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. The analytic backbone is the Schoenfeld (1976) RH-conditional 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 (section 8), 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. The concentration of the proof on these seven small indices is recorded separately and 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 an analytic route that differs from the one taken here, in that their argument depends 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. We explicitly do not promote the global theorem above its RH-conditional status. The reconstructed proof is supported by a certified enumeration certificate and a lightweight checker (tools/check_certificate_v2.py, 7/7 blocks C1–C7 PASS) whose residual set is exactly the singleton $\{21\}$, in agreement with the unconditional small-index sub-claim and with the original Pomerance–Spicer conclusion. Note on this deposit (version 2, snapshot c101). This is an unfinished working draft, deposited as-is at the author's request. Relative to version 1: the LaTeX rendering of the abstract and of the analytic bound is now correct, and the paper is 77 pages (previously 81). Still outstanding: the PDF was built with citation processing disabled, so bibliographic citations appear as unresolved keys (e.g. @Dusart2010) and the reference list is absent; the title page carries unfilled author/date placeholders.