The Kinds of Stillness on the Spiral: A Taxonomy of λ = 0 across the Euler Universe

The special value λ = 0 of the complex winding e^((λ+i)θ) — winding without growth — is not one phenomenon but several, distinguishable by why the growth rate vanishes and which rate is vanishing. This working paper collects the Euler Universe series’ scattered marginal cases, adds the black-hole cases, and applies the series’ failure-mode discipline to the result: it dissolves one candidate (the TOV “smear,” which proves epistemic rather than ontic) and quarantines the one genuine extension (the thermodynamic, imaginary-time case at extremality) into a flagged tier outside the Lorentzian core. Four core types are defended — definitional, dead, maintained, and critical (with Type I / Type II sub-flavors anchored in the Chandrasekhar limit, the photon sphere, and Choptuik critical collapse); a fifth, thermodynamic type is admitted only as an extension under a printed caveat. No quantity is derived; the contributed-number column reports none throughout. The parent Euler Universe cosmology is reprinted as falsified; the logarithmic spiral is retained only as an interpretive lens over established physics. Claude (Anthropic) was used as an analytical and writing tool and is not credited as an author.