no_classical_mediator_under_T0T8
plain-language theorem explainer
Under any T0–T8-consistent substrate, a density-only (CPTP-classical) gravitational channel response is forced to vanish on every eight-tick signal. Gravity and quantum-channel workers cite this as the Track 2.D no-classical-mediator statement. The proof is a one-line application of the Track 2.C density-only impossibility theorem once the substrate is identified with a recognition-coupled factorization.
Claim. Let $F$ be a substrate consistent with the T0–T8 forcing chain (matter side the recognition update, joint substrate the binary tensor product, joint operator factorizing on pure tensors). If the channel response $R_C$ of $F$ is density-only—i.e., $R_C(c\cdot\psi)=R_C(\psi)$ whenever $\|c\|=1$—then $R_C(\varphi)=0$ for every eight-tick signal $\varphi$.
background
Track 2.D of the quantum-gravity master plan asks whether a nontrivial CPTP-classical gravitational mediator can sit inside a substrate forced by T0–T8. The forcing chain (Foundation.UnifiedForcingChain) pins the matter side to the recognition substrate Signal8 with the cyclic-shift update (T7 eight-tick octave, T8 spatial dimension three, T6 $\varphi$-self-similarity). A T0–T8-consistent substrate is defined here as a recognition-coupled factorization: matter equals the recognition update, the joint space is the binary tensor product, and the joint operator factorizes on pure tensors.
A response is density-only when it is invariant under unit-modulus complex rescaling of the signal. That is the structural footprint of a CPTP-classical readout from the density matrix alone, since $|\psi\rangle\langle\psi|$ is phase-invariant. Track 2.C already proved that, on any recognition-coupled factorizable joint substrate, a density-only channel response collapses to the zero map. The present theorem simply specializes that closure to the T0–T8-named substrate type.
proof idea
Term-mode one-liner. Introduce an arbitrary eight-tick signal $\varphi$, then apply track2C_channel_eq_zero_of_density_only to the underlying recognition-coupled factorization of $F$, the density-only hypothesis, and $\varphi$. No extra algebraic work: T0T8ConsistentSubstrate is definitionally RecognitionCoupledFactorization, so the Track 2.C hypothesis matches exactly.
why it matters
This is the core Track 2.D no-classical-mediator theorem: under T0–T8, no nontrivial density-only gravitational channel is admissible. It feeds the one-statement packaging (no_classical_mediator_one_statement), the existential no-go (no_T0T8_substrate_with_nontrivial_classical_mediator), the contrapositive nontrivial-impossibility form, and the certificate bundle noClassicalMediatorCert. Downstream, rs_amplitude_channel_unique in the BMV falsifier band quotes it as the density-only half of the claim that the RS gravitational channel is uniquely amplitude-linear.
Framework landmarks: T0–T8 forcing (especially T6 $\varphi$, T7 eight-tick, T8 $D=3$) uniquely names the RS substrate, so Bohmian continuous trajectories and Diósi–Penrose stochastic collapse are excluded as substrate axioms rather than merely as rival channel models. The module status is structural theorem (zero sorry, zero RS-internal axiom). Remaining scope is the broader master-plan Track 2.D closure beyond the substrate-internal no-go.
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