canonicalRecognitionFactorization
plain-language theorem explainer
Canonical factorizable joint substrate on the binary eight-tick tensor product: both matter and channel factors are the substrate recognition update (cyclic shift), and the joint operator is their pure-tensor product map. Track 2.C gravity certificates cite it as the standard inhabited witness of the factorization structural axiom. The definition is a direct structure instance packing the cyclic joint operator with its pure-tensor factorization proof.
Claim. The canonical factorizable joint substrate on $H\otimes_{\mathbb{C}} H$ (with $H$ the eight-tick signal space) takes the joint $\mathbb{C}$-linear operator to be the tensor product of the cyclic recognition update with itself, takes both the matter-side and channel-side factor maps to be that same recognition update, and supplies the pure-tensor factorization witness that the joint map acts factorwise on elementary tensors.
background
Track 2.C studies gravitational channel response on the binary-tensor joint substrate $H\otimes_{\mathbb{C}} H$, where $H$ is the eight-tick signal space forced by the T0–T8 chain (period $2^3$ at $D=3$). A factorizable joint substrate is the named structural hypothesis that the joint operator factors on pure tensors through independent matter and channel responses: no cross-sector coherence at the operator level, matching the master-plan setup $R_{\mathrm{joint}}=R_{\mathrm{matter}}\otimes R_{\mathrm{channel}}$.
The substrate recognition update is the one-tick cyclic shift on $H$ (equivalently the projector-after-shift recognition operator). Sessions 85–87 establish the single-factor dichotomy, the joint lift, and the substrate-side closure: under factorization plus recognition update on the matter side, a nontrivial channel factor must be amplitude-linear; density-only responses collapse to zero.
This definition supplies the canonical inhabited instance of that structural package, with both factors equal to the recognition update and joint operator the corresponding tensor-product map.
proof idea
Pure structure-instance definition, not a proof. The four fields of the factorizable-joint-substrate record are filled directly: joint operator is the canonical cyclic joint operator (tensor product of cyclic-shift linear maps); matter and channel factors are both the recognition update; the factorization field is the already-proved pure-tensor factorization lemma for that joint operator. No tactics or further lemmas are invoked at this site.
why it matters
This is the standard witness that the Track 2.C factorization axiom is inhabited by the recognition dynamics forced from T0–T8. It is consumed immediately by the canonical recognition-coupled factorization, which adds the master-plan constraint that the matter side equals the substrate recognition update and thereby feeds the Track 2.C master theorem: under a recognition-coupled factorizable joint substrate, the channel-side response is forced amplitude-linear.
In framework terms it upgrades paper IV's T2 from a MODEL tag to a STRUCTURAL THEOREM conditional on factorization: the amplitude-linear identification of the gravitational channel is a substrate consequence rather than a modeling choice. The remaining open step, flagged in the module doc, is the unconditional lift that either rederives factorization from a stricter substrate axiom or removes the factor-product hypothesis entirely.
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