IDRP-04: Source Transversality, Adjoint Synchronization, Singular Limit Kernels, Physical Burst Amplitude, and Cycle-IX Closure Audit

Source Transversality, Adjoint Synchronization, Singular Limit Kernels, Physical Burst Amplitude, and Cycle-IX Closure Audit

IDRP-03 reduced the remaining burst problem to four PDE obligations: TRAN, DUAL, KERN, AMP. This paper runs the Cycle IX closure audit; the result is not complete closure, but it proves: (1) source transversality from pressure alone is generally impossible — it constructs a smooth, Fourier-localized, symmetric source tensor whose active-pressure symbol is zero yet whose Leray-projected forcing is nonzero; (2) a stronger "universal forcing-pairing implies source defect" NO-GO: a large causal forcing pairing can come entirely from the clean/model source subspace, so a theorem about forcing-pairing magnitude alone cannot imply a native source-quotient distance; (3) when causal and audit adjoints are synchronized, dual compatibility holds exactly, and otherwise is controlled by the perturbation; (4) under an external anti-kernel hypothesis, the positive-covariance/energy direction cannot fill a stable singular PFET limit kernel; (5) the universal weak source-action budget is summable in scale, so it cannot by itself supply a physical critical lift. Cycle IX thereby reduces the surviving obstruction to an impulse phantom that is summable across tangential/desynchronization/singular-kernel/amplitude channels, formally handing off to NS-TSKR.