Gait cost and Froude scaling

# 10. The walk–run transition is Froude

**Verdict: Derivation consistent with the observation; not an independent measurement**

*Working test · not peer reviewed · AI-assisted · no clinical or safety advice*

## The hypothesis, as published

> The walk-to-run transition occurs at a critical Froude number rather than at a
> critical speed, so leg length sets the speed at which people switch.

> Prediction: transition speed should scale with the square root of leg length.
> Refute: transition at a fixed speed across body sizes would remove the ratio.

## What was tested

This is a derivation with a consistency check, not a new dataset. Under dynamic
similarity both gait costs are functions of the Froude number Fr = v²/(gL) alone.
Walking is modelled as an inverted pendulum: cost per unit distance is the work
of raising the centre of mass over the stance arc, divided by the step length.
Running is modelled as a spring-mass system: elastic energy per step scales with
the square of the vertical excursion, and cost per unit distance is approximately
linear in Fr.

The walking curve is anchored to human cost-of-transport values at 1.4 and
2.0 m/s, and the running curve to values at 3.0 and 5.0 m/s. The calibrated model
is then solved for the crossover Fr, and the crossover is converted to a speed
for leg lengths from 0.45 to 1.25 m.

## Results

| Quantity | Value |
|---|---|
| Walking coefficient A | 25.95 J kg⁻¹ m⁻¹ |
| Running intercept B | 4.22 J kg⁻¹ m⁻¹ |
| Running slope C | 0.276 J kg⁻¹ m⁻¹ per Fr |
| Crossover Froude number | 0.423 |
| Predicted transition speed at L = 0.90 m | 1.93 m/s |
| Observed human transition (benchmark) | 2.0 m/s |
| Log–log slope of transition speed vs leg length | 0.5000 |
| Slope with a non-similar step-energy term | 0.525 |

The model crosses over at Fr ≈ 0.42 and puts the human
transition at 1.93 m/s for a 0.90 m leg,
against the observed ~2.0 m/s. The square-root
law is exact in the similar model (0.500).

The sensitivity is the informative part. A fixed energy cost per step — a term
that does not scale with Fr — breaks similarity and rotates the slope to
0.525. That is a measurable deviation from 0.5, which means
the prediction is only as good as the similarity assumption behind it. The
observed agreement at human leg length therefore supports the model but does not
prove that Fr is the controlling variable across species.

## Sensitivity

- Anchoring the running curve at 2.5 and 4.5 m/s instead of 3.0 and 5.0 moves
  the crossover by less than 0.03.
- A 10% change in the walking coefficient moves Fr* by about 0.02.
- Adding a step-energy term of 0.05 J kg⁻¹ per step shifts the slope to
  0.525, as reported above.

## What this test cannot see

It has no new measurements. The anchors are textbook cost-of-transport values,
and the model is constructed so that the scale-invariant part produces the
square-root law by design. The test therefore checks internal consistency and
agreement with one well-known human transition speed; it does not test the law
against children, other animals or pathological gaits. A real test would regress
measured transition speeds on measured leg lengths across those groups, and that
is the experiment this note recommends rather than performs.

## Files

| File | What it is |
|---|---|
| `hypothesis.md` | the published entry, quoted verbatim |
| `run.py` | the derivation, runnable |
| `data.csv` | transition speed by leg length |
| `results.json` | every number quoted above |
| `figures/fig1.png` | cost curves and the leg-length scaling |
| `SOURCE.md` | anchors and method |

**Source.** Derivation with textbook cost-of-transport anchors. No new data.