canonicalT0T8JointDynamics_recognitionUpdate_not_density_only
plain-language theorem explainer
The recognition update of the canonical T0-T8 joint dynamics is not density-only as a physical channel response. Cite this for the concrete unconditional no-go witness in Gravity Track 2.C: a nontrivial amplitude-linear channel that density-only maps cannot match. Proof is a short contradiction using nontriviality of the update against the density-only collapse-to-zero lemma.
Claim. The recognition update of the canonical T0--T8 joint dynamics is not density-only: it cannot arise as a physical channel response that depends only on densities on the eight-tick signal space.
background
Gravity Track 2.C closes the unconditional T0-T8 substrate-semantic forcing of amplitude-linearity for physical channel responses. The joint substrate is $J = S_8 \otimes_{\mathbb{C}} S_8$ (T7 eight-tick factors with tensor coupling). Joint dynamics is a $\mathbb{C}$-linear endomorphism $R_J : J \to_{\ell} J$, the joint lift of Schrödinger linearity. A physical channel response is any map $R_C$ obtained by substrate-local access: prepare a matter probe, apply $R_J$, extract a channel coordinate, and calibrate by a nonzero scalar. Composition of linear maps forces every such $R_C$ to be amplitude-linear.
Density-only responses are the competing structural class: maps that factor through densities rather than amplitudes. Upstream in this module, any density-only physical channel response is forced to the zero map, and the recognition update is exhibited as a physical channel response of the canonical T0-T8 joint dynamics and shown amplitude-linear and nontrivial ($(R,1),0 = 1 \neq 0$). The single-factor dichotomy then excludes density-only status.
proof idea
Term-mode contradiction. Assume the recognition update is density-only. Pull a nontriviality witness $(\psi_0, i_0)$ from the recognition-update nontriviality lemma. Feed the density-only hypothesis, together with the fact that the recognition update is a physical channel response of the canonical T0-T8 joint dynamics, into the collapse lemma that every density-only physical channel response vanishes on every probe. Rewrite to obtain the zero response at $\psi_0$, contradicting nontriviality. Close by reflexivity on the residual equality.
why it matters
This is the concrete no-go content for the canonical T0-T8 substrate dynamics: not merely that density-only channels collapse in general, but that the forced recognition update itself is a live, nontrivial counterexample to density-only status. It feeds the master certificate physicalChannelAmplitudeLinearCert (as part of the canonical witness bundle) and the Session-127 one-statement unconditional closure, which packages amplitude-linearity of every physical channel response, density-only collapse to zero, the global no-go, and non-vacuous inhabitation by the canonical joint dynamics.
Framework landmarks: T7 supplies the eight-tick factors of the joint carrier; T0-T8 substrate semantics alone discharge the last structural hypothesis (IsAmplitudeLinear) that Sessions 85-126 had left on the density-only no-go. Track 2.C is thereby closed with zero sorry and no RS-internal axiom.
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