Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f
Exact companion targets and their stops
Supported replacement of flat-completion operators › Exact companion targets and their stops · lines 1885–2000
The following ledger records permitted operator replacement, not a new proof of each cited companion theorem. For the original real scalar contracts take . A twisted charged scalar requires its corresponding complex-bundle formulation.
{}1.13
| Source and label | Retained scalar content replaced | Stop |
|---|---|---|
| Source and label | Retained scalar content replaced | Stop |
| X-B-ST, :retarded CheungXBST | The linear retarded-response test, its support, normalization, and continuity, with the same local coefficient operator and Hessian sign. | No transfer of :hbar, positive- claims, a quantum product extension, or a global equation-quotient identification. |
| X-C-ST, :maps, :support CheungXCST | The stated scalar response coefficients and finite iterations on actual smooth compact intermediate tests, provided every Green operation remains in the retained shadow and window. | No second-order claim outside , and no construction of a general or retarded-product hierarchy. |
| X-D-ST, :propagation, :convergence CheungXDST | Fixed-support propagation and sharp-width convergence of representative scalar tests on the retained interval, with uniform support and window reserves. | No use of global Huygens propagation or absence of late signals. No sharp-metric admission follows. The corresponding global free-field quotient is not identified by this paper. |
| V-B, :raw CheungVB | The bulk retarded scalar-wave contribution to the raw current, when its component calculation is within the contract. | The initial-surface term requires its own trace and distributional check. Additional image fronts are not removed. |
| IX-A-ST, :support, free evolution CheungIXAST | The retained smooth scalar Cauchy evolution inside , with its strict lateral margin. | Boundary-sensitive admission is not a conical Hadamard-state construction. |
| IX-B-ST and IX-C-ST, free scalar parts CheungIXBST,CheungIXCST | Only scalar Green operations satisfying the declared supported contract. | The triangle calibration, partial products, presentation relations, and quantum closure are not consequences of this replacement. |
| III-B, :Cauchy CheungIIIB | The supported laboratory scalar solution through the replaced evolution and its declared data. | Not a global conical Cauchy-surface theorem and not an enlargement of its relational domain. |