track2CCert

Track 2.C studies gravitational-channel responses on the eight-tick signal space (the binary-tensor model $S_8\otimes_{\mathbb{C}}S_8$). Two structural predicates compete: amplitude-linearity (complex-homogeneous, phase-equivariant response) and density-only behaviour (CPTP-classical density-matrix readout). Session 85 proves they are incompatible on a single factor except at the zero map.

Sessions 86–87 lift the story to a joint substrate. A recognition-coupled factorizable joint substrate is a $\mathbb{C}$-linear joint operator that factorizes on pure tensors through factor-wise responses, with the matter factor fixed to the substrate recognition update (cyclic shift). Under that axiom the channel factor is forced amplitude-linear, and any density-only channel collapses to zero.

The certificate structure packages five theorem-grade clauses: the single-factor dichotomy, channel amplitude-linearity, density-only impossibility, the existence-form no-go, and existence of a canonical witness. The module doc states this upgrades paper IV's T2 from a modeling choice to a structural theorem, still conditional on the named factorization axiom.

One-line structure inhabitant. Each field is filled by a named upstream result:

• single-factor dichotomy is eq_zero_of_isAmplitudeLinear_isDensityOnly (Session 85: test amplitude-linearity against density-only at the unit-modulus scalar $c=-1$);
• channel amplitude-linearity is track2C_channel_isAmplitudeLinear (forcing direction under recognition coupling);
• density-only impossibility is track2C_channel_eq_zero_of_density_only;
• no nontrivial density-only channel is track2C_not_exists_nontrivial_density_only_channel (existence-form no-go via the previous lemma);
• canonical witness is the pair ⟨canonicalRecognitionCoupled⟩, the cyclic-shift factorization on both factors.

No new reasoning occurs; the def only assembles already-closed anchors.

This is the aggregation point for Gravity Track 2.C in the binary-tensor model. Downstream, track2CCert_inhabited is the one-line Nonempty witness built from this def, so any consumer that needs a single inhabited Prop bundle cites here.

Per the module doc, the package upgrades paper IV's T2 from MODEL to STRUCTURAL THEOREM: identifying the gravitational channel with the amplitude-linear extension is no longer a modeling choice but a substrate consequence, conditional on the factorizable joint-substrate axiom (matter and channel sectors evolve under independent dynamics that do not mix at the operator level). Density-only CPTP-classical readouts are ruled out except at zero.

The anti-retreat principle is not yet fully met: the binary-tensor structural theorem is strictly stronger than MODEL but strictly weaker than an unconditional theorem. The lift to general joint operators on $S_8\otimes_{\mathbb{C}}S_8$ without the factor-product assumption remains future work. Within RS landmarks this sits on the eight-tick octave (T7) substrate used throughout the gravity channel development.