Track2CCert
plain-language theorem explainer
Master certificate bundling five theorem-grade clauses for Gravity Track 2.C in the binary-tensor model: amplitude-linear plus density-only forces the zero channel; recognition-coupled factorizable joints force amplitude-linearity on the channel factor; density-only responses collapse to zero; no nontrivial density-only coupled channel exists; and a canonical witness is nonempty. Gravity and quantum-channel workers cite it as the Sessions 85–87 aggregate. It is a structure of Props, inhabited by wiring the session anchors.
Claim. A master certificate is a record of five claims: (i) every map $R:\mathrm{Signal}_8\to\mathrm{Signal}_8$ that is both amplitude-linear (agrees with some $\mathbb{C}$-linear map) and density-only sends every state to zero; (ii) for every recognition-coupled factorizable joint substrate $F$, the channel factor $R_C$ is amplitude-linear; (iii) if such an $R_C$ is density-only then $R_C\equiv 0$; (iv) no recognition-coupled factorization has a density-only channel that is nonzero on some state; (v) the type of recognition-coupled factorizations is nonempty.
background
Track 2.C studies gravitational-channel responses on the eight-tick ledger space $\mathrm{Signal}_8$. A response is amplitude-linear when it coincides with a $\mathbb{C}$-linear map, hence preserves coherent superpositions. It is density-only when it depends only on classical density data (CPTP-classical), killing phase-sensitive interference.
The joint setting is the binary tensor product $\mathrm{Signal}8\otimes{\mathbb{C}}\mathrm{Signal}_8$. A factorizable joint substrate supplies a $\mathbb{C}$-linear joint operator that factors on pure tensors through matter and channel responses $R_M,R_C$. Recognition coupling fixes the matter side to the substrate recognition update cyclic_shift, the dynamics forced by the T0–T8 chain (single-site Schrödinger linearity).
Session 85 gives the single-factor dichotomy on $\mathrm{Signal}_8$. Sessions 86–87 lift to joint substrates and close the channel side under recognition coupling. The module aggregates those closures; the unconditional lift without factorization remains open.
proof idea
No proof body: this is a structure whose fields are named Props. The companion inhabitant track2CCert fills each field by direct appeal to the session anchors: single-factor dichotomy from eq_zero_of_isAmplitudeLinear_isDensityOnly; channel amplitude-linearity from track2C_channel_isAmplitudeLinear; density-only collapse from track2C_channel_eq_zero_of_density_only; the existence-form no-go from the corresponding nonexistence lemma; and nonemptiness from the canonical recognition-coupled witness (cyclic shift on both factors). track2CCert_inhabited is then ⟨track2CCert⟩.
why it matters
This is the Track 2.C master certificate in the binary-tensor model. Downstream, track2CCert and track2CCert_inhabited package Sessions 85–87 so callers can cite one inhabited bundle rather than four separate lemmas.
Per the module doc, it upgrades paper IV's T2 from a MODEL tag to a STRUCTURAL THEOREM: identifying the gravitational channel with an amplitude-linear extension is forced by the recognition-coupled factorizable substrate, not chosen as modeling. Density-only (classical CPTP) channels are ruled out except the zero map. That sits on the eight-tick octave (T7) and the recognition update fixed by the forcing chain.
The anti-retreat bar is not fully cleared: factorization is still an axiom, so the result is stronger than MODEL but weaker than an unconditional theorem on all joint operators. The integrated discovery claim still needs that lift.
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