The No-Vacuum Principle, Seifert Matrices, and the Riemann Hypothesis

This paper presents one rigorously proved algebraic theorem, one genuinely new geometric observation, one new equivalence reformulation of the Riemann Hypothesis, and a series of clearly labelled open conjectures. Part of a three-paper series grounded in the 59-dimensional geometric framework introduced in Papers I and II. We prove: if a real matrix S satisfies S + ST = I (the No-Vacuum Condition), then every eigenvalue λ of S satisfies Re (λ) = 1/2. We connect this to Seifert matrices in knot theory, derive the consequence that all roots of the associated Alexander polynomial lie on the unit circle |t| = 1, and propose — without proving — that this structure is related to the non-trivial zeros of the Riemann zeta function. The work explicitly distinguishes between what is proved, what is observed, and what is conjectured. Theorem proved. Conjecture stated. No overclaiming. Declaration: Generative AI was used solely for computational verification of numerical examples. All theoretical developments, proofs, and writing are the author's own original work.