Branch-Weight Gravitational Witnesses: Cumulants, Coherence Recovery, and Classical-Record Bounds

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Abstract
Varying the probabilities of the branches in a massive spatial superposition provides a controlled way to test gravitational response models. We develop this comparison for fixed source configurations and calibrated probe measurements. The between-branch covariance obeys a pairwise-distance law on the probability simplex and has rank at most one less than the number of occupied branches. Higher cumulants and ensemble-size scans constrain the same response model, although their statistics can also be produced by classical branch mixtures. Coherence recovery adds information when its permitted operations and accessible records are specified independently. For a branch-preserving measurement that produces an irreversible classical record and drives a common feed-forward response, we derive an upper bound on recovered source coherence in terms of record overlaps. A calibrated response diameter converts this into a response--recovery inequality. A second bound relates recovery to the conditional response variances and remains applicable to unbounded feedback with finite second moments. An explicit binary model saturates both inequalities. Coherent momentum transfer provides a contrasting example in which the probe statistics remain branch resolved while an eraser measurement restores source coherence. These comparisons require separate measurements of conditional probe noise, trajectories, and calibration parameters. The bounds apply to the stated classical-record channel; their application to a gravitational theory requires a derivation of that channel from the theory. Branch-weight scans therefore support quantitative tests of specified models without providing a theory-independent criterion for quantized gravity.

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