T0T8ConsistentSubstrate_inhabited
plain-language theorem explainer
The space of T0–T8-consistent substrates is nonempty. An explicit witness is the canonical recognition-coupled factorization with cyclic-shift updates on both matter and channel factors. Track 2.D gravity work cites this to ground the no-classical-mediator no-go on a concrete model rather than an empty hypothesis class. The proof is a one-line term that packages that canonical witness.
Claim. The type of T0--T8-consistent substrates is inhabited: there exists a recognition-coupled factorization in which the matter side is the recognition cyclic-shift update, the joint substrate is the binary tensor product, and the joint operator factorizes on pure tensors.
background
Track 2.D of the quantum-gravity plan closes a substrate-internal no-go: under the T0–T8 forcing chain, no nontrivial CPTP-classical (density-only) gravitational channel response is admissible. The module status is a structural theorem with zero sorry and no RS-internal axioms.
A T0–T8-consistent substrate is defined as a recognition-coupled factorization: matter side is the recognition update cyclic_shift, the joint substrate is the binary tensor product, and the joint operator factorizes on pure tensors. Upstream, T0–T8 (UnifiedForcingChain) force eight-tick discreteness (T7), spatial dimension $D=3$ (T8), and $\varphi$-self-similarity (T6), so the matter side is the recognition substrate rather than an arbitrary continuum or collapse model.
Track 2.C already showed that, on any such recognition-coupled factorizable joint substrate, a channel response cannot be both nontrivial and density-only. This declaration only settles that the hypothesis class is not vacuous.
proof idea
One-line term proof of inhabitation. The constructor of Nonempty is applied to the explicit term canonicalRecognitionCoupled, the canonical recognition-coupled factorization (cyclic-shift on both matter and channel factors). No tactics or intermediate lemmas are needed beyond that witness.
why it matters
Without an inhabited substrate class, the Track 2.D no-go would be vacuously true and scientifically empty. This theorem supplies the concrete RS witness so downstream certificates can quantify over real models.
It feeds noClassicalMediatorCert, which packages three claims: no density-only channel under T0–T8, no T0–T8 substrate with a nontrivial classical mediator, and forced amplitude-linear channel response. That certificate is the reviewer-facing composite for the headline that CPTP-classical mediator hypotheses are incompatible with T0–T8 (via the binary-tensor factor-product axiom of Track 2.C).
Framework landmarks: T7 eight-tick octave and T8 $D=3$ pin the matter side; the argument answers the objection that Bohmian or Diósi–Penrose substrates sit outside RS by noting those axioms already violate T0–T8 (continuous trajectories vs T2 discreteness; stochastic collapse vs T1 ledger linearity).
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.