Self-Referential Fibration Figure 3: Order Effects of Cosmological Comoving Truncation—Window-Operator Commutator and Fisher Adjudication

Whether the order of multilayer-window composition affects cosmological power
spectrum estimation is a question that can be adjudicated separately by analytic
structure and by signal-to-noise readings. In the spectral-level operator framework,
the redshift window induces a Markov convolution whose kernel is the Fourier power
of the selection function, while the scale window induces the multiplier S2; the dif
ference between the two ordering protocols is given exactly by the commutator.
The difference spectrum has a closed convolution expression, and its entire struc
ture is determined by three inputs: the kernel, the scale window, and the input
spectrum; there are no residual terms beyond this structure. The condition for
the commutator to vanish is that S2 is constant within the effective bandwidth of
the kernel, rather than that the scale window is globally trivial. The sup norm
of the commutator is controlled by a local bound that separates the accounting
of the kernel’s sidelobe mass from the oscillation amplitude of the scale window
within the kernel bandwidth. Under the parameterized simulation setup, the flat
region single-bin signal-to-noise reading is below the resolvable threshold at the
existing survey volume, and exceeds the “clearly resolvable” decision line at the
next-generation volume (this decision-line reading is a parameterized conditional
reading; its registration and backfill channel are given in Sec. 4.1 and Appendix
B.1, and it is not cited separately before backfill is complete); the critical volume is
algebraically inferred back from the volume scaling of the signal-to-noise ratio. The
peak position falls near the upper bound of the decision window, and its quantita
tive attribution must pass window-scan acceptance; before acceptance is complete,
the peak scale and the critical volume are window-dependent readings, not physical
constants independent of the prescription. The observability conclusion is sensitive
to the single-bin prescription, Gaussian error bars, the choice of decision line, and
the scale-window parameterization registration; the closure of the structural layer
and the numerical shift of the decision layer must be cited separately