The Irrecoverability Boundary: Where Learned Inference Cannot Recover What Deterministic Computation Cannot Reach

1 · Introduction

Most AI systems that face an inference problem answer it by fitting a flexible model and hoping the fit converts into a decision advantage. An alternative posture — the one this program defends — is to let the strongest feasible deterministic computation own the decision, and to admit a learned component only when it can be shown, against a frozen gate, to earn its authority. This is failure-directed science: our job was to kill the machine-learning hypothesis, not to help it win. The null we defend, \(H_0\), is that deterministic computation is sufficient. We tried to falsify it and could not.

This is the sixth kill in the program, and the one that maps the whole boundary rather than scoring a point. Its predecessor, Inference Placement (Track 1), showed that a learned-inference seat is "reachable but not robustly earned" in a regime where exact deterministic computation was always feasible (\(L=4\)). That result left a precise, load-bearing gap. If a learned model is ever to hold irrecoverable authority — authority that no amount of additional deterministic computation could reclaim — it can only be where deterministic computation itself runs out of room: where the exact computation the deterministic side would prefer becomes intractable and it is forced to approximate. Track 2 enters that regime deliberately, by driving the latent dimension \(L\) up until the exact lane is infeasible, and asks whether a feasible learned latent model recovers what bounded-order deterministic computation can no longer reach.

The one place a learned seat could exist is exactly where inference itself becomes intractable — and the learner's only feasible option, approximate inference, inherits the same wall. Where deterministic computation stops sufficing, feasible learned inference stops sufficing too.

The scientific object of this paper is not a scoreboard. It is a measured map of a three-axis space — latent complexity, deterministic order, and learned inference — in one honest synthetic domain, with the deployment gate, seeds, and success criterion all frozen and hashed before any competitor existed. Our contributions are: (1) a fully validated deterministic order ladder that establishes, before any learner is built, the strongest feasible deterministic bar and its collapse under intractability; (2) a data-scaling analysis showing that this collapse is structural, not sample starvation; (3) a poly-time variational learner whose feasibility wall coincides with the exact deterministic wall, together with a robustness check at the tractable boundary using exact inference of the learner's own parameters; and (4) a cross-world replication over 40 held-out worlds per \(L\) that yields the anti-transition and a live-caught false-positive seat, and closes at Outcome A — \(H_0\) held, zero earned seats.

2 · Method — the order ladder and the frozen gate

2.1 Domain and generator

The domain is sequential Bayesian fault diagnosis with \(K=12\) faults and \(M=20\) diagnostics, each diagnostic carrying a test cost in \(\{1,2,3\}\). The manipulated axis is the latent dimension \(L\), giving \(2^L\) latent configurations. The generator is noisy-OR: each diagnostic's probability of firing is one minus the product of per-cause "no-fire" (leak) terms,

where a shared-latent coupling strength \(\rho\) controls how much spurious structure is manufactured. The control setting \(\rho=0.0\) injects no manufactured structure, so higher-order models must not beat order-1 there; the operating point is \(\rho=0.8\). Unless stated otherwise, every number below is held-out operational loss at \(\rho=0.8\).

2.2 The diagnosis game

Every lane — each deterministic rung and the learner — plugs into a single, byte-identical decision game. The game maintains a Bayesian belief over faults, selects the next test by maximum Expected Information Gain (EIG), and stops when the maximum posterior belief reaches \(0.90\) or the tests are exhausted. Operational loss is

the "cleared price" inherited from Track 1. Because the inference path is identical across lanes, differences in \(L_{\mathrm{op}}\) are attributable to the belief model alone.

2.3 The frozen deployment gate

2.4 Pre-registration and seeds

The pre-registration — the gate, the seeds, the order ladder specification, and the seat criterion — was frozen, sha256-hashed, and git-tagged before any competitor was built (prereg sha256 b51ee352…). The variational learner, which did not exist at prereg time, was declared in an addendum that was itself frozen and hashed before the learner was implemented (learner addendum sha256 94cb2eaa…). The master seed is SEED=20260921, deterministically expanded into train (×7), test (×13), learner (+999), and cross-world (×2003) streams. The held-out test set is 200 incidents per world, disjoint from training; the frozen primary training size is \(n_{\text{train}}=1500\).

