33.54 turns the convergence–X-ray offset into a tightening court: after one smoothing scale, one center definition, one phase anchor built from velocity bimodality plus shock / cold-front geometry, a preregistered fixed-τ law, and blinded separation of ΔκX measurement from TSP estimation, post-pericenter mergers must show a signed ΔκX that relaxes toward a floor with a population-level τ that stays relatively narrow, survives axis-rotation, recentering, and pipeline swaps, and tightens when the joint phase key is added; under V08-compatible tighten, this remains only a merger-memory ruler rather than a total cluster-dynamics or ontology verdict.
Use this section as a compact machine-readable EFT reference.
33.54 treats the convergence–X-ray offset as a response variable with memory rather than a static geometric snapshot. The admissible claim is that ΔκX starts larger soon after pericenter and relaxes toward a floor with one common or narrowly distributed timescale. The court object is therefore a merger-memory ruler, not merely the observation that some mergers look displaced.
The measurable package is deliberately small and auditable. ΔκX is the signed projected offset under one smoothing scale and one center definition. TSP is estimated with one unified method and a two-key phase anchor built from velocity bimodality plus shock or cold-front geometry. A roughness or surge index U may explain scatter or initial amplitude, but it may not replace time as the horizontal axis.
The model and workflow are tight by design. A primary fixed-τ law and one robustness alternative are preregistered, with Δ0 and Δ∞ carried explicitly and “one cluster, one τ” freedom disallowed. Blinding separates the phase team from the lensing/X-ray measurement team, while arbitration fits the track and summarizes the τ distribution. Holdouts and at least two lensing plus two X-ray pipelines prevent one processing choice from owning the conclusion.
Controls must break fake relaxation tracks. High-eventness post-pericenter mergers should show the clearest early offsets, while pre-merger or not-yet-passed systems should not. Randomizing TSP bins, rotating the merger axis, or randomly re-centering the system must erase the track. Small smoothing or centering changes may move amplitudes, but they may not reverse direction.
Support requires right-sign regression, a convergent population-level τ that replicates in holdouts, and a joint phase anchor that tightens rather than worsens the track. Falsification follows from wrong-sign or null regression, no τ convergence, failed phase anchoring, or systematics that rewrite the fit. The named adversaries are center-definition drift, projection and phase misclassification, and sample-selection bias.
So the chapter closes only one tightened κ–X merger-memory ledger aligned with V08-8.6 / 8.12 / 8.13. It may not certify a universal merger clock or rewrite cluster ontology. Its clean onward value is to pass a bounded phase-ruler grammar to the next spatial-covariance and particle courts.