Small-Amplitude Coupled Backgrounds in Fixed-Ambient Normal-Graph Dynamics: Backreaction, Mixed Response Rank, and Local Identifiability

This paper develops exact small-amplitude coupled backgrounds within the fixed-ambient normal-graph realization of the OFT/VGM research programme. It follows the physical-admissibility framework of Paper XXII and the nonlinear local Cauchy theory of Paper XXIII, examining which perturbations a finite detector protocol can distinguish around a nonzero coupled solution. Small scalar-field Cauchy data generate a family of exact admitted solutions on a common time slab. Field-sign symmetry organizes the expansion: the scalar field begins at first order in amplitude, while the normal displacement begins at second order. The leading normal response is sourced by the trace of the scalar stress tensor, and the resulting metric perturbation produces the next scalar-field correction. A homogeneous massive mode on a de Sitter slab provides an explicit nonzero-backreaction example. Linearization about this family produces two off-diagonal mixing channels. A selected normal input, scalar input, proper-time clock, and compact scalar detector define a two-by-two operational response matrix. Nonzero vacuum diagonal channels imply persistence of rank two for sufficiently small background amplitude. An explicit short-time protocol shows that both diagonal and both mixed coefficients can be nonzero simultaneously. The paper separates response rank on a selected input plane from observability of the full field space. It constructs protocol-relative rank loss without loss of geometric or hyperbolic admissibility and derives the bound that N scalar readouts can regularly identify at most N independent local parameter combinations. Practical conditioning, larger detector families, and empirical calibration remain subsequent steps. This deposit contains the English manuscript, revised version 1.1 dated 8 November 2026, its XeLaTeX source, and the accompanying Peaceful Civilian Use Licence. Version 1.1 updates the published references to Papers XXII–XXIII, makes parameter-regularity and variational-normalization assumptions explicit, corrects the Cauchy-to-readout composition, and strengthens the rank-loss and mixed-channel arguments. Series: OFT/VGM Research Programme — Paper XXIV.DOI: https://doi.org/10.5281/zenodo.24059315