A Single-Orbit Bridge Lemma for the Riemann Hypothesis. Localizing NBBD Boundary Defects to a Cross-Orbit Orthogonality Condition.

This consolidated technical note (v2) presents a conditional structural result on the Riemann Hypothesis (RH), with an unconditionally-proven single-orbit core. It is not a proof of RH.

We introduce a ℤ/2-orbit decomposition of the non-trivial zeros of ζ(s) under the β-reflection involution σ: ρ = β+iγ ↦ (1−β)+iγ. Each orbit's Chebyshev-type contribution splits into a symmetric part S_O (trivial representation) and an antisymmetric part A_O (sign representation), with the algebraic biconditional A_O ≡ 0 ⟺ β = ½ — the first observable detecting off-critical zeros algebraically, without asymptotics.

Main results:

• Unconditional: the single-orbit bridge lemma ‖A_O‖² ≤ ‖2S_O‖² (asymptotic squared-norm ratio → ¼), the first unconditionally-proven component of this orbit-decomposition path.
• Conditional: RH follows from cross-orbit orthogonality of the {S_O} — a Schur-complement lower bound of Montgomery-grade difficulty, conjecturally implied by Montgomery-type spacing genericity (not proven equivalent to Montgomery pair-correlation).
• Honest negatives: the fixed-η scalar energy route (closed), the growth-bound reduction (retracted), and the envelope-persistence proxy (failed, recorded as dark data).
• Adversarial empirical lab: synthetic resonant zero configurations drive the diagnostic ratio to ≈0.02, showing why a non-resonance / Montgomery-genericity hypothesis is necessary, not merely sufficient.

The contribution is to localize the remaining difficulty to one precisely-formulated, classically-hard object (cross-orbit orthogonality / Schur-complement bound), and to record honestly which routes did not work.

This record extends the prior NBBD Inventory (DOI: 10.5281/zenodo.19102214), which is the unconditional foundation; the "Relation to the NBBD Inventory" section states precisely what is inherited, superseded, and new.

The note was produced with an autonomous multi-model AI research fleet under the author's direction; per arXiv/journal norms, AI systems are not authors of record. The human author takes full responsibility for the mathematics. The model families used are disclosed in the Acknowledgments; full attribution is in the project repository.