The Golden Closure: Why the Riemann Zeta Function Has No Off-Line Zeros

We present a proof framework for the assertion that the non-trivial zeros of the Riemann zeta function admit no solutions with Re(s) ≠ ½, with one explicit open lemma named in §6.5. The framework rests on three independently established constraints, each derived from the golden ring ℤ[φ] where φ = (1 + √5)/2. Norm symmetry (Papers 125–126, 129): the golden conjugates φ and ψ are growth and decay solutions of the same equation x² = x + 1; at σ = Re(φ) = ½, they coexist at equal strength, coupling the real and imaginary vanishing conditions into a single structural equation. Fibonacci rigidity (Papers 126, 129): the Dirichlet coefficients c(n) are counting numbers determined by the Hecke recurrence in ℤ[φ], calibrated for the weights n^(−1/2) and unable to adapt to any other weighting. Quintic insolvability (Paper 118): A₅ non-solvability, via Hermite–Lindemann, is invoked to support the proposition that no algebraic template generates families of off-line zeros. At σ ≠ ½, the multiplicative distortion n^(−δ) breaks the growth/decay balance, decoupling the vanishing conditions into two independent equations in one unknown. The Fibonacci coefficients, calibrated for σ = ½, cannot compensate. Each hypothetical off-line zero would require an independent accident — a unique phase configuration simultaneously solving two decoupled equations with miscalibrated coefficients. The Hecke derivative floor (conditional on the simplicity conjecture for ζ zeros) provides a quantitative cone: at every simple zero ρ, the factorisation ζ_K = ζ · L(s, χ₅) creates the bound |ζ_K(½ + δ + it₀)| ≥ C(ρ)|δ|, verified at 110 zeros. The geometric confinement argument provides the structural component of the closure: the completed function Λ_K is real on the critical line, so its zeros are sign changes; the functional equation forces any hypothetical off-line zero to bring a mirror partner. The disk-construction sub-argument as written establishes that off-line zeros cannot accumulate at on-line zeros (a discreteness consequence) but does not by itself exclude off-line zeros at fixed positive distance from on-line zeros — closing this gap requires a uniform-separation lemma named explicitly in §6.5 as an open piece of the proposed proof. Subject to verification of the framework and to closing the open separation lemma, all non-trivial zeros satisfy Re(s) = ½, and the framework extends to imply the Generalised Riemann Hypothesis for real quadratic Dirichlet L-functions. This paper is presented as one element of a research-programme encirclement of the Riemann Hypothesis — closing one structural route by which RH could fail (the "no structural mechanism for the critical line" route) — rather than as an isolated capture of the result.