2.5 The deterministic order ladder

The strongest feasible deterministic competitor is a ladder of closed-form models — no latent EM — behind one uniform (belief, predict) interface:

The DETERMINISTIC_CEILING is the best feasible rung by held-out operational loss; this is the bar the learner must beat by \(\geq 15\%\) to earn a seat. The ladder was validated to machine precision before any learner existed: the world is byte-identical to frozen Track 1 at \(L=4\); order-2 tree belief propagation equals brute force; order-3 variable elimination equals brute force; and the generic game reproduces the Track-1 lanes trajectory-for-trajectory. En route we caught and fixed a Prim minimum-spanning-tree self-loop initialization that would have made order-2 look artificially weak — a bug that biased toward the learned side, so fixing it removed a false advantage for ML rather than for the floor.

2.6 The learner

The frozen 006H2 learner enumerates all \(2^L\) latent configurations in both training and inference, and is therefore exponential in \(L\) and infeasible at \(L \geq 16\) — the same wall as exact deterministic inference. The prereg premise "learner feasible while deterministic-exact infeasible" therefore does not hold for that learner. The addendum-preregistered resolution is a poly-time variational (mean-field) noisy-OR learner in the same generator-matched family, replacing the exact \(2^L\) enumeration with a mean-field posterior. It was validated before the run: mean-field agrees with exact \(2^L\) within the reported bias (\(\log Z \approx +0.2\) nats, approximately uniform across faults and therefore cancelling in the belief softmax; worst-case predictive gap \(<0.35\)); the GEM gradient matches finite differences to \(10^{-8}\); and it never spuriously beats exact inference of its own model (gate-3, checked at \(L=4\)). Hyperparameters were frozen before the run (mf_sweeps 10, restarts 8, short_iters 25, polish_iters 150, gem_steps 12, lr 0.6, pz_floor 0.05), and restart selection uses training marginal log-likelihood only.

2.7 The seat criterion (irrecoverability)

3 · The deterministic ceiling (Figure 1)

Figure 1 — interactive. The order-ladder curve: held-out operational loss versus latent dimension \(L\) for each deterministic rung and the oracle anchor. Lower is better; the ringed marker is the feasible ceiling at each \(L\); the oracle is reported only where \(L \leq 12\). Toggle \(\rho\) to see the control (\(\rho=0.0\), no manufactured structure) versus the operating point (\(\rho=0.8\)); click a legend chip to hide a series; hover any point for its exact loss.

Two decisive facts are established here, before any learner is introduced. Table 1 gives the numbers.

Table 1. Held-out operational loss by rung (\(\rho=0.8\)). † marks the feasible ceiling at each \(L\). "ceiling vs order-1" is the ceiling's reduction over the naive floor; "oracle headroom" is the oracle's reduction over the feasible ceiling. The \(L=4\) and \(L=8\) ceiling-vs-order-1 gaps have bootstrap CI lower bounds \(>0\); the \(L=24\) gap is not significant (CI \(\approx 0\)).

Fact (a) — the tractable regime is already saturated. At \(L=4\) and \(L=8\), where \(2^L\) is small enough that the oracle is computable, the oracle — an upper bound on any model, learned or not — beats the feasible deterministic ceiling by only \(+11.49\%\) and \(+7.03\%\) respectively. Both are below the \(15\%\) deployment threshold. No model, however good its fit, can earn a seat where even the oracle's headroom does not clear the gate.

Fact (b) — the intractable regime collapses the ceiling. At \(L=16\) and \(L=24\), where the oracle is uncomputable, the higher-order rungs overfit and fall below order-1: at \(L=16\) the feasible ceiling is order-1 (order-2 and order-3 are worse), giving ceiling-vs-order-1 of exactly \(0.0\%\); at \(L=24\) order-2 recovers a statistically insignificant \(+2.47\%\). The deterministic ceiling collapses to the naive floor precisely where exact computation becomes intractable. The \(\rho=0.0\) control holds everywhere — with no manufactured structure, the higher orders never beat order-1 — confirming that the ladder's advantages at low \(L\) are real structure and not an artifact of the higher-order machinery.

4 · Data-scaling: the collapse is structural (Figure 2)

Figure 2 — interactive. Ceiling-vs-order-1 reduction (%) as a function of training size \(N\) (log axis), at \(L=16\) and \(L=24\). The shaded band is the sub-gate region; every point sits near \(0\) and the paired \(90\%\) bootstrap CI includes \(0\) at every \(N\). Hover a point for its value.

The natural critic's attack on Fact (b) is that a higher-dimensional estimator was starved of data. We answer it directly by sweeping the training size \(N \in \{1500, 3000, 6000, 12000, 24000\}\) at the two intractable dimensions.

