amplitudeLinearForcedStructuralCert_inhabited
plain-language theorem explainer
The structural certificate for amplitude-linear forcing on the gravity quantum channel is inhabited. Under the canonical recognition-coupled factorization (factor-product joint substrate with cyclic shift on the matter side), the channel response is forced amplitude-linear and any density-only response collapses to zero. Master-theorem consumers cite this to discharge the amplitude-linear hypothesis input. The proof is a one-constructor term wrapping the prebuilt structural certificate.
Claim. The structural certificate is nonempty: there exists a record asserting that the channel response $R_C$ of the canonical recognition-coupled factorization is amplitude-linear, that if $R_C$ is density-only then $R_C\varphi=0$ for every eight-tick signal $\varphi$, and that the master-theorem hypothesis of unconditional amplitude-linear forcing is witnessed by that same structural content.
background
Track 2.C/2.D in the gravity quantum-channel layer asks when a recognition-coupled channel response is forced to be amplitude-linear rather than density-only. The local setting is a named factor-product hypothesis: a joint substrate that factors, with the recognition update (cyclic shift) acting on the matter side. The canonical recognition coupling supplies an explicit such factorization.
The certificate structure packages three claims: amplitude-linearity of the canonical channel response $R_C$; the collapse implication that any density-only $R_C$ is the zero map on eight-tick signals; and an inhabitant of the master-theorem hypothesis input for unconditional amplitude-linear forcing. Upstream work (Sessions 85-88, 94) already proved the substantive forcing and collapse under that factorization; this module only packages them as a structural witness.
The eight-tick signal type and the spatial dimension $D=3$ sit in the usual RS forcing chain (T7 octave, T8/T9 dimension). The module status is structural theorem: zero sorry, zero RS-internal axiom, with the factor-product hypothesis still present.
proof idea
One-line term proof. The certificate value is already constructed in-module from the canonical recognition coupling and the Session 88/94 forcing results; the theorem simply wraps that value in the Nonempty constructor. No new algebraic work occurs here.
why it matters
This declaration is the inhabited structural witness that the gravity master theorem can consume for its amplitude-linear hypothesis input. It closes the packaging step for Track 2.C/2.D under the canonical recognition-coupled factorization: channel response forced amplitude-linear, density-only responses collapsed to zero, and the master hypothesis record filled.
No downstream dependents are wired yet; the immediate consumer is the Session 97 master theorem hypothesis list. Framework landmarks in play are the eight-tick octave (signal domain) and the recognition update on the matter factor. The fully unconditional Track 2.C/2.D closure, which would retire the factor-product hypothesis entirely from a stricter substrate axiom, remains explicit future work. Anti-retreat is satisfied because the witness names the factorizable joint-substrate hypothesis rather than hiding it.
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