33.34 turns lensed-gravitational-wave multi-image timing into an image–image residual court: only if δt_ij remains dispersion-free under band splitting, closes into stable image-level constants τ_i, tracks the environment ranking J_i, reproduces across waveform and lens-model pipelines, and beats band/image/environment permutations can the section pass; under V08-compatible tighten, this is only a severe multi-path residual ledger, not a lensing-cosmology or propagation verdict.
Use this section as a compact machine-readable EFT reference.
33.34 recasts lensed-gravitational-wave timing from a single fitted delay into an image–image residual sequence. The key is not a prettier fit, but whether δt_ij can be simultaneously dispersion-free, loop-closing, and environment-ordered; under compat adjudication, the chapter is tighten and serves only as a high-pressure residual court under the first-line path test.
The court compresses the chapter into five readouts: observed delays Δt_obs,ij, macro-model delays Δt_lens,ij, residuals δt_ij, image-level constants τ_i, and ray-path environment indices J_i. What matters is not one photogenic residual, but whether these quantities can be written as a stable τ_i − τ_j loop-closure structure while remaining consistent across low- and high-frequency bands.
The workflow runs two frozen tracks in parallel: a gravitational-wave timing track and an electromagnetic or host-localization lens-model track. Image labels, band labels, and environment labels stay blinded; J_i only produces sealed ranking cards, and measurement teams may not retune model families or closure tolerances after unblinding.
Support is narrow. Band-label, image-label, and environment-label permutations, injection–recovery runs, timing-ledger audits, and wave-optics microlensing controls must all destroy false structure, while the genuine candidate survives as dispersion-free, loop-closing, and environment-ordered in at least two waveform pipelines and two lens-modeling pipelines.
The hard boundary is triply severe: band inconsistency, closure failure, or no stable τ_i–J_i link each kills the claim. Macro-model degeneracies, time-delay-distance freedom, detector timing errors, and frequency-dependent microlensing are old ledgers that must be allowed to fully absorb the residual before anything remains admissible.
The chapter ends not with a verdict on lensing cosmology or propagation as a whole, but with one high-pressure residual ledger under the first-line path court: if dispersion-free closure and environment ordering survive, the section sharpens V08-8.4; if not, it must collapse back to upper limits or falsification and then route onward to the clock-network audit in 33.35.