Split: Raw Height Comparison and Néron-Symbol Calibration
Tosho Lazarov Karadzhov · Zenodo (CERN European Organization for Nuclear Research) · 2026
This preprint continues the BSD-IV split sequence after the primitive-frame and actual coefficient-calculus manuscripts. The preceding split manuscripts reduce and discharge the raw Mordell--Weil verification residue for the source-complete primitive spectral extractor. The present split isolates the remaining height front:\[q^{\raw}_{E,\spec}=B_E^{\arith,*}\langle\, ,\,\rangle_{\NT}.\] The manuscript defines the raw height comparison defect\[\Delta^{H,\raw}_E(v,w):=q^{\raw}_{E,\spec}(v,w)-\bigl\langle B_E^{\arith}v,B_E^{\arith}w\bigr\rangle_{\NT},\]and records the exact local Neron-symbol calibration theorem needed for its vanishing:\[q^{\raw}_{E,x,\spec}(v,w)=\lambda^{\Ner}_{E,x}\bigl(B_E^{\arith}v,B_E^{\arith}w\bigr)\quad\text{for every place }x.\] It proves that exact local Neron-symbol calibration implies\[E_H^{\ver}=0,\]regulator agreement\[\Reg_{\spec}(E)=\Reg(E),\]and, together with the source-complete Mordell--Weil bridge front, the vanishing of the raw BSD-IV verification ledger\[R^{\ver}_{\mathrm{BSDIV}}(E)=(0,0,0,0,0)\]under the stated calibration hypothesis. The manuscript is status-explicit. It does not claim to complete the full Birch--Swinnerton--Dyer conjecture, nor does it close BSD-IV by a normalization convention. Its purpose is to isolate the final height-comparison front after the Mordell--Weil bridge reduction and to make the exact local Neron-symbol calibration burden referee-readable.