Cyclotomic IMC Closure and the Rank-2 Generalized Perrin–Riou Bridge

The main result of this piece is establishing that "the full cyclotomic Iwasawa Main Conjecture for (389.a1, 11) is no longer an open gate"—the Burungale-Castella-Skinner theorem applies to elliptic curves with p>3 good ordinary reduction and residual irreducibility, but requires an additional image condition: there exists σ fixing ℚ(μ_p^∞) such that T/(σ-1)T≅ℤ_p. The document explicitly constructs the unipotent element u=[[1,1],[0,1]] using the maximal 11-adic image (GL₂(ℤ₁₁)) to verify this condition—deducing from the Weil pairing that det u=1 means the corresponding σ fixes the entire cyclotomic extension, and directly calculating (u-1)T=ℤ₁₁e₁ yields T/(u-1)T≅ℤ₁₁, completing the proof. Combining this with the previously closed Sha[11^∞]=0, rank=2>0, and residual irreducibility automatically satisfying the BKS standard hypothesis package, the document deduces P5-ALG₁₁=CLOSED UP TO UNIT (the algebraic Iwasawa-to-Bockstein relation is closed up to an 11-adic unit). The document also clearly delineates two derivative directions that must not be confused—the cyclotomic p-adic derivative D²_cyc L₁₁(E) and the complex spectral parameter derivative d²/ds² L(E,s)|_{s=1}—no Kurihara derivative or Iwasawa characteristic element will automatically identify the two; this is exactly the missing rank-2 Generalized Perrin-Riou (GPR) comparison. The document marks P5-GPR11=OPEN, which is currently the main conceptual barrier. It additionally isolates a purely finite computational Bockstein non-zero gate P5-BOC-NZ11; the included SageMath script is honestly marked as PENDING_LOCAL_SAGE_REPLAY because the execution environment lacks SageMath, not pretending it is already completed.