Structural Classification of the Exterior-Exponent Collapse Mechanism

Paper I studies a class of candidate finite-time collapse mechanisms for the three-dimensional incompressible Navier–Stokes equations. In this class, an axisymmetric collapsing vortex is coupled to a non-axisymmetric interaction field whose Reynolds stress acts on the mean flow. The class is organized by a single parameter, the exterior circulation exponent α. The results hold under explicitly stated premises: a source-free leading core, a strained-core width law, saturated self-similar supply and axial balance. Circulation growth and vanishing core energy hold together exactly when 0 < α < 1/3. The cut-dependent collapse Reynolds number cancels from this criterion. For the specified interaction-wave family, covariance realizability is governed by a circular cone aligned with the radial shear. Its exterior parameter satisfies v_s − 2 = α. An exact finite perturbation constructs an active midplane layer. A linear lift gives a stress-free global midplane preparation under a quantified non-resonance condition. Continuation in height is a coupled boundary-value problem. Its axis fold has a quasi-elliptic (2,6) core. The full frozen-window resolvent is bounded below by a quantity that tends to infinity at large height frequency; no power-law rate is claimed. Scope: this paper does not prove that Navier–Stokes solutions blow up. OpenAI's external construction corresponds to α = 2h < 1/50, and it realizes the class only if that construction is correct. Finite-frequency non-resonance, range closure, variable profiles, height continuation and full Navier–Stokes verification are left to Paper II, which is in preparation. No priority is claimed. Files: 00 – Paper I 01 – Claim ledger: the current status of every claim, including corrections to Attachment A 02 – Attachment A: the complete technical manuscript, whose theorem numbers Paper I cites. Superseded intermediate statements are kept and marked. 03 – Attachment B: cone realizability and the midplane construction 04 – Attachment C: global quasimodes of the frozen axis-fold operator 05 – Attachment D: compactness closure and the proof of Theorem 9.60 06 – Attachment E: where the paper's statements about OpenAI's construction appear in its Lean formalization. This is based on reading the Lean source; the Lean build was not run. Code availability: the supporting calculations are at https://github.com/LarCorps/ns-exterior-exponent-calculations. Numerical results are audits and are not used in place of proofs. Use of AI tools: large language model tools, including Anthropic's Claude and OpenAI Astra/Sol, assisted with derivations, symbolic and numerical audits, source verification, and drafting and editing. The author directed the research and takes full responsibility for the content.