T1 — the race: crossover vs quadratic-fit root-finder. Verdict: the fit wins.
Reproduce: ../cosmic-web/.venv/bin/python scripts/run_p5_race.py (~75 min). Results →
artifacts/p5_race.json, figure → public/img/posts/forbidden-directions-race.png.
Protocol pre-registered in PROGRAM-P5-litmus.md.
Result
Median relative error of the estimated pair separation (same noisy gridded B to both; crossover builds its own potential by the ray gauge; fit = divergence-free quadratic LSQ + Newton roots; κ frozen from one disclosed calibration):
| leg | sep | σ=0 | σ=0.1 | σ=0.25 | |||
|---|---|---|---|---|---|---|---|
| xover | fit | xover | fit | xover | fit | ||
| pure | 0.40 | 0.21 | 1e-11 | 0.07 | 0.003 | 0.06 | 0.004 |
| pure | 0.20 | 0.011 | 1e-8 | 0.04 | 0.004 | 0.14 | 0.014 |
| pure | 0.10 | 0.02 | ~0 | 0.47 | 0.023 | 1.15 | 0.029 |
| contam | ~0.56 | 0.32–0.45 | ~1e-4 | 0.33–0.41 | ≤0.002 | 0.29–0.40 | ≤0.004 |
The quadratic fit wins every cell, by one to four orders of magnitude. Per the pre-registered rule, the scale crossover is demoted from candidate tool to conceptual observable. This is R2’s lesson confirmed a second time on our own strongest claim.
The anatomy (why, honestly)
- Smooth backgrounds are what polynomial fits absorb. The “contaminated” leg’s far point-charge field is analytic over the ball; its Taylor series converges fast, so a quadratic model captures it almost perfectly — the fit stayed at ~1e-4 error with 25% contamination. A background that actually breaks a polynomial fit must vary on the probe scale — and then the crossover’s trajectories integrate through the same roughness. There is no obvious middle regime, and we did not find one.
- The crossover needs a base point; the fit doesn’t. The background shifted the true pair midpoint off the crossover’s probe point, inflating its errors (0.3–0.45) — a genuine deployment cost (scanning base points multiplies its already ~10⁴× higher compute).
- Estimator calibration is a real weakness: the frozen κ transferred imperfectly across separations (21% systematic at sep 0.4, ~1–2% at 0.2/0.1) because the knee is a broad feature, not a sharp point.
- At the smallest separation under noise (pure, sep 0.10, σ≥0.1) the crossover degrades catastrophically (0.47–1.15) while the fit holds at ~2–3%.
What survives, stated precisely
- Q = 7 at the fold point is a statement about the field’s intrinsic geometry (homogeneous dimension of the tangent structure), not an estimate in competition — it survives untouched, as does the 2/3/4 plateau dictionary as physical meaning.
- The crossover remains the conceptual explanation of what “an unresolved pair” looks like to scale-covariant geometry — the right mental model, demonstrated with a universal dilation collapse (S1).
- For quantitative pair metrology on data: fit a polynomial field model and root-find it. The division-of-labour table gains its final row.
The programme-level moral, twice measured
Integrated/geometric observables beat naive pointwise practice and lose to matched statistical estimation — classification (R2) and now pair metrology (T1). The sub-Riemannian framing’s durable contributions are the ones with no estimation competitor: existence/order laws (Q = k+5, the fold’s Q = 7), exact analytic structure (δ = 1 − (2/π)K(2ε), the critical gradient ε = 1/2), and physical dictionaries — not estimators.