Riemann's Line Without Riemann's Proof: A Localization of the Riemann Hypothesis · The Critical Line Forced as Locus at Theorem Grade, the Open Question Localized to a Single Conserved Coordinate Bracketed Between Two Theorem-Grade Walls

This paper consolidates what the analytic structure of the Riemann zeta function settles about the location of its nontrivial zeros and what it does not, and it locates the open part exactly. The critical line is forced as a determinate locus at theorem grade, residence-independently: it is the fixed-point set of the anti-holomorphic involution s ↦ 1 − s̄ under which the modulus of the completed function is invariant, and this is corroborated by the canonical self-adjoint dilation generator on the multiplicative Haar space, whose spectrum is real and vertical by the spectral theorem and whose abscissa is placed at one half by the involution. The location of the zeros on that locus, residence, is open. The Riemann Hypothesis is reformulated losslessly as a causal-support condition on one tempered distribution, and the obstruction to that condition is isolated, by the even-odd parity decomposition, to a single object: the odd-symmetric component of the boundary distribution, equivalently the imaginary part of the normalized logarithmic derivative on the line, which is logically equivalent to the hypothesis. This object is bracketed between two theorem-grade walls. On the even side the functional equation, an involution acting on the line as the parity reflection, is provably silent on the minus-one eigenspace where the hypothesis lives, a conservation law rather than a contingent barrier, witnessed in the wild by the Davenport-Heilbronn function, which carries the identical functional-equation shape and possesses off-line zeros. On the odd side the natural de Branges positivity overshoots into a condition false for zeta, by the theorem of Conrey and Li. The even channel is shown blind to off-line zeros, so the residence-encoding even atoms record only the on-line zeros and the independent open coordinate is exactly one object wide, the odd channel. The contribution is a localization and a synthesis, not a proof: no proof of residence is claimed, the location of the obstruction is the result, and the closure requirement is identified as a second structural symmetry from outside the prime-zero ledger, the analogue of the Frobenius action that closes the proven function-field case. The analytic spine above is self-contained and stands alone. The paper then carries a second, explicitly interpretive layer, the author's Trisduction framework, which maps where this open object sits in a verification architecture and reads the same localization in a constructive register; that layer is marked throughout at the grade it earns, is not load-bearing for any analytic result, and is separable from the spine by a reader who wants only the mathematics.