BDH-CQ: In-Context Learning with Recurrent Latent Reasoning — part 3/3

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The present study establishes this account for ARC-like visual reasoning. The ConceptARC ontology organizes the learned capability profile, while post-freeze generators test generalization and localize current boundaries in consistency, composition, conditional execution, and output construction.

### 9.1 Outlook

BDH-CQ points toward a broader family of BDH-based reasoning systems in which memory, adaptation, and inference are part of the same computational fabric. The immediate direction is to scale BDH-CQ itself. The current result comes from a compact model and leaves a large cost budget for increasing capacity while retaining a favorable operating point. Larger models and longer training runs will test whether BDH’s favorable sequence-model scaling carries into in-context latent reasoning and whether the capability boundaries identified here move predictably with scale.

The second direction is breadth. ARC-AGI-2 is the next visual-reasoning target ( Chollet et al., 2025 ) ; the present diagnostics provide a concrete development agenda around output construction, conditional binding, demonstration coverage, and multi-operator composition. Sudoku and other constraint-satisfaction domains offer complementary tests of long-horizon latent refinement ( Pathway Research, 2025 ) . Language and mathematical reasoning will test whether the same recurrent memory can acquire tasks from textual demonstrations while retaining BDH’s sequence-model capabilities. Because BDH layers support language modeling as well as latent reasoning, future systems may also combine continuous internal computation with verbalized steps when communication, verification, or tool use requires them.

Finally, BDH-CQ keeps the strengths of language models, but pushes beyond token-by-token processing toward parallel latent reasoning. BDH-CQ is one instance of this direction rather than its endpoint. Because BDH layers support language modeling as well as latent visual reasoning, future BDH-based systems could combine continuous and verbalized chains of thought in one recurrent architecture. Combining such systems could retain the bandwidth and efficiency of latent computation while decoding intermediate language when communication, verification, or tool use benefits from it. This direction is motivated by evidence that language and nonlinguistic reasoning are partly separable in humans ( Fedorenko and Varley, 2016 ) , without assuming that the models reproduce the corresponding biological mechanisms.

### 9.2 Scaling model size

The BDH-CQ architecture scales naturally to large model sizes, admitting tensor sharding patterns inherited from the BDH architecture that make it particularly easy to train at 1T scale. Early experiments confirm Transformer-like scaling laws apply during pretraining at scales from 1B to 600B parameters, while preserving the latent reasoning capabilities specific to BDH-CQ.

## 10 Conclusion

BDH-CQ demonstrates that in-context learning and recurrent latent reasoning can coexist in a compact, practical system. Demonstrations modify recurrent memory at inference time, and the resulting task is solved through iterative continuous computation rather than a verbalized chain of thought. On ARC-AGI-1, this design reaches 29.5% pass@2 for $0.00070 per task, establishing a new state of the art in benchmark cost efficiency. The controlled experiments further show that the system binds new mappings, extrapolates several learned operators, benefits predictably from demonstration coverage, and exhibits structured rather than uniform limits under composition. Together, these results establish in-context latent reasoning as a concrete path toward systems that learn new tasks from context while reasoning beyond the token stream.

## Appendix A Evaluation set analysis

### A.1 Evaluation sets and surface descriptors

The concept-organized experiments above isolate individual transformations and structural demands. We complement them by profiling the public ARC-AGI-1 evaluation set, a generated set calibrated to match its measured distribution, and a generated set stratified by required operation. We report this quantity as exact whole-task solve rate rather than test-pair accuracy.

Table 6 summarizes the three cohorts.

Simple surface properties provide little explanation for success. We evaluate 39 bucketings of grid size, color count, object count, and demonstration count on the public set and 38 on the calibrated generated set. Nine public-set bucketings exceed the permutation search null, while none does so on the generated set. Grid size is the strongest individual descriptor but remains weak, with reported pseudo- R 2 = 0.072 R^{2}=0.072 on public tasks and 0.010 0.010 on generated tasks. Thus, the measured differences in success cannot be reduced to one coarse surface descriptor.

### A.2 Transformation mechanics

We next group tasks by the operation required to solve them, which we call the task’s mechanic . Unconstrained generation, without controlling the requested transformation mechanic, produces roughly 80 rotation tasks for every three gravity-and-stacking tasks, leaving rare mechanics too sparsely sampled to estimate. We therefore generate approximately balanced samples across 16 mechanics, yielding 1,131 tasks. The resulting profile is shown in Table 7 .

Mechanic is a useful diagnostic stratifier for observed solve rate in this generated set. The solve rates span 65.7 percentage points, from 68.6% for flood fill to 2.9% for gravity and stacking. In 20,000 permutations of solved labels at fixed group sizes, the 95th-percentile spread is 27.0 points and no permutation matