Is the walk-run transition number kinematic or biological?

Object. Animals across four orders of magnitude in body mass switch from walking to running at roughly Fr = v²/gL ≈ 0.5, and that number is used to read locomotion speeds off fossil trackways.

The corrected ceiling. Doing the force balance throughout stance rather than at midstance, contact is lost at the end of stance and the constraint is Fr_mid ≤ 3cos(alpha) − 2 with alpha the half-step angle. Converted to the average-speed units the literature quotes, the ceiling at observed gait geometry is about 0.62, not the textbook 1.0. Animals transition near 0.45 — roughly three-quarters of it, not half.

Differential. (1) Kinematic: the transition is the ceiling, and reported spread reflects spread in relative step length. (2) Biological: a metabolic crossover, no dependence on step length. (3) Both: a constant safety factor. (4) The spread is measurement heterogeneity — Fr is linear in leg length, and hip height, leg length and effective pendulum length differ by tens of percent.

The discriminating test is one scatter plot against data that already exists: observed transition Froude against 3cos(alpha) − 2 in average-speed units, per species. Slope 1 means kinematic; slope 0 means biological; a constant ratio below 1 means a safety factor.

Why this is blocked on literature rather than compute. The arithmetic is trivial. What it needs is per-species speed, stride length, and leg length under one consistent definition — and a 20% definitional difference moves Froude by 20%, which would swamp the effect being measured.

Structural constraint worth stating. Tendon compliance and metabolic cost cannot appear in any dimensionless group built from velocity, gravity and length alone, since those three admit exactly one such group. Any explanation of that kind requires adding a variable, and the added variable is necessarily material or metabolic.

Prior: relative stride length at the transition clusters near 2.0 across species, and a quantity that does not vary cannot explain one that does.

Verdict — partial (2026-09-14, literature-assembly pass, no run). A reading pass, not run 017, produced a working verdict on the cursorial-mammal + human core.

The working table (one convention, primates excluded):

The reading, no fit required:

• The textbook 0.5 is the human number, not universal — horses transition at 0.35.
• Branch 1 (kinematic ceiling) is disgraced by the case's own prior. It needs Fr to track relative stride length, but rSL at transition clusters near 2.0 across species; a quantity that does not vary cannot explain one that does. The horse/human gap is not stride-geometry-driven.
• Branch 3 (constant fraction of the ceiling) is weakened. Against the ~0.62 contact-loss ceiling, horses sit at 56% and humans at 81% — not a constant fraction.
• Branch 2 (biological constant) is favoured. The transition Froude is near-constant, and its residual variation is not explained by either kinematic account.

The definitional trap is real, and located. The primate paper (Fig 1) shows the same species differ significantly from non-primates at a given Froude when the characteristic length is limb length, and the difference disappears under hindlimb COM position — same data, two definitions, opposite conclusion, the COM definition the outlier. Hip height and greater-trochanter-to-ground are the same measurement for a straight-legged stance, so the human (Kram) and horse (Griffin) numbers are already on one convention. So the case's "a 20% definitional difference would swamp the effect" is real for primates and not for the cursorial + human core — which is exactly Alexander & Jayes's own scope restriction. The primate paper is the documented instance that the definitional risk is real.

Caveats, load-bearing. Three data points, and the primate exclusion is doing real work — the clean discrimination depends on removing the COM-definition outliers. Now blocked on more cursorial species to firm up the 0.35–0.50 spread, not on a definition question.

Sources. Alexander & Jayes 1983, J. Zoology 201:135–152 (the anchor); Griffin et al. 2004 (horse walk–trot); Kram, Domingo & Ferris 1997 (human walk–run).