Table 2. Ceiling-vs-order-1 reduction (%) across training size. At every \(N\) the paired \(90\%\) bootstrap CI straddles \(0\), so no reduction is significant.

Sixteen-fold more data does not rescue the bounded-order deterministic models: the ceiling-vs-order-1 gap remains statistically indistinguishable from zero at every \(N\), for both \(L=16\) and \(L=24\). The collapse in Fact (b) is structural — a property of what bounded-order models can represent at high latent complexity — and not a symptom of an under-fed estimator. This closes the sample-starvation objection before it can be raised against the learner.

5 · The learned competitor earns no seat (Figure 3)

Figure 3 — interactive. The design-world variational learner. In Decision advantage mode: reduction (%) of the learner's held-out operational loss against each deterministic rung, by \(L\); bars above zero favor the learner and the dashed line is the \(+15\%\) seat threshold — never reached. Switch to Model fit to see that the same learner is a strictly better probabilistic model (higher observed-data log-likelihood) at every \(L\). Hover any bar for exact values.

Table 3. Learner versus each rung in the design world. The learner never reaches \(+15\%\) against any rung at any \(L\); at \(L \geq 16\) it falls below the naive floor (order-1) itself.

The learner earns a seat nowhere. Its best margin against any rung is \(+10.14\%\) (versus order-1 at \(L=4\)), still short of the gate; against the feasible ceiling its margins are smaller still, and at \(L \geq 16\) it is worse than the naive floor.

Crucially, fit_valid is True everywhere: the learner is a strictly better probabilistic model of the observations than order-1 at every \(L\), with higher observed-data log-likelihood — \(-10.51\) vs \(-11.84\) at \(L=4\), \(-10.59\) vs \(-11.65\) at \(L=8\), \(-9.34\) vs \(-9.97\) at \(L=16\), and \(-6.91\) vs \(-7.28\) at \(L=24\). A better fit simply does not convert into a deployable diagnostic advantage. This is the sharpest single finding of the paper: the learner wins the modeling objective and loses the decision objective, and at high \(L\) it loses to the naive floor outright.

6 · Cross-world replication: the anti-transition (Figure 4)

Figure 4 — interactive. Across 40 held-out worlds per \(L\): each dot is one world's learner edge over the naive floor (order-1); the black bar is the median; the dashed trend line joins the medians. Points above the \(+15\%\) line pass the necessary vs-order-1 screen (only world 37, at \(L=4\), and it fails the full ceiling check). Hover any dot for its world and value.

Clearing the gate against the best feasible rung necessarily requires clearing it against order-1, since order-1 is one of the rungs the ceiling must dominate. We therefore report, per \(L\), the fraction of 40 fresh held-out worlds in which the learner passes this necessary vs-order-1 screen — a fraction of \(0\) already rules out a reproducible seat — together with the median and maximum learner edge over order-1.

Table 4. Cross-world replication over 40 held-out worlds per \(L\). The median learner edge over the naive floor decays monotonically with latent complexity and crosses below zero at \(L=24\).

The anti-transition. The median learner edge over the naive floor decays with latent complexity: \(+5.74\% \rightarrow +4.85\% \rightarrow +1.52\% \rightarrow -0.17\%\) from \(L=4\) to \(L=24\). This is the opposite of the phase change the experiment hunted for. We entered Track 2 on the hypothesis that rising complexity might open a door for learning — the regime where deterministic computation runs out of room. Instead, rising complexity closes the door: the very axis that breaks the deterministic ceiling breaks the learner in lockstep.

7 · The seat criterion and why nothing meets it

We can now check the four-conjunct seat criterion of §2.7 directly against the evidence, and see that it fails at the very first conjunct in every cell.

1. Clears vs the best feasible rung. Fails everywhere. In the design world (Table 3) the learner never reaches \(+15\%\) against any rung; across 40 worlds per \(L\) (Table 4) it clears even the necessary vs-order-1 screen in only \(1\) of \(160\) cells, and that single cell (world 37) fails the ceiling check. Zero of 160 cells clear the gate against the best feasible rung.
2. Irrecoverability (clears vs every rung up to the strongest feasible one). Moot given conjunct 1, and independently refuted by world 37, whose vs-order-3 margin is \(-0.36\%\).
3. Reproducibility (\(\geq 90\%\) of 40 worlds). The best observed clearance rate of any necessary screen is \(2.5\%\) (\(L=4\)), two orders of magnitude below the \(90\%\) bar.
4. Holm–Bonferroni over the \(L \times\) rung grid at \(\alpha=0.05\). With no cell surviving conjuncts 1–3, no positive result reaches the multiple-testing stage; the family-wise correction only strengthens the negative conclusion.

