The Secret of a Half

Abstract

The real part 1/2 occurs in the theory of the Riemann zeta function as the symmetry axis of its non-trivial zeros. This monograph asks a narrower question than the Riemann Hypothesis itself: can the distinguished role of the half-axis be explained by a common structural mechanism joining binary complementarity, information balance, two-channel interference, spinorial phase, and the anti-linear symmetry of the completed zeta function? The exact part of the programme is developed first. Binary Shannon entropy has its unique maximum at σ = 1/2, with value ln 2. A normalized equal-gain two-channel amplitude A(σ, ϕ) = √ σ + e iϕ√ 1 − σ vanishes if and only if σ = 1/2 and ϕ ≡ π (mod 2π). The involution J (s) = 1 − s has fixed set ℜs = 1/2. If the spinorial sign is parameterized by e 2πiσ, then the sign −1 also selects σ = 1/2 in the open unit interval. These statements combine into an exact “triple coincidence” theorem inside the explicitly defined binary-spinor model. A non-metaphorical bridge to zeta theory is provided by the Dirichlet eta function, η(s) = X n≥1 (−1)n−1n −s = (1 − 2 1−s )ζ(s), with η(1) = ln 2. Within the open critical strip, the eta and zeta zero sets coincide because the binary prefactor has no zeros there. This establishes a genuine relation among alternating binary sign, ln 2, and the non-trivial zeta zeros. It does not determine the horizontal location of those zeros. The central conditional theorem is then stated. If there exists a canonical, involution- covariant Hilbert-space state map whose fixed readout vanishes exactly with ξ(s) and whose normalized channel weights are ℜs and 1 − ℜs, then every non-trivial zero lies on ℜs = 1/2. The theorem is short; the construction of such a map is the entire unresolved burden. Several tempting shortcuts are shown to fail: symmetry alone produces zero quartets rather than fixed points, eta alternation adds a separate line of prefactor zeros, unequal channel metrics move the balance point away from one half, and a pointwise normalized factorization can be engineered non-canonically. The monograph closes by formulating operator, positivity-kernel, de Branges, Li-coefficient, and theta-kernel routes as concrete research programmes. Numerical calculations are used only as regression tests for identities and software, never as substitutes for proof.