Audit which searches need prior-support handling (coverage after Prodigy)

[Jammy2211]

Coverage follow-up to the prior-support work. Everything measured so far came from
one search (MultiStartProdigy) on one cell (imaging/mge hst) —
autolens_profiling#131, write-up results/notes/clipper_campaign/RESULTS.md. This
asks which other searches share the failure mode.

Prompt: PyAutoMind/draft/feature/autofit/prior_exit_sampler_coverage.md

Run this after the Prodigy step-scaling task
(PyAutoMind/draft/feature/autofit/per_parameter_step_scaling.md), so the audit can
measure both levers rather than only the clipper.

The mechanism

The MAP objective is -2 * (log_likelihood + sum(log_prior)); a UniformPrior is
-inf outside its box. The gradient searches step in physical space with
nothing holding them inside, and prior box widths on a real model span 40x
(einstein_radius U(0,8) against bulge.centre U(-0.1,0.1)), so a globally-scaled
step is a wall-crossing for the narrow-box parameters. Lanes do not start near the
edges — _broad_starts draws in the unit cube at [0.15, 0.85] — they walk there.

Already immune, per the phase-1 record (verify, don't re-derive)

• Nested samplers (Nautilus, Dynesty) work in unit-cube coordinates and
  cannot leave prior support. This is why Nautilus is the campaign's reference bar.
• MCMC (Emcee, Zeus) reject -inf proposals so the walker stays put.
  "Rejection is the restoring mechanism that gradient methods lack."
• NUTS diverges rather than dying — different mechanism, its own task.

So the exposed family is the gradient / MLE searches.

To audit

search	expected exposure	why
MultiStartAdam	high	fixed learning_rate=0.01 in PHYSICAL units, no adaptation
MultiStartLion	high	fixed learning_rate=0.001, sign-based
MultiStartADABelief	high	fixed rate, same exposure
MultiStartProdigy	measured	baseline in #131
LBFGS / AbstractBFGS	different shape	takes clipper, hands bounds to scipy — check enforcement actually happens
BlackJAXNUTS	out of scope	divergence, not lane death — confirm and file separately

Load-bearing hypothesis: the fixed-rate optimizers should be affected MORE than
Prodigy, not less. Prodigy at least estimates its own step scale; Adam, Lion and
ADABelief take a literal constant in physical units, so the 40x disparity cannot be
absorbed anywhere. If that holds, the Prodigy numbers are the optimistic end of
the range and #131's "clipping is cosmetic" verdict may not transfer.

Deliverables

1. Coverage table per search — is prior exit reachable, already mitigated, by what
   mechanism, and is that mechanism enforced or merely assumed.
2. A cheap empirical check per exposed search on imaging/mge hst with
   ClipperPriorBox on, recording clip rate and the alive-versus-step curve
   (the clip rate is the quantity comparable across searches; raw lane counters are
   survival integrals and are not).
3. A per-search recommendation on both levers — clipper default and step scaling
   — feeding phase 3.

Grading rules (inherited, all paid for)

• Grade on the alive-versus-step curve, not the percentage — the lane counters
  are survival integrals.
• At least two seeds. Identical settings swung the Prodigy result 171,000
  nats between seeds 0 and 1. Use the search's seed argument; seeding
  random/numpy reaches only the initializer.
• A ClipperPriorBox arm reporting zero clips has not exercised the clipper — a
  broken arm, not a null result (unless the model is genuinely unbounded, where zero
  is the PASS).
• Name the step budget with every number: at 105 steps clipping was worth 114 nats
  on Prodigy; by 3000 it was worth zero.

Out of scope: flipping any default (phase 3), NUTS divergence, and the
seed-dependence investigation — all filed separately.

[Jammy2211]

This narrows: the other families already have the equivalent

Code-verified survey of what each search family does about per-parameter scale:

search	step mechanism	adapts per-param scale?
Nautilus / Dynesty	proposals in the unit cube	scale-free by construction
Emcee	affine-invariant stretch move	yes, provably
Zeus	ensemble slice sampling, tune=True	yes — slice expand/contract self-tunes per direction
BlackJAXNUTS	blackjax.window_adaptation (.../blackjax/nuts/search.py:248)	yes — step size and mass matrix
LBFGS / BFGS	quasi-Newton inverse-Hessian approximation	partial — learns curvature, starts from the identity
Drawer	independent draws from the priors	n/a, no stepping
MultiStart{Adam,Lion,ADABelief,Prodigy}	fixed/global step in physical space	no — the exception

So this issue narrows from "audit which samplers need prior-support handling" to:

1. Confirm and document that the others already have the equivalent — the
   table above, in the module docs.
2. Check the two partial cases: Zeus, and L-BFGS's badly-scaled transient
   before curvature accumulates (it passes no analytic jac, so scipy
   finite-differences on both branches).
3. Do not spend GPU budget re-testing Emcee. Under affine invariance
   rescaling is a mathematical no-op there.

The gradient-family gap is being closed in #1483 (a prior-derived diagonal
preconditioner — the direct analogue of NUTS's mass matrix and L-BFGS's inverse
Hessian).

Separately, and do not conflate: the prior-support half of this issue has a
real hole — UniformPrior bounds are not enforced in the objective on the NumPy
path at all, so Emcee / Zeus / Drawer / LBFGS never see the -inf this issue
assumes they reject. Filed as #1484. A search can be immune to the scaling problem
and exposed to the enforcement one — Emcee is exactly that.