The slot-stationarity collapse boundary in learned variable-cardinality semantic schemas — a negative result

Status

Closed. A two-part negative result.

• Part 1 (single seed): a decisive failure with a sharply diagnosed mechanism. Training itself was healthy — total loss descended monotonically — but the learned-schema encoder collapsed to a degenerate cluster structure for a clearly identified structural reason. The failure is informative, not noise.
• Part 2 (three repair mechanisms across three seeds each): no mechanism cleared the upgrade bar. The collapse boundary stands as a boundary finding, now sharpened against three principled mitigations.

Part 1 — The slot-stationarity collapse

The design under test

The work pursued a learned semantic schema with variable cardinality: multiple encoders (text, visual) each emit a fixed-width set of entity slots, supervised against a ground-truth set of entity prototypes via a permutation-invariant set-reconstruction loss (per-batch assignment between encoder slots and ground-truth prototypes). The intent was to deliberately remove the canonical-ordered slot assignment used in the prior phase, so that cardinality could be discovered rather than fixed in advance.

What happened

The schema-discovery hypothesis — predicting an entity count within a tolerance band of ground truth — failed decisively: predicted entity counts were far below ground truth, with a near-zero match rate on the validation set. A decode-side threshold sweep ruled out the clustering layer as the culprit: loosening the cosine cluster threshold all the way to zero made the collapse worse, confirming the collapse lives at the embedding level, not the decoder.

The cross-encoder cardinality-match hypothesis "passed" only trivially: both encoders agreed because both converged to the same wrong answer. This is degenerate agreement, not mechanism validation. A downstream reasoning hypothesis failed against its gate, though the reasoner did extract some above-chance signal from soft edge endpoints.

Root cause: non-stationary per-slot supervision

The mechanism is precise and is the core finding:

1. Per batch, permutation-invariant matching assigns a given ground-truth prototype to some slot — say one slot in one batch.
2. In a differently shuffled batch, that same prototype may be assigned to a different slot, and the first slot may receive a different prototype.
3. So each slot's supervision target embedding is non-stationary across training — sometimes one prototype, sometimes another.
4. The encoder's loss-minimum for a slot is therefore the mean of all targets ever assigned to it — approximately the mean of all prototype embeddings.
5. All slots receive roughly the same average target, so all slots converge to near-identical embeddings; cosine clustering merges them into one or two groups; variable-cardinality decode then produces one or two entities regardless of true cardinality.

This is a fundamental defect of slot-permutation-invariant matching combined with continuous-target supervision. The prior phase did not suffer it because canonical-ordered supervision gave every slot a stationary assignment — exactly the property removed here in pursuit of learned cardinality.

What the single-seed evidence does establish (positives)

• Trainer infrastructure is sound. Total loss descended cleanly; all loss components trained without divergence; the temperature anneal worked. Only the loss design is at fault.
• Per-component observability is informative. The matched-slot embedding sub-loss and the relation-embedding loss both reached near-zero — the encoder did learn to place embeddings near prototypes. The single broken dimension is "the encoder uses its slots distinguishably." Everything else worked.
• The reasoner partially learns despite collapse — above-chance accuracy, likely via positional information retained through the soft edge endpoints.

Pre-declared scope

Claims: naive permutation-invariant set-reconstruction causes representation collapse in this specific small-scale setup, and the collapse is mechanistically attributable to slot-permutation non-stationarity. Does not claim: that all learned-schema designs are doomed (only this naive design), that variable-cardinality state is impossible, or multi-seed stability of the collapse (single seed here).

Part 2 — Three principled repairs, all fall short

The second part attacked the boundary structurally at the matching layer only — encoders, reasoner, and prototype table frozen; only the assignment loss module swapped. The goal was to upgrade the boundary from boundary-only to boundary-plus-repair. Three structurally distinct candidates spanning a (hardness × temporal-scope) axis were tested:

• A — Slot-anchored cross-entropy: each slot owns a slowly trained learned anchor embedding; prototypes are assigned deterministically by affinity to anchors (hard, every step).
• B — Soft optimal-transport with anchor prior: soft optimal-transport assignment with a per-slot anchor-prior term added to the transport cost (soft, every step).
• C — Persistent-warmup assignment: the ground-truth-to-slot mapping is frozen for a warmup window, after which soft reassignment is unlocked (hard initially, then free).