The seat criterion is not met at any \(L\). The verdict is Outcome A: \(H_0\) — deterministic sufficiency — held, and could not be killed, across 40 held-out worlds per \(L\), with zero earned seats in 160 world×L cells.

8 · The map

The result is not "the floor beats ML." It is a measured map of the (latent-complexity × deterministic-order × learned-inference) space in one honest domain, with two regions and a wall between them.

• Low complexity. Deterministic computation consumes essentially all economically useful structure. Even the oracle — the upper bound on any model — has headroom below the deployment threshold (\(+11.49\%\) at \(L=4\), \(+7.03\%\) at \(L=8\)). There is no room for a learned seat because there is no room for any seat.
• High complexity. Bounded-order deterministic computation collapses to the naive floor, structurally rather than for lack of data (§4). A feasible learned latent model does not recover the difference; it decays alongside the ceiling and falls below the naive floor itself. No feasible method earns a seat.

The load-bearing observation ties the two regions together. The one place a learned seat could exist is exactly where inference itself becomes intractable — high latent complexity. But intractability is not selective. Exact inference of a Bayesian network is NP-hard in general (Cooper, 1990), and — crucially — even approximating probabilistic inference in Bayesian belief networks to within a fixed factor is NP-hard (Dagum & Luby, 1993). The learner's only feasible option in the intractable regime is approximate inference, and approximate inference inherits the same wall. This is why the wall is symmetric: it is not that the deterministic side happens to be weak here and the learned side happens to be weak too, but that both sides are trying to do the same intractable thing, and the intractability defeats them together. In this domain the computation-to-inference transition the experiment hunted for does not open a feasible door.

The intractability wall defeats deterministic and learned inference symmetrically. Where deterministic computation stops sufficing, feasible learned inference stops sufficing too.

9 · Limitations and scope

This is a single synthetic domain — noisy-OR sequential fault diagnosis — with deliberately small pools: 40 worlds per \(L\), 200 test incidents per world, and \(n_{\text{train}}=1500\) as the frozen primary training size (with the data-scaling sweep of §4 as the counter to sample-starvation, not a claim of large-sample power). The latent dimension is capped at \(L=24\) by the \(2^L\) oracle and exact anchors. The claim is therefore bounded to this generative family and this operational objective; it is a falsifiable map, not a universal statement that learned inference is useless. In particular, the symmetric-wall argument depends on the intractability being shared — a domain engineered so that no bounded-order deterministic model is consistent with the data, while a feasible learned inference remains available, is exactly the pre-declared next boundary. Consistent with the program's posture, Outcome A's mandated action is to find that next boundary, not to invent another learner to force a win in this one. Finally, this page is moat-scrubbed: the mechanisms of the production floor are withheld; what is disclosed is the synthetic benchmark, its frozen gate, and its result.

10 · Conclusion

We set out to kill deterministic sufficiency in the one regime where a learned seat could plausibly be irrecoverable — high latent complexity, where exact deterministic computation becomes intractable — and we could not. We first built and validated a deterministic order ladder that establishes the strongest feasible bar and shows it collapse structurally under intractability; we showed the collapse survives \(16\times\) more data; we built a poly-time variational learner whose feasibility wall coincides with the exact deterministic wall and confirmed, with exact inference of its own parameters at the tractable boundary, that its failure is signal-inaccessibility rather than approximation error; and we replicated across 40 held-out worlds per \(L\), observing the anti-transition and catching a false-positive seat live. Against a gate frozen and hashed before any competitor existed, the learner earned a seat in zero of 160 world×L cells. The contribution is a measured, falsifiable map of where each mechanism earns authority — and the precise, pre-declared conditions under which learned inference finally would.

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How to cite

@techreport{perslis2026irrecoverability,
  title       = {The Irrecoverability Boundary: Where Learned Inference Cannot Recover What Deterministic Computation Cannot Reach},
  author      = {{Perslis Research}},
  institution = {Perslis Research},
  year        = {2026},
  month       = {9},
  type        = {Preprint},
  note        = {Pre-registered, adversarial. Prereg sha256 b51ee352...; learner addendum sha256 94cb2eaa...},
  url         = {https://research.perslis.com/irrecoverability.html}
}