Paper Pseudocode and Current Official Code Audit

The conclusion is stated upfront: the current GitHub implementation is not a simple transcription of the paper's pseudocode, but incorporates several important certificate strength corrections, centered around a gate to "prevent overclaiming $\operatorname{BSD}(E,2)$." The key distinction is that 2-descent directly controls $\dim_{\mathbb F_2}\Sha(E)[2]$, not $\operatorname{ord}_2\#\Sha(E)$ — even if the calculated $\Sha[2]$ dimension is 2, there could still be $\Sha[2^\infty]\cong(\mathbb Z/4\mathbb Z)^2$ making $\operatorname{ord}_2\#\Sha=4$. Therefore, the official program's check_BSD_at_2 only accepts when both $\operatorname{ord}_2\#\Sha_{\mathrm{an}}=0$ and the descent strictly yields $\Sha[2]=0$ simultaneously — this aligns with the warning in Remark 2.19 of the paper that a positive analytic 2-adic valuation requires higher 2-power descent. The same discipline applies to the conditions for $E'$: the program reads $\Sha_{\mathrm{an}}(E')$, but only uses it as an input for the descent gate, without directly passing it off as the actual group order. The document also records important engineering details: the descent backend sequentially tries PARI, mwrank, and Sage native, with mwrank placed in an independent process with a wall-clock timeout because it might hang; the criterion for $\mathcal S\ne\varnothing$ is changed to the deterministic filter from Proposition 2.16 of the paper, no longer relying solely on bounded prime search as the primary certificate; once testing-only flags like skip_filter_S or skip_BSD_at_2_check are enabled, the output can no longer be called complete theorem-qualified curves and must be written into the metadata. Finally, it points out a gap still pending audit in Phase 1 v0.2: Algorithm 1 initially uses the LMFDB rank field, then actually determines the analytic rank by excluding central vanishing via the special value condition — during a full reproduction, the algebraic rank field, analytic rank field, and special value nonvanishing should be explicitly saved separately to avoid silently conflating the two types of rank.