The Uniqueness of the Sheldon Prime via Explicit Bounds for the n-th Prime

We reconstruct the proof of the uniqueness of the Sheldon prime using explicit, verifiable bounds for the $n$-th prime $p_n$ (Rosser–Schoenfeld 1962; Dusart 2010) combined with the elementary product and mirror properties of Pomerance–Spicer (2019), thereby replacing the inverse logarithmic integral $\operatorname{li}^{-1}$ on which the original argument relies. Main theorem (RH-conditional). Under the Riemann Hypothesis, $p_{21}=73$ is the unique Sheldon prime: it is the only prime $p_n$ for which the product property $D(p_n)=n$ and the mirror property $\operatorname{rev}(p_n)=p_{\operatorname{rev}(n)}$ hold simultaneously. The RH-conditioning enters through the explicit Schoenfeld (1976) bound $|\pi(x)-\operatorname{li}(x)|<\frac{\sqrt{x}\,\ln x}{8\pi}$ for $x\ge 2657$, which the present reconstruction invokes for the analytic window above the small-index regime; the global theorem is therefore declared here only under RH, and is not claimed to be unconditional. Unconditional sub-claim (separately labelled). For the seven indices $n\le 29$ flagged as inherited_unconditional in the audit schema, the search-space elimination is carried out by exact OEIS A000040 data alone and does not invoke any RH-dependent bound; in particular, the elimination of every index $n\le 29$ with $n\neq 21$ is unconditional. This must not be conflated with the global RH-conditional statement above. Relation to Pomerance–Spicer (2019). Pomerance and Spicer established the same uniqueness statement unconditionally along a different analytic route, depending on bounds for $\operatorname{li}^{-1}$. The present paper does not reproduce their unconditional theorem; it provides an independent, audit-friendly reconstruction whose unambiguous residual is the survivor $n=21$, and whose global conclusion is gated by the RH-conditional Schoenfeld bound. The reconstructed proof is supported by a certified enumeration certificate and a lightweight checker (7/7 blocks C1–C7 PASS) whose residual set is exactly the singleton $\{21\}$. Note on this deposit. This is an unfinished working draft, deposited as-is at the author's request. The PDF was built with citation processing disabled: bibliographic citations appear as unresolved keys (e.g. @Dusart2010) and the reference list is absent. Some LaTeX macros in the title-page abstract and main-theorem statement are not rendered. The title page carries unfilled author/date placeholders. A corrected version is expected to be deposited as a new version of this record.