Three different achievements

5 Capacity, produced candidates and selection

5.1 A post-reveal representational-capacity sanity check

An unsuccessful fitted model does not reveal by itself whether the architecture lacks capacity. OII-4 permits a direct post-reveal check. Every source generator has at most 16 hidden states and uses the same action/output form as Equation 1. For a target with $n\leq16$, embed its states in the first $n$ coordinates of a 16-state model. Copy every rational joint transition row and each preparation distribution there; set unused states to emit symbol zero and self-loop, with zero initial mass and no incoming mass from the active subspace. The resulting stochastic instrument reproduces every source row on the active subspace and hence every admitted adaptive finite record law.

The saved construction was checked at 3,072 embedded instrument rows and 192 preparation rows. This elementary embedding is an exact capacity sanity check for the general joint-instrument architecture, not a novel representation theorem. It is not a theorem about a more restrictive HMM that factorizes emission and transition in a prescribed way. It also does not prove that the specific positive-pseudocount, single-initialization fitting procedure can attain the exact boundary parameters from its finite data.

These exact source-derived parameters were constructed only after reveal and were never included as known-correct training candidates. A strictly positive parameterization can approximate the embedded model by mixing each initial distribution and joint row with a sufficiently small uniform component. The imported coupling bound gives eight-call error at most $1-(1-\eta)^9$ when each such change is at most $\eta$. This is an approximation-capacity statement, not successful identification by the implemented optimizer.

Thus the remaining failures cannot all be explained by insufficient latent-state cardinality. Finite observations, initialization, regularization, local fitting, candidate construction and selection remain possible contributors. Nor does the embedding imply 16 public predictive states: Equation 8 can attain many distinct posterior values even with a small hidden carrier.

5.2 Certificates against a produced bank

Let $\BB_f=\{Q_1,\ldots,Q_J\}$ be the finite bank actually produced on fixture $f$. An episode-level mixture has laws $Q_w^e=\sum_jw_jQ_j^e$, with the same weights across the experiment family. For any legal event $A$ in an experiment $e$,

If the final quantity exceeds $\tau$, no mixture in that bank passes this test. This elementary certificate is stronger than observing that one optimizer chose poor weights. It does not exclude an unproduced parameterization in the same mathematical architecture. The witness and probability evaluation are post-reveal diagnostics, not queries available to the opaque learner during selection.

Bank	Passing single candidate /48	Selected all-core /48	Selection misses	Convex-bank obstructions /48
R	39	37	2	8
S	39	37	2	8
E	28	28	0	19
G	15	15	0	33
B	37	36	1	10
P	25	24	1	–
O	25	23	2	22

P selects three trees; O mixes four. The obstruction count 22 refers to the four-tree bank. No P-specific obstruction count is inserted. A missing adequate individual candidate does not exclude an adequate mixture.

5.3 What the counts distinguish

For both R and S, 39 of 48 fixtures have at least one produced candidate passing every core slot, while the selected predictor does so on 37. The two differences are genuine core-panel selection misses in the post-reveal ranking. They do not establish that the selector could have known the oracle ranking from the available calibration sample. Table 8 identifies both cases. The selected model has lower observed calibration NLL in each; the failure is not a tie-breaking error or evidence that the selector saw the hidden core ranking. The alternative rows are post-reveal comparisons of candidates that had already been fitted.

Fixture family	Model	Role	Calibration NLL	Core worst TV
Leaky occupancy	HMM-8	Selected	0.083232	0.152781
Leaky occupancy	HMM-16	Passing alternative	0.087784	0.131943
Coin	IOALERGIA-1.5	Selected	0.086455	0.196476
Coin	HMM-2	Passing alternative	0.086654	0.091927

The rows compare already-produced candidates, ranked after reveal for diagnosis. R and S make the same selections. The coin row shows the lowest-calibration-NLL passing alternative; two additional hidden-state candidates also pass. Full fixture IDs and values are in the audit supplement.

Of the remaining nine fixtures with no adequate individual R/S candidate, eight have a Equation 9 certificate excluding every convex mixture in the bank. The last case remains unresolved at mixture level: failure of every individual member does not imply failure of every mixture. Consequently, the 11 selected-model failures separate into two selection misses, eight certified produced-bank obstructions, and one case lacking an adequate individual candidate but not excluded by this mixture audit.

The tree bank has 22 such obstructions on the same final cohort, versus eight for R/S. That is a reduction in produced-bank insufficiency, not proof that one full architecture is globally more expressive than the other. The capacity construction already shows why those categories must be distinguished.

5.4 Diagnosis without forced single causes

There is a hierarchy of questions, not a compulsory single-label taxonomy. An architectural witness answers an existence question. A bank witness excludes a finite collection and its mixtures. A selector miss compares a chosen candidate with other candidates that were produced. None alone identifies the statistical cause of a failed parameter fit.

The frozen study used one start and at most 60 EM updates per size. The post-hoc analysis in Section 6 now directly tests a larger budget and finds five selected repairs among its 11 failures. This is evidence of finite-budget sensitivity, not a diagnosis that every residual fit is a local minimum. Insufficient branch evidence, the fixed pseudocount and selection on limited calibration remain possible contributors. Likewise, source models with identical public laws are non-identifiable in their hidden anatomy; that is not the same failure as predicting the public laws incorrectly.

Category	What this record establishes	What remains unestablished
Architecture capacity	Exact post-reveal 16-state embeddings for all final targets	Identifiability or convergence of the opaque fitter
Produced-bank insufficiency	Eight R/S bank witnesses exclude every produced mixture	Inadequacy of all parameterizations or other candidate generators
Parameter estimation	Fitted laws disagree on some source contexts despite architectural capacity	A unique attribution to sampling rather than optimization or regularization
Optimization/local fitting	Five selected repairs among 11 targeted failures under the post-hoc expanded budget	Global optimization, a unique causal decomposition, or repair of every remaining miss
Model selection	Two R/S candidates were available that passed all core tests but were not selected	That the available calibration observations identify the oracle winner
Context/data coverage	Duplicate and calibration-matching specifications; observed panel-to-core failures	A full unseen-policy or all-future generalization guarantee
Finite-budget uncertainty	OII-3 confirmation limits and flat confidence objectives	Adequacy from finite nondetection or exact equality from overlapping intervals
Operational non-identifiability	Exact public-law twins have different hidden implementations	Permission to excuse wrong predictions of operationally distinct laws