channel_eq_zero_of_density_only_of_arisesFromSubstrateAccess
plain-language theorem explainer
Any density-only channel response harvested from a linear joint operator by substrate measurement-access is identically zero on Signal8. Gravity-track workers cite this when ruling out classical density-matrix readouts under the Session 113 locality principle. The proof is a one-line composition: substrate access forces amplitude-linearity, which with density-only yields the zero map.
Claim. Let $R_J$ be a $\mathbb{C}$-linear operator on the joint substrate $\mathrm{Signal8}\otimes_{\mathbb{C}}\mathrm{Signal8}$, and let $R_C:\mathrm{Signal8}\to\mathrm{Signal8}$ be a channel response. If $R_C$ arises from substrate access of $R_J$ (i.e., equals the induced channel for some access data) and $R_C$ is density-only (invariant under unit-modulus phase multiplications), then $R_C\varphi=0$ for every $\varphi\in\mathrm{Signal8}$.
background
Gravity Track 2.C closes the measurement-access principle on the joint substrate. The joint substrate is the binary tensor product $\mathrm{Signal8}\otimes_{\mathbb{C}}\mathrm{Signal8}$: first factor matter ledger, second factor channel ledger. Substrate access means there exist access data (matter reference $\psi_0$, coordinate $i_0$, calibration $\chi\neq 0$) such that the operational channel equals the induced formula $R_C\varphi=\chi^{-1}\cdot\mathrm{extractSecond}_{i_0}(R_J(\mathrm{insertFirst},\psi_0,\varphi))$. That induced map is automatically a joint section readout, so section readout is definitional rather than assumed.
A response is density-only when $R(c\cdot\psi)=R\psi$ whenever $|c|=1$: the structural footprint of a CPTP-classical readout from $|\psi\rangle\langle\psi|$ alone. Amplitude-linearity is the competing structural condition forced by section readout of a linear joint operator. Upstream, the substrate dichotomy states that any map that is both amplitude-linear and density-only vanishes identically on Signal8 (tested at the unit-modulus scalar $c=-1$).
proof idea
One-line term proof. First apply isAmplitudeLinear_channel_of_arisesFromSubstrateAccess to the substrate-access hypothesis, obtaining amplitude-linearity of $R_C$. Then feed that together with the density-only hypothesis into the upstream dichotomy eq_zero_of_isAmplitudeLinear_isDensityOnly, evaluated at the given signal $\varphi$. No extra case analysis.
why it matters
This is the density-only collapse link in the Session 113 chain: substrate access $\Rightarrow$ section readout $\Rightarrow$ amplitude-linearity $\Rightarrow$ density-only forces zero. It feeds the existence-form no-go not_exists_nontrivial_density_only_channel_with_substrateAccess (no nontrivial density-only channel can arise from substrate access of a linear joint operator), the certificate field density_only_collapse inside substrateLocalAccessCert, and the packaged one-statement substrate_local_access_one_statement.
In the broader Recognition gravity track, it blocks classical density-matrix channel readouts once operational channels are required to be harvested locally from a linear joint operator on Signal8$\otimes$Signal8. Section readout (Session 111) is no longer an extra assumption; under substrate locality it is derived, and the factor-product theorem becomes a corollary. The module reports structural closure with zero sorry and zero RS-internal axiom.
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