Krein-Space Structure of the Non-Backtracking Matrix above the Kotani–Sunada Radius: Definite Type, Conditioning, and Graph-Uniform Power Bounds

This preprint studies the non-backtracking matrix of a finite simple graph through the indefinite inner product induced by arc reversal. Beyond the Kotani–Sunada radius, it proves definite type and semisimplicity and derives quantitative control of complete outer spectral subspaces. Exact Gram identities lead to spectral-projection norms and explicit graph-uniform power bounds without a minimum-degree, connectivity, global diagonalizability, or spectral-gap assumption. Applications include uniform walk growth, percolation bounds and Bethe-Hessian inertia formulas. The v1.0 public package contains the current manuscript and LaTeX source, reproduction code and numerical reference data, frozen graphs, exact certificate evidence, and literature comparison records. Detailed postal addresses and obsolete private manuscript copies are excluded. The historical finite audit used Part IX v1.1; that method dependency is not silently reassigned to the new Part IX v1.2 release. Release: Version 1.0, initial manuscript 14 October 2026; literature revision and public packaging 15 October 2026. Current manuscript: 33 pages, 47 references. This is a new, independent preprint record. The PDF and complete source/reproduction ZIP are supplied; current file hashes are listed inside the archive. Verification scope: fresh standard-library exact replay accepted 39 integer-pencil inertia certificates on 13 graphs, 13 scalar certificates, 30 outward constants, and 77 finite/limiting radial records. All 24 selected mutations were rejected. Stored evidence and verifier code were preserved. The previous 1,281-graph numerical campaign is retained history, not a new rerun for this public release. Finite acceptance is distinguished from the analytic uniform proofs and from general software soundness. Historical proof-method dependency: https://doi.org/10.5281/zenodo.23340664 (Part IX v1.1). AI assistance and author responsibility: GPT and Claude models were used substantively in proof development, verification code, numerical computation, literature checking, drafting and revision. Bin Seol directed the research and experiments, managed the data and evolving definitions, and selected, reviewed and revised the final claims. He is the sole human author and takes responsibility for accuracy, integrity, source attribution and originality. Computational checks are distinguished from mathematical proofs. Related methodological release: https://doi.org/10.5281/zenodo.23384719 (Part IX v1.2). This related release does not replace the historical v1.1 method input identified above.