TEBAC BSD-IV: Final Closure and Export Manuscript

This preprint is the final closure and export manuscript for Module BSD-IV in the TEBAC Birch--Swinnerton--Dyer program. Its purpose is to collect the BSD-IV split sequence into a short referee-facing handoff document and to state precisely what the arithmetic-bridge module exports downstream to BSD-V. The manuscript records the source-complete raw verification closure of BSD-IV:\[E_D^{}=E_B^{}=E_T^{}=E_M^{}=E_H^{}=0,\]and hence\[R^{}BSDIV(E)=(0,0,0,0,0)\]in the source-complete raw verification sense. The exported BSD-IV package consists of the arithmetic bridge\[B_E^{}:(-1){~}E( Q)⊗Z R,\]the rank bridge\[(-1)=rankE( Q),\]the raw height comparison\[q^{}E,=B_E,*\, ,\,_{},\]and the regulator identity\[_{}(E)=(E).\] This upload should be read as the BSD-IV arithmetic-bridge export document for the downstream BSD-V assembly. It does not claim to prove the full Birch--Swinnerton--Dyer conjecture by itself, and it does not replace the terminal leading-term assembly that belongs to BSD-V.