Unit-Level Generalized Perrin–Riou Minimal Gate for $389.a1$ at $p=11$

Following up on the two obstacles left by document 04 (Bockstein non-vanishing P5-BOC-NZ11, rank-2 GPR comparison P5-GPR11), this document first closes the former using externally published data: Mazur-Stein-Tate computed the cyclotomic 11-adic regulator for the first rank-2 curve 389A, and the table shows R₁₁≡4 (mod 11) is non-zero—this non-vanishing property remains invariant under any non-zero normalized scaling, so the document only uses the non-vanishing property itself (not the specific numerical value) to elevate P5-BOC-NZ11 to CLOSED_BY_PUBLISHED_COMPUTATION. The second is more fundamental: the document proves that the full rank-2 GPR equality is overly strong for a single-prime valuation proposition, and defines a weaker unit-level gate uGPR₁₁: b_BSD∈ℤ₁₁×κ_∞ (only requiring the two vectors to generate the same integral lattice, without needing to know the exact unit multiple). Theorem 5.1 strictly proves P5-LAT₁₁⟺uGPR₁₁. Because Λ₁₁ is a free rank-one module over the discrete valuation ring ℤ₁₁, Proposition 6.2 further decomposes uGPR₁₁ into two bits: P5-INT11 (b_BSD lies within the lattice) and P5-PRIM11 (b_BSD does not lie in 11 times the lattice, i.e., a non-zero mod-11 residue). The document also personally audits three superficial shortcuts—Fouquet 2025's Hecke-algebra ETNC (only covers non-zero special values, excluding the rank-2 central point), Bullach-Honnor's refined Mazur-Tate (explicitly states refined congruences still require BSD or GPR or height comparison extensions), and classical high-rank p-adic BSD (controls the leading term of the p-adic L-function, not the complex derivative L''(E,1)/2!)—none of which can substitute for the missing two bits.