TEBAC BSD-IV Absolute Hard-Input Discharge Addendum

This preprint is an absolute hard-input discharge addendum for Module BSD-IV in the TEBAC Birch--Swinnerton--Dyer program. Its purpose is to attack and discharge the five theorem-level hard inputs H1--H5 required for absolute raw BSD-IV verification closure, rather than merely recording source-complete or conditional closure. The addendum treats the following five inputs: \[H1:\ central-image no-loss,\]\[H2:\ support-boundary cancellation,\]\[H3:\ neutral terminal exactness,\]\[H4:\ generator-complete central realization,\]\[H5:\ quadratic height normalization.\] The manuscript records original-source discharge mechanisms for these inputs inside the BSD-IV arithmetic-bridge framework. In particular, it establishes the central detector-saturation route for H1, support-boundary cancellation through primitive Cech-exact support sources for H2, neutral terminal exactness in the fixed terminal complex for H3, generator-complete central realization for H4, and local raw-height source-normality against the Neron--Tate height for H5. The resulting BSD-IV export package is the absolute raw verification package\[E_D^{}=E_B^{}=E_T^{}=E_M^{}=E_H^{}=0,\]equivalently\[R^{}BSDIV(E)=(0,0,0,0,0),\]together with the arithmetic bridge, rank bridge, raw height comparison, and regulator identity needed by the downstream BSD-V terminal assembly. This upload should be read as a BSD-IV hard-input discharge and export addendum. It is not a BSD-V assembly manuscript and it does not by itself claim to complete the full Birch--Swinnerton--Dyer conjecture. The completed leading-term formula, period/Tamagawa/torsion constant ledger, and final BSD assembly remain the responsibility of BSD-V.