Corollary: Time-only readings of the scale tensor

Assume Gate V. In the time-only reading and in the time-only ${\chi }_{\mathrm{I}}\equiv 1$ reading of Section sec:to-timeonly, for every $\mu >0$ and every admissible $\varpi$, the collapsed scale tensor ${\mathrm{Collapse}}_t(C_{\mathrm{scale}}(\mu ))$ is not variational. (The proof combines the archived top-order coordinates with the exact kernels of Theorem thm:to-timeonly and Proposition prop:to-chiplateau; it is complete relative to the reported computations of Lemma lem:to-comp; audit 3 review.)

Proof. The top-order part $C_{\mathrm{top}}$ of $C_{\mathrm{scale}}$ is a nonzero vector of the $D=4$ block (all its slots have $|\gamma |=N$, hence $D=4-N+|\gamma |=4$), and its $N=0$ coordinate is $-2i{\pi }^2{c}^2\ne 0$ (archived in wpB/_wpB_state.json; re-checked by regression_M.py). $H_t$ is homogeneous in $D$ (Lemma lem:to-comp: the blocks have disjoint rows), so the $D=4$ component of $H_t(C_{\mathrm{scale}})$ is $H_t(C_{\mathrm{top}})$, whatever the $\varpi$-dependent lower-order parts are. In the time-only reading $\ker H_t$ is $0$ at $D=4$ (Theorem thm:to-timeonly), so $H_t(C_{\mathrm{top}})\ne 0$. In the time-only ${\chi }_{\mathrm{I}}\equiv 1$ reading $\ker H_t$ at $D=4$ is spanned by vectors supported on $N=4$ (Proposition prop:to-chiplateau); $C_{\mathrm{top}}$ has a nonzero $N=0$ coordinate, so again $H_t(C_{\mathrm{top}})\ne 0$. By Lemma lem:helm, applied with the time-only adjoint of Lemma lem:to-intertwine, a collapsed tensor with nonzero adjoint defect is not an Euler tensor.