A continuous critical-branch obstruction to C*-algebraic Sendov conjectures

A cubic over C([0,1],C) has roots of norm at most 9/10, a completely split derivative with two distinct continuous critical branches, and strictly positive continuous convex weights. Nevertheless every critical element has supremum distance greater than 21/20 from a fixed root. This refutes the published C*-Sendov Conjectures 3 and 4 as one common outcome. The scalar Sendov problem and the separate density-of-invertibles formulation are not formally claimed resolved. AI-assisted preprint by Myeongjin Kang, with no affiliation. Internal AI review is not external journal peer review. The accompanying Lean certificate covers the precise scoped main statement; inspect the formal endpoint, assumptions and provenance in the supporting package. Prior-art investigations are documented bounded searches, not worldwide-priority certification. Original prose is offered under CC BY 4.0 and original code/Lean under MIT; third-party rights and existing notices are preserved. The public supporting ZIP contains original proof sources and verification evidence, with bibliographic links and hashes instead of redistributed third-party source manuscripts. Recorded prior-art/source-access limits: No prior source-conjecture counterexample found in inspected primary/citing/author/selection literature. Subscription indexes, exhaustive non-English coverage unavailable; bounded screen does not certify worldwide priority. Parent addendum qualifies initial failed supplemental audit logs and numerical one-branch indentation; independent strict136replay and parent extracted replay clean.