₁) classifies the convergence of all continued fractions involved: super-exponential at t = 1 (the exponential node and the e-rule), geometric with rate |(√t − 1)/(√t + 1)| for t ∉ (−∞, 0] (the golden rule sits at t = 5), and non-geometric on the cut t ≤ 0. For the logarithmic node, t equals the argument y of ln y. (ii) Each of the three rules is an EML node at a definite place of this axis; the π-rule is the first member of Brouncker's family that reappears inside the logarithmic node at y = −1. (iii) On the branch cut y < 0 we prove an exact formula for all convergents of the Euler–Padé continued fraction of ln y in terms of Legendre polynomials and Ferrers functions of the second kind, and deduce an interference law: the convergents form a standing tangent wave whose centre line is Re ln y = ln|y| and whose amplitude is Im ln y = π, with phase step (ϑ + π)/2, cos ϑ = (y + 1)/(y − 1). The Poincaré root of the recurrence becomes −ζ with ζ = eiϑ, and y = ((1 + ζ)/(1 − ζ))². The pattern is periodic exactly when ζ is a root of unity; for rational y only at y ∈ {−1, −3, −1/3}, and the golden ratio enters at y = −φ³/√5 and y = −√5 φ³. Off the cut the same wave is damped; on the unit circle each step is an exact quarter turn. Angle halving, i.e. the EML square root, moves π from the cut into nested-radical rates tan(π/2ᵐ⁺²). For irrational ϑ/π the convergents on the cut are Cauchy distributed with median ln|y| and scale π: a Lorentz profile whose position and width are the real and imaginary parts of ln y. For composite trees we prove that the error decays at the rate of the slowest logarithmic node, and, assuming Schanuel's conjecture, that the golden operator φˣ − log_φ y cannot reach its own scale ln φ in its first layer. On the cut the convergents turn about the fixed point ln y of an elliptic Möbius rotation whose multiplier is the Poincaré root, and every bounded holomorphic mean of the wave returns ln y: this is how EML reaches iπ in one step while the continued fraction circles it. The local centres of the wave are computed to first order, and their window means approach ln y at the rate k⁻² for every y < 0, with an explicit oscillating constant of modulus between π/(2 sin²ϑ) and 3π/(2 sin²ϑ). Off the cut the error of the logarithmic node is 2πρᵏ⁺¹(1 + c_k/k), where c_k continues the phase correction on the cut, and the rate of a tree is decided by a leading amplitude that can cancel: at the omega constant a tree with two tied nodes converges at the squared rate. The exponential node has its own wave: on the imaginary axis the diagonal Padé approximants of eˣ stay on the unit circle, and their error is exactly the phase 2δₙ(θ/2) with tan δₙ = jₙ/yₙ, the scattering phase shift of a hard sphere; equivalently, the n-th convergent of Lambert's continued fraction for tan z is tan(z + δₙ(z)). Both nodes are half-line transforms of the two factors of the Rayleigh expansion eⁱᶻᵘ = Σ(2n+1)iⁿjₙ(z)Pₙ(u): on the cut the even convergents of ln y are the phases of ∫₀^∞ jₙ(s)eⁱᵘˢ ds = iⁿ(π/2·Pₙ(u) + i·Qₙ(u)), and on the circle the hard-sphere matrix element of the exponential node is the quotient of ∫₋₁^∞ and ∫₁^∞ of Pₙ(t)eⁱᶻᵗ dt. As a consequence the n-th Lambert stage contains exactly the Bernoulli numbers B₂, …, B₂ₙ and misses the next one by the leading coefficient of δₙ. Finally, the purely angular operator arccot x − cot y yields the field operations, contrary to a remark in the literature, but no closed tree is non-real, and neither eˣ nor ln x is a tree function. Every result carries one status label; all computations are reproduced by the accompanying scripts. EML operator; continued fractions; Padé approximation; logarithm; branch cut; Legendre polynomials; Ferrers functions; Brouncker continued fraction; Wallis product; golden ratio; roots of unity; Niven theorem; Schanuel conjecture; Cauchy distribution; limit-periodic continued fractions; Möbius transformation; Poisson kernel; Legendre functions; Birkhoff–Adams asymptotics; omega constant; spherical Bessel functions; Bessel polynomials; Lambert continued fraction; hard-sphere phase shift; Rayleigh expansion; Hankel functions; Bernoulli numbers; Risch structure theorem