The Chiral Topological Ledger: Gravitational-Anomaly Baryogenesis and Infrared Tensor Matching

We formulate a chiral tensor-ledger framework in which a parity-odd primordial tensor sector sources charge asymmetry through the anomalous Ward identity of the active chiral spectrum. A helicity-asymmetric tensor background, hR ≠ hL, produces a nonzero expectation value of the gravitational Pontryagin density, ⟨R R̃⟩. This quantity enters the mixed gravitational anomaly and can source a pre-sphaleron B−L asymmetry. Electroweak sphalerons subsequently convert this asymmetry into a baryon asymmetry according to the standard equilibrium conversion factor. For the Standard Model field content without active right-handed neutrinos, the active-spectrum coefficient is Cgrav (B−L) = −3; if anomaly-canceling right-handed neutrinos are active during the relevant epoch, the gravitational B−L source cancels. The baryogenesis calculation is therefore reduced to one dynamical input: the helicity-resolved tensor history hR, L (k, τ). Once this history is specified by a parity-violating tensor completion, the anomaly coefficient, thermal washout rate, and sphaleron conversion determine the surviving baryon yield. We derive the corresponding helicity integral required to reproduce ηBᵒbs, thereby converting the mechanism into a falsifiable target correlated with parity-odd CMB observables and a helicity-resolved stochastic gravitational-wave background. As a separate infrared application, we show that a late-time macroscopic tensor remnant admits an exact matching target: an auxiliary tensor dielectric with cubic norm potential V (H) = cH^ (3/2) integrates out to reproduce the deep-MOND scalar functional. Together, the two branches define a claim-tiered tensor-ledger architecture: the baryogenesis branch is a falsifiable anomaly-source condition, while the infrared branch is an independent matching target for possible late-time tensor response.