Non-slot-indexed set losses (e.g. Chamfer / Frechet variants) were deliberately excluded — they do not preserve the slot-indexed state representation needed downstream — and deferred to future work.

Five strict, pre-registered hypotheses

1. Slot-identity persistence across batches.
2. Reconstruction loss does not ascend during the anneal window (the canonical collapse signature from Part 1).
3. Per-slot assigned-prototype entropy inside a diagnostic band (neither collapsed-to-one nor near-uniform).
4. Cross-encoder set-equivariance recovered without the degenerate low-cardinality agreement.
5. No regression when the repaired encoder is dropped into the downstream cursor-routing policy.

Closure rule: a mechanism upgrades the boundary only if at least 4 of 5 hypotheses pass (majority across seeds for each).

Result: every mechanism caps at 2 of 5

Per-mechanism majority outcome (three seeds each):

The mechanism story behind the numbers:

• A and B achieve perfect persistence but a degenerate collapse. Persistence is maximal and entropy is essentially zero on every seed — the anchors win so hard that each slot collapses to a single prototype. That is the opposite failure from Part 1 (over-rigid rather than mean-of-all), but it still fails the entropy band and still fails set-equivariance (cross-encoder same-count agreement far below gate, with predicted counts scattered). Trivial persistence with zero entropy is not a real fix.
• C avoids the degenerate collapse (entropy lands in-band) but cannot hold slot identity once warmup ends — persistence decays substantially, and post-warmup the encoder collapses to a single entity. Warmup buys stationarity that does not survive the unlock.

So the three structural axes fail in different ways — hard anchoring over-rigidifies, soft anchoring over-rigidifies, temporal anchoring decays — and none clears the bar. This is a stronger boundary statement than Part 1 alone: the boundary now stands against three principled, structurally distinct mitigations, each adversarially characterized rather than merely "not tried."

The drop-in gate is architecturally undefined

The strictest gate — dropping the repaired encoder into the downstream cursor-routing policy — turned out to be architecturally undefined, not merely failed. The repaired encoder emits a whole-situation graph-set encoding (entity embeddings, soft edge endpoints, relation embeddings, used gates). The downstream policy consumes a per-step cursor tuple (current entity, requested relation, step index, activated count, candidate count) embedded fresh each routing hop. There is no socket connecting the two representations, and the policy evaluator has no encoder-injection seam. A genuine drop-in would require a new state-bridge adapter (future work). Rather than fabricate a bridge or fake metrics, the evaluation returns an honest "architecturally undefined" status, counted as a fail (drop-in survival untestable, therefore not demonstrated). This cross-frontier seam gap is recorded as a separate finding: drop-in compatibility between the graph-set encoder and the per-step cursor representation is undefined under the frozen designs, relating directly to architecture coherence.

Pre-declared scope (failure case)

Claims: the three structural axes (hard anchor / soft anchor / warmup-temporal) all fail at this scale; the slot-stationarity collapse is reinforced as a hard architectural boundary for slot-indexed variable-cardinality learned-schema work at this scale. Does not claim: that all learned-schema designs are doomed (non-slot-indexed Chamfer/Frechet variants remain unexplored), or that the broader research goal is impossible.

Findings recorded

• Slot-stationarity collapse boundary: a multi-encoder system with variable-cardinality slot output, trained against permutation-invariant set-matching supervision, exhibits representation collapse when slot identity is not anchored; the encoder optimum under non-stationary per-slot supervision is the mean of all targets ever assigned to that slot. Mitigations (canonical-ordered assignment, per-slot anchor priors, warmup-persistent assignment) characterized as insufficient at this scale.
• Set-reconstruction collapse: the specific instantiation of the boundary under permutation-invariant set-reconstruction supervision.
• Cross-frontier drop-in is architecturally undefined: the seam between the graph-set encoder and the per-step cursor representation is undefined under the frozen designs (an architecture-coherence boundary).

Why this is worth publishing

A clean, mechanistically attributed negative is more durable than a fragile positive. Part 1 converts "learned variable-cardinality schema didn't work" into "permutation-invariant set-reconstruction without slot anchoring has a named, reproducible collapse mode." Part 2 then shows the boundary survives three principled repairs, each failing in a characteristic and different way. Together they define an operating limit rather than report an absence of success.