job_type = ms reports no fit at all when its first augmented model is non-finite, although the start point certifies fine (gntr reports it)

[wshlavacek]

What happens

A job_type = ms run whose first augmented model is non-finite reports no fit at all,
even when the start point itself certifies perfectly well. The run ends with:

Start 1 of 1: homotopy over 1 stage(s), inner_failed: best certified objective none, 2 evaluations
  stage trace: m=4: inf
No simulation completed, so there is no best fit to report.

job_type = gntr from the identical start point, same config, reports the fit normally:

Stop criterion satisfied with objective function value of -150.70078270720597

The two disagree about whether there is a result, and ms is the one that is wrong: the
point does score, and ms can prove it — see below.

Why

Reproduced on Borghans_BiophysChem1997 at a start drawn from the prior box. Rebuilding the
m=4 stage at that exact point:

1. unsegmented simulation: OK, 112 rows
2. seeded knots (log10):
   experiment1@1/4        [ 0.766 -2.836  4.133]
   experiment1@1/2        [ 0.766 -2.836  4.133]     <- all three identical
   experiment1@3/4        [ 0.766 -2.836  4.133]
3. objective at the seeded start: value=inf  finite=False
   equality finite=False  defect_norm=inf
4. per-segment simulation at the seeded start:
   segment 0: OK   end_state=[5.8307e+00 1.4589e-03 1.3584e+04] finite=False
   segment 1: OK   end_state=[5.8322e+00 1.4595e-03 1.3586e+04] finite=False
   segment 2: OK   end_state=[5.8322e+00 1.4595e-03 1.3586e+04] finite=False
   segment 3: OK   end_state=[5.8322e+00 1.4595e-03 1.3586e+04] finite=False
5. certificate: accepted=True objective=-150.70078278843627 detail='single-shoot reconstruction'

Three things are going on, and only the third is a bug:

1. The trajectories integrate; the forward sensitivities overflow. Every segment returns
   OK (so backend.simulate's own finiteness check passed on the trajectory) but
   SegmentTrace.is_finite() is False, which tests d_end_param / d_end_ic. Z_state
   reaches 1.4e4 here. Refusing the point is correct — an assembly over non-finite
   sensitivities would produce a garbage local model.

2. Seeding fell back to nominals, which is why all three knots carry the identical vector:
   _seed_knots returns [nominal.copy() for _ in knots] when the seeding trajectory is not
   finite. That is the documented fallback and is also fine.

3. The run then reports nothing, although it has a certified score available. In
   AugmentedLagrangian.run:

   model = subproblem.at(u)
   outer_evals += 1
   if not model.is_finite():
       stop_reason = 'inner_failed'
       break                       # <-- returns before _certify is ever called
   ...
   record = self._certify(iteration, u, model, multipliers.penalty, optimality)

   The break precedes _certify, so no CertifiedIterate is produced, nothing is offered
   to CertifiedBest, on_iterate never fires, and nothing reaches
   MultipleShootingAlgorithm._record_iterate — hence an empty trajectory and
   "No simulation completed". Yet step 5 above shows problem.certify() at that very point
   returns accepted=True, objective=-150.70078, agreeing with gntr to 7 significant
   digits. Certification does not need the augmented model at all: it discards the auxiliary
   states and re-simulates unsegmented.

Why it matters beyond one start

• On a single-start run this is the whole result. refine_method = ms (Add sensitivity-aware multiple shooting for ODE parameter estimation #563 arm 4) is
  single-start by construction — _resolve_n_starts returns 1 for an injected start — so a
  refine that lands here silently contributes nothing, and the run reports the search's
  fit with no indication the refine produced nothing usable.
• On a multi-start run other starts paper over it, so it is easy to miss.
• It is not rare on a stiff oscillator: 1 of 8 oscillating box draws hit it in a small paired
  experiment.

Suggested fix

Certify before bailing. The reported parameters at u are certifiable whether or not the
augmented model at u is finite, and the certificate is the run's honest answer for that
point. Roughly: on the non-finite branch, build the CertifiedIterate from
self.problem.certify(layout.reported_of(u)) with defect_norm = inf, offer it to best /
shared_best / on_iterate as usual, and then break with stop_reason = 'inner_failed'.

That would have this run report -150.70078 — the same number gntr reports — instead of
nothing.

A test at the layer level can pin it without a simulator: an offline TranscriptionProblem
whose objective_at returns a non-finite model while certify succeeds should still produce
OuterResult.best is not None.

Reproducing

Found while running the #563 acceptance work. The start is a prior-box draw on
Borghans_BiophysChem1997; the distinguishing property is a point whose whole-horizon
trajectory integrates but whose forward sensitivities are non-finite. Happy to attach the
exact parameter vector and the two confs if useful.