Strict connection–Dirac spectral dominance for every finite graph

For every nonempty finite graph simplicial complex and each permitted k, the sum of the k largest connection-matrix eigenvalues exceeds the corresponding signed-Dirac sum by at least sqrt(2)-1. Isolates and disconnected graphs are included. This proves the entire dimension-at-most-one case of Knill's conjecture, not the higher-dimensional statement. All incidence, spectral and literal face-family bridges are proved in Lean; no optimal gap is claimed. 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 all-k signed comparison located in inspected primary/citing/author sources or supplementarywebsearch. Generic hydrogen/KyFan/SchurHorn/Loewner facts credited. Coveragebounded; noabsolutepriorityclaim. Canonicalorientationformalized; nooptimalgapclaim.