The volume–determinant inequality for alternating ARA braid closures

I prove the volume–determinant inequality for a family of alternating braid closures with arbitrarily many strands, arbitrarily many motifs, and independent integral twist exponents at least three. Each motif uses the adjacent generators in increasing, decreasing, and increasing order. No periodicity of the exponents or parity condition is imposed, and knots and links are both included. For a closure with m motifs, the proof gives 2π log det L − vol(S3 ∖ L) > m/50. A single motif is treated by a marked belted-sum decomposition of a partially augmented parent. For repeated motifs, positive transfer matrices give a uniform two-motif bound. Reciprocal spectra settle odd strand numbers; a retained unit mode, positive compound matrices, and a width-truncation argument settle even strand numbers. Exact integer and rational certificates, local triangulations and verification programs are specified in the appendices and supplied with the sources. The source archive includes the LaTeX sources, verification programs, local triangulations and exact certificates. Licensing. Article text, original figures and tables, LaTeX sources, documentation, and certificate files are licensed under CC BY-NC 4.0 to the extent that copyright or similar rights apply. The original Python programs in code/*.py, including their reproduction in the article's code listing, are licensed under the MIT License. Third-party software and publications retain their own licenses. See LICENSE.md and the accompanying license texts.