no_classical_mediator_one_statement
plain-language theorem explainer
Under the T0–T8 forcing chain, every consistent gravitational channel response is amplitude-linear, and any density-only (CPTP-classical) response collapses to the zero map; equivalently, no nontrivial classical mediator exists on such a substrate. Quantum-gravity and channel theorists cite this as the Track 2.D one-statement headline. The proof is a three-component term packaging the amplitude-linear forcing, the density-only collapse, and the existential no-go.
Claim. For every substrate $F$ consistent with the T0--T8 forcing chain (recognition-coupled factorizable joint substrate), the gravitational channel response $R_C$ is amplitude-linear: it agrees with some $\mathbb{C}$-linear map on eight-tick signals. Moreover, if $R_C$ is density-only (invariant under unit-modulus phase multiplications), then $R_C(\varphi)=0$ for every signal $\varphi$. Equivalently, there is no T0--T8-consistent substrate whose channel response is both density-only and nontrivial.
background
Track 2.D of the quantum-gravity master plan asks for a substrate-internal no-go: under T0–T8, no nontrivial CPTP-classical gravitational channel is admissible. The local setting is a structural theorem (zero sorry, zero RS-internal axiom) that composes T0–T8 substrate forcing with the Track 2.C factorizable-joint closure.
A T0–T8-consistent substrate means: the matter side is the recognition update (cyclic shift on the eight-tick ledger Signal8), the joint substrate is the binary tensor product, and the joint operator factorizes on pure tensors. Amplitude-linearity means the channel response agrees with some $\mathbb{C}$-linear map, so it preserves coherent superpositions. Density-only means invariance under unit-modulus complex scalings: the structural footprint of a CPTP-classical readout from the density matrix alone.
Upstream, the forcing chain (T0–T8) pins eight-tick discreteness (T7), $D=3$ (T8), and $\varphi$-self-similarity (T6), forcing the matter side onto the recognition substrate. Track 2.C already shows that on a recognition-coupled factorizable joint, a channel cannot be both nontrivial and density-only.
proof idea
Term-mode packaging of three already-proved siblings into a single conjunction. The first conjunct is channel_forced_amplitude_linear_under_T0T8, which applies the Track 2.C amplitude-linearity result to every T0–T8-consistent substrate. The second is a thin lambda that feeds density-only hypotheses into no_classical_mediator_under_T0T8, itself a one-line appeal to the Track 2.C zero-response lemma. The third conjunct is exactly no_T0T8_substrate_with_nontrivial_classical_mediator, the existential form of the same no-go. No new algebra is performed; the declaration is the reviewer-facing product of those three facts.
why it matters
This is the Track 2.D one-statement theorem (partial closure form) named in the module doc and the quantum-gravity master plan §4. It bundles the positive claim (channel forced amplitude-linear) with the negative claim (density-only responses are trivial) and the existential no-go, so a single citation carries the full headline.
Framework landmarks in play: T0–T8 forcing (especially T7 eight-tick octave and T8 spatial dimension $D=3$) uniquely identify the RS recognition substrate, so alternative substrates are not free parameters. The module doc uses this to answer the reviewer objection that Bohmian or Diósi–Penrose models sit outside RS: continuous Bohmian trajectories violate T2 discreteness, and stochastic Diósi–Penrose collapse violates T1 ledger superposition preservation. Under T0–T8 the substrate axiom and a nontrivial classical mediator are incompatible.
No downstream Lean users are recorded yet; the declaration is the terminal packaging for Track 2.D partial closure rather than an intermediate lemma.
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