Two Paths to Iron: The Turning Point in Fusion and Fission

Euler's identity e^(iθ) = cos θ + i sin θ is not merely the namesake of this series; in the nucleus it is the working equation. Part I reads the spatial motion: with the Schrödinger wavenumber k² = (2m/ħ²)(E − V), real k gives ψ ~ e^(ikr), Euler's rotation (the i-motion), and imaginary k → iκ gives ψ ~ e^(−κr), pure dilation (the λ-motion); the classical turning point E = V is where the two exchange roles, and the Gamow factor e^(−∫κ dr) is accumulated λ. Fusion and fission are then the same i→λ→i winding traversed in opposite directions — light nuclei climbing to iron, heavy nuclei descending to it. A worked example from AME2020 masses shows both paths arriving at the same state-function maximum of the binding energy per nucleon — iron's number, 8.79 MeV/nucleon (⁵⁶Fe; 8.795 for ⁶²Ni) — where dB/dA changes sign. Part II reads the energy. The binding difference ΔB those reactions release disperses into the largest accessible reservoir of microstates until rest — the reservoir principle of "Entropy on the Spiral" — and iron's binding maximum is a reservoir floor, the inward (λ < 0) drain the parent framework already names in its Gamow states. The spatial moat and the temporal decay are one parameter: the barrier's accumulated κ sets the rate Γ, which the entropy paper reads as λ = −Γ/2ω. A matched entropy calculation (ΔS = Q/T) for the same reactions shows the per-nucleon entropy ceiling is set by the binding-curve climb and vanishes at iron from both sides. Iron is then one rung of an inward ladder — atom, nucleus, stellar core, black hole — each a different physics reaching the same kind of floor, with the cosmic de Sitter horizon the single outward ceiling. The unity is of form, not of object: the lens re-describes WKB, the Gamow factor, and the liquid-drop model; it fixes no λ, predicts no cross-section or half-life, and the scales do not couple. The parent cosmology's falsification record is reprinted in full.