The same angle is not the same angle: what a knee angle is attached to, and what it costs to forget it

This record contains the third paper in a series on the domain of validity of derived quantities in movement analysis, together with its appendix and a deployment note. The paper asks what a knee flexion angle is attached to. The same joint, in the same person, on the same day, gives different numbers depending on the plane the motion is projected onto, the points the segments are defined from, and the speed at which the person walks. It separates two questions: what the construction guarantees — an exact invariance under positive amplitude, constant offset and phase shift — and what the number is empirically attached to: out-of-plane geometry, marker placement, and speed. The archive is self-verifying. Two commands, neither needing the data: python third/check_numbers.py python third/check_pair.py Expected output: PASS unmatched total 0 across 5 documents, and PASS 5/5. Contents: third/manuscript.pdf (14 pp.), third/appendix.pdf (3 pp.), instruments/README.pdf (10 pp.), instruments/GAUGE.pdf (18 pp.), verification/README.pdf (4 pp.), markdown sources, the declarative number pool, the checkers, and figures. Text licensed CC BY 4.0; code MIT.