Branch-Weight Gravitational Witnesses: Cumulant Geometry, Recoverability, and Finite-Record Classical Bounds

Most proposed table-top tests of gravity in the quantum regime use equal-amplitude spatial superpositions and seek either gravitationally induced entanglement or collapse-motivated loss of interference. This paper develops a complementary theory-classification framework in which branch weights are scanned as operational control parameters. A massive source is prepared in a two-branch state\[\ket{\psi_p}=\sqrt p\ket{L}+\sqrt{1-p}\ket{R},\]or, more generally, in an \(m\)-branch state\[\ket{\psi}=\sum_{a=1}^{m}\sqrt{p_a}e^{i\theta_a}\ket{a}.\]The elementary two-branch second cumulant \(p(1-p)(\Delta q)^2\) is treated as a baseline branch-resolution law, not as a quantum-gravity signature. The main object is an enlarged witness geometry combining mean response, branch-induced covariance and higher cumulants, operational recoverability, and excess-noise or irreversibility coordinates. The \(m\)-branch extension yields simplex and covariance-rank constraints; the \(n\)-particle extension distinguishes independent ensembles from collectively entangled branch states by \(n\) versus \(n^2\) scaling. Recoverability is formulated as a resource-theoretic witness convention: adaptive recovery may use probe instruments and classical feed-forward but may not introduce an independent branch-basis coherence resource; a separate joint-recovery functional captures coherent erasure using accessible source-probe correlations. Worked models illustrate deterministic expectation-value gravity, coherent branch-mediated interaction, effective irreversible branch-basis dephasing, and noisy classical measurement-feedback dynamics. A finite-record, calibrated measurement-feedback bound shows that, within that simulator class, reproducing nontrivial branch-conditioned response requires record distinguishability; the associated record overlaps bound adaptive recovery of source coherence, with response-calibrated corollaries when the feed-forward range is bounded. The resulting framework is not a new microscopic theory of gravity. It is a model-discrimination scheme for locating gravitational response models in a witness space generated by branch-weight, branch-number, and ensemble-size scans.