Cosmological Redshift without Expansion: Multiplicative Beat Accumulation in a Topological Vacuum

Preprint — not peer reviewed. Fourth paper of a series. In the framework introduced in the first paper, the vacuum is a topologically structured medium with a spatially uniform breathing mode, and light is the half-frequency parametric companion locked to that breathing. This paper follows the locked propagation over cosmological path lengths, with space static throughout: no expansion, and consequently no dark energy, is invoked at any point. The central results: (i) a classification theorem — observed achromaticity together with per-beat composition forces the step map to be linear, hence the accumulation multiplicative, giving 1+z = e^(D/R_H) with the Hubble law as its low-z limit and H_0 = c/R_H as a conversion fixed by one calibration, in structural analogy to c and G; the distance law carries zero free functions and zero shape parameters. (ii) A waveform dilation theorem — redshift and (1+z) time dilation are Fourier-dual aspects of one coherent spectral scaling; coherence itself is the dividing line from the historical tired-light class, which produces redshift without dilation and is excluded by supernova light curves. (iii) A coherence theorem — the redshift-driving pump is spatially uniform, so the per-beat deflection vanishes by translation symmetry: blur does not scale with redshift, confirmed by a numerical single-mode confinement test. (iv) The photometric sector follows with no adjustable content: a Tolman surface-brightness exponent whose broadband form −1.3 lands on the JWST-era measurement −1.28 ± 0.05 with zero parameters (stated as an economy result, not a refutation of expansion, since evolution within the standard model can reach the same data); a distance-duality slope of 1 against the measured 1.17 ± 0.09; and, against the Type Ia supernova Hubble diagram, the same rigid curve reproduces the qualitative dimming that motivated dark energy while leaving a residual 0.1–0.2 mag shape difference relative to ΛCDM that survives our processing-step tests and is registered openly as the sharpest current exposure of the proposal. What this paper does not claim is stated in the body: no metric field equations; structure formation, BAO, and nucleosynthesis open; the origin, absolute scale, and normalization history of the microwave background open (the paper uses only the spectral-shape covariance of T(z)); the effective propagation equation — the bridge from the microscopic field equation to the per-beat step — registered as owed. Declarations. The author is an independent researcher without institutional affiliation. The derivations and all numerical analyses new to this paper were proposed and executed by an AI assistant (Anthropic Claude) and adjudicated item by item by the author, who takes full responsibility for the content; this is declared in full within the manuscript. All analysis scripts are archived with the research record.