gauss_iod picks the first, middle and last observation, which is too wide on long arcs

[matthewholman]

gauss_iod takes seq[0][0], the midpoint and seq[0][-1] (iod.py:104-106).
On a 67-day arc that hands a short-arc approximation three points a month apart.

Fitting 500 Rubin-only arcs (MPC observatory code X05) with the #465 close-Earth
root deferred leaves five flag=3 failures. They are not the information-starved
cases — for the discoveries they have longer arcs than the converged ones,
40 d median against 18 d. Choosing a triplet by time span instead:

object	arc	first/mid/last	~2 d	~5 d	~10 d	~20 d	MPCORB a	|da/a|
K25OI3O	40.1 d	flag 3	ok	ok	ok	ok	2.1913	6.0e-4
K25MU7E	38.1 d	flag 3	ok	ok	ok	flag 3	2.7095	3.9e-3
K25MY9V	67.1 d	flag 3	flag 3	ok	ok	ok	44.9821	4.6e-3
K21RF8T	11.0 d	flag 3	flag 3	ok	ok	ok	2.6692	4.1e-3
K14E02J	11.3 d	flag 3	ok	ok	ok	ok	2.2394	4.7e-3

A ~5-day triplet converges all five on the catalogue orbit, including a TNO at
45 au from a Rubin-only arc. With #465 and a 5-day triplet the sample is 500/500.

The usable span is two-sided and depends on the object: a = 2.71 fails at 20 d
(too long for the series), a = 45 fails at 2 d (too little curvature). So a fixed
constant is not the answer either.

Cheapest fix that needs no theory: try triplets at several spans and pool the
candidates. _pick_best_root already selects among multiple candidates on
chi-square, so the machinery exists.

[matthewholman]

Measured, on a 2000-object cold start.

The premise holds. Replacing first/middle/last with a triplet targeted at a
fixed 60-day span takes convergence from 95.55% to 98.65% at 1.37x runtime;
a cost-ordered ladder on top of it reaches 99.50%. Across three populations
(2243 objects) that is 92.78% -> 98.31%, NEOs gaining most at 65% -> 87%.

Two natural fixes both fail, in opposite directions:

• Pooling every span is a net loss at a fixed budget: 95.55% -> 81.65%,
  timeouts 31 -> 349.
• Densest chunk as seq[0] instead of longest-span gains 4 and loses 2 on
  the same 2000. The losses have a dense but temporally collapsed chunk (39
  observations over 4.0 days), leaving Gauss no curvature.

So neither span nor observation count is the criterion alone. Worth noting the
target is 60 days rather than an angle: 29 degrees of mean anomaly for an NEO,
0.2 for a TNO, and it works for both.