The card and the curve: a discrete summary needs a stable alphabet

This is the fourth paper in a series on the domain of validity of derived quantities in movement analysis, together with its appendix, four figures, the derived artefacts, three frozen pre-specifications and a standalone checker. The third paper in the series found that a knee angle is not one quantity: across five sources the knee angle splits into two clusters depending on how the joint centre is estimated. If the units do not travel between datasets, what can travel? The answer offered here is a procedure — detect events, take one lap, quotient the gauge, choose a coordinate, build a reference, score — together with a rule for when to refuse. The claim is about representation, not estimation: a discrete card is a summary of a curve, and a discrete summary is lossless only when its alphabet is stable. When the alphabet shifts, the curve still carries information and the summary does not. Two tasks are compared, walking and sit-to-stand, over four public datasets. The card wins against equally spaced nodes on walking (delta +0.1313) and loses on sit-to-stand (delta −0.3122); the dense arc-length comparator beats chance on both (Q = 0.502 and 0.767). Three further tests were run after the first draft. A14 measures the row condition on held-out repetitions of the same person: walking C2 = −0.0638 (95% CI −0.0789 to −0.0476), sit-to-stand C2 = +0.0065 (CI −0.0473 to +0.0509). A15 repeats the walking result on a second source (GaitRec-VR, C2 = −0.0957), with verdict-level agreement only. A16 replaces the binary question "does it hold across people" with a decomposition of what is shared by the task, what belongs to one's own reference, and what remains. Ablating the row-to-node assignment removes 0.194 of the card's delta on walking. The archive is self-verifying. One command, needing no data and no third-party package: python fourth/check_release.py prints PASS 8/8 and exits 0. It checks every printed decimal against the shipped artefacts, figure and table ordering, cross-references, placeholders, frozen pre-spec hashes and licence files. Exit codes 0/1/2 are part of the interface. No clinical, torque, metric-skeleton, identification or cross-session claim is made. ITW (CC BY-NC 4.0) is the one commercial red light in the licence audit and is not used in this paper; the four datasets used are cited, not redistributed. Text is CC BY 4.0; the shipped checker is MIT. CITATION.cff carries all three earlier records as references.