Audited Note on a Negative Framework as a Complement to the Riemann Hypothesis

This note revises, using a deliberately non-assertive strategy, two manuscripts concerning negatively oriented arithmetic, a left theta series, a zeta construction in negative coordinates, a prime-gap functional, and an angular parametrization. Its purpose is to leave a reusable record: what can be calculated directly, what depends on transported classical results, what remains conditional, and what should be treated only as a conjecture or future task. The analysis confirms that the projection P−(a) = −|a|, the oriented operations, the separation between orientation and magnitude, the negative Gamma identity under a declared branch, convergence of the theta tail, and the elementary formulas for the gap functional are coherent mathematical materials within their stated domains. It also fixes two necessary precautions. First, the notation n^{−s} cannot be used simultaneously for two different conventions. Second, the Mellin transform of the theta tail with exponent s/2^{ − 1} is directly related to ζ(s) in ℜ(s) > 1; the negative variable appears only after an explicit reparametrization. To retain completeness without overclaiming, the notation Z−(s) = e^{iπs}ζ(−s) is introduced and, only as a transport from classical theory, Ξ−(s) = ξ(−s). The identity Ξ−(s) = Ξ−(−1 − s) is then explained as a consequence of ξ(s) = ξ(1 − s). Its fixed axis ℜ(s) = −1/2 is a reflected coordinate, not a second critical line of the original zeta function. Under s = −ρ, locating the zeros of Ξ− on that axis is equivalent to the Riemann Hypothesis; it is not a proof. The audit therefore yields a cautious classification: the integrated framework can be preserved as a structural Toy Model and conditional research program. The note prioritizes verifiable results and a future agenda with success criteria, without presenting reparametrizations of classical objects as new theorems. Keywords: Riemann Hypothesis; mathematical audit; negative coordinates; transported zeta; theta; Mellin transform; prime gaps; Toy Model.