Manuscript XVI-D@1 · XVI-D.tex · section · b73c6f4a
Status and minimal next steps
Status and minimal next steps · lines 1529–1615
XVI-D@1#prob:legs
{}1.1
| Item | Exact status |
|---|---|
| All-smooth zero-data uniqueness | Proved locally without spatial decay: Lemma lem:uniqueness. |
| Source support and compact-domain distinction | Proved for : Proposition prop:source-domain. At the source is zero. |
| Polarization and complexification | Proved from diagonal response existence and local uniqueness: Lemmas lem:realB and lem:BC. |
| Witness response and traceless contraction | Proved: Propositions lem:identify and lem:contraction. |
| Scalar completion | Verified on compact-interior forcing: Proposition lem:scalar. |
| Coordinate Cauchy problem | Candidate exists and is unique: Proposition lem:Xcand. No nonexistence is asserted. |
| Restricted auxiliary extension | Constructed: Proposition prop:restricted-extension. Not silently identified with inherited notation. |
| Inherited coordinate identity | Conditional: Proposition prop:coordinate-compatibility. The compact-source theorem alone is insufficient. |
| Root and auxiliary witness values | XVI-E supplies the root reading; XVI-F-v3 proves . The obsolete unresolved residual is not retained. |
| High-frequency estimates | No new full inherited-insertion estimate here. Later fixed-data scalar estimates are recorded separately. |
| Class kernel and quantum admission | Not constructed by this manuscript; programme unresolved. |
Verify, for the inherited retarded prescription used in (eq:completion-legs), one of the following:
an applicable smooth-source definition or extension that gives a smooth zero-data solution at ;
agreement there with the explicitly constructed restricted extension; or
a larger-domain theorem covering the real response family needed before polarization.
The new restricted extension already proves that such an extension is mathematically possible on the witness source space. The remaining question is its match to the inherited prescription and, when required, the prepolarization family.
The hypothesis of Proposition prop:coordinate-compatibility is sufficient for the direct witness reading. Family-level polarization additionally requires the relevant diagonal coordinate fields and linearity on their domain.
Status of that inherited identification in this manuscript: Unresolved.