P5 Rank-2 Scalar Collapse and Archimedean Comparison Boundary

After importing Sha[11^∞]=0 from Document 01, the 11-primary contributions on the arithmetic side (Sha, Tamagawa, torsion) of the strong BSD leading coefficient formula are all trivial, so the problem converges to a real quantity: B_∞(E) := [L''(E,1)/2!] / [Ω_E·Reg^NT(E)]. The document proves Proposition 2.1: once B_∞(E) has a canonical rational realization, the 11-primary valuation proposition of strong BSD is equivalent to v_11(B_∞(E))=0. But it then points out a more fundamental problem—B_∞(E) is initially just a real number, and there is no canonical v_11 operation for general real numbers, so the proposition must first pass the "rationality" gate (P5-RAT: B_∞(E)∈ℚ×) before reaching the "valuation" gate (P5-VAL11). The document audits four possible shortcuts one by one (Burns-Kurihara-Sano's Generalized Perrin-Riou, Castella-Hsieh's rank-2 Kato classes, Chan-Ho Kim's higher Gross-Zagier, Kim-Pollack's refined Tamagawa conjecture), concluding that they all provide powerful Selmer/discrete-side tools, but none provides the missing archimedean comparison of "complex higher derivatives → classical Néron-Tate determinant". Using LMFDB display values yields B_∞^disp(E)≈1.0000000000000000000003, making the natural candidate B_∞(E)=1, but the document explicitly emphasizes that this is not a proof—arbitrarily strong real approximations do not imply exact equality unless there is a discreteness/rationality theorem. The document provides a concrete escape route (Lemma 7.1): if one can prove B_∞(E)=a/b with a denominator bound B, combined with sufficiently precise interval arithmetic |B_∞(E)-1|<1/(2B²), one can rigorously force the equality using rational separation.