Uncertainty on the Spiral: The Real and Imaginary Parts of the Quantum Exponent

Quantum mechanics carries two uncertainty relations usually taught as one. This paper reads them as the real and imaginary parts of a single complex exponent e^(λ+i)θ. Position–momentum (ΔxΔp ≥ ℏ/2) is an i-phenomenon — Fourier conjugacy, substance-independent, importing ℏ/2 without deriving it; the slogan that uncertainty is "the equation of state of the atomic realm" is named and refused. Energy–time is different: not a conjugate-operator relation (t is not an operator; Pauli) but lifetime–linewidth, ΔE ≈ ℏΓ, the complex energy E₀ − iΓ/2 of Gamow and Weisskopf–Wigner — the inward λ-drain at the atomic floor of the nesting ladder, with Γ = −2ωλ from Entropy on the Spiral. This is the real equation of state the exponent discloses: a relation carrying a measurable rate. The framework is then stress-tested where most exposed — number–phase uncertainty against the absence of a self-adjoint phase operator (Dirac; Susskind–Glogower; Pegg–Barnett) — and the result is logged as a wall, not rescued. One organizing arrangement, one logged obstruction, no new number. The parent cosmology's falsification is reprinted.