Abstract
A coarse description of a conservative amplitude dynamics is not determined by an energy compression alone. We develop a conditional account of refinement, unresolved contrasts, and effective preparation for finite weighted graphs and a supplied one-dimensional local chain. Norm conservation, covariance, and additive refinement permit different site measures. Repeated-value refinement closes exactly only when the measure-normalized external rates agree within each coarse block. When they do not, eliminating the complementary coordinates yields coherent memory and an initial-data forcing term, not stochastic noise or irreversible decoherence. A three-site example separates return to the coarse subspace at one time from autonomous coarse evolution. On a uniform pair-refined cycle, a contrast of vanishing norm retains finite energy feedback; a corrected nearest-neighbor generator has a fixed-band, finite-time second-order compressed estimate. Static elimination induces both stiffness and a generally nondiagonal metric. For uniformly positive nonuniform couplings, the normalized static graph has fourth-order low-energy eigenvalue accuracy, but its general prepared-state estimate is separately only first order. A contracting inverse series controls finite-range approximation, with additional costs for normalization and evolution. First-resolvent dressing improves an isolated low mode’s state and phase ceiling to fourth order under quantitative spectral separation. The contribution is the source-bound combination of these conditional statements and obstructions, not a new general Schur, projection-memory, or subspace-perturbation mechanism. Geometry, measure, couplings, preparation, clock, action scale, and any empirical readout interpretation remain supplied.
Where it sits in the release
The coarse-dynamics paper of the branch. Geometry, measure, couplings, preparation, clock, action scale, and any empirical readout interpretation remain supplied rather than derived.
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