62 / C6-M: Carrier Completeness, Spectral/Pressure Visibility, and Nested-Rebinding Rigidity

The question C6-L handed off was: is low carrier visibility only because the current three labels ($TS$, $GP$, $HF$) are not enough, rather than the singular mass genuinely being uncapturable? C6-M is the first round to prove that $L^3$ is not the only carrier channel. Using a Littlewood–Paley decomposition, it builds the critical spectral phase-space probability measure $d\Sigma_n(q,x)=2^q|\Delta_qU_n|^2/\sum_j2^j\|\Delta_jU_n\|_2^2\,dx$, whose spatial marginal $\sigma_n$ is the $\dot H^{1/2}$ analogue of C6-L's mass measure $\mu_n$, and defines the spectral visibility $\Omega_{DH,n}=1-d_{TV}(\sigma_n,\eta_n)$. The Spectral Singular-Carrier Extraction Theorem (C6-M.1) proves that when $\Omega_{DH,n}\ge\omega_H>0$, one can extract a carrier that simultaneously carries a defect label and a fixed fraction of the diverging $\dot H^{1/2}$ critical energy, so an $L^3$-spectator profile can still be a $\dot H^{1/2}$-visible carrier — the Labeled Spectral-Carrier Trichotomy (C6-M.2) splits each label into four classes: V33 (visible in both channels), V3 ($L^3$ only), VH (spectral only), and V0 (Strong Spectator). It then also promotes pressure to a formal carrier channel: using the Calderón–Zygmund pressure kernel it builds a directional far-field source functional, defines the pressure capacity $C_P=\int|a_P|$ and coherence $\Gamma_P=R_P/C_P$, the Pressure Alignment Identity (C6-M.3) and the aligned-source probability $\pi_P^+$, and then, taking the overlap $\Omega_{3P}^+$ with $\mu_3$, the Pressure-Coherent Singular-Carrier Theorem (C6-M.4) is the series' first pressure-channel singular-carrier extraction theorem. But the pressure side is not unlimited: the Separated Far-Pressure Capacity Bound (C6-M.5) proves $C_P^{far}\lesssim d^{-5}\|v\|_2^2$, giving quantitative pressure decoupling for a profile that is sufficiently far away (distance $d$) and $L^2$-bounded — narrowing the spectator loophole down to profiles that are near the core, insufficiently separated, or carry high-weight pressure capacity. The other half of the round attacks the sub-scale-rebinding problem left by C6-L: it proves the exact retention identity $\beta_m=\beta_0\prod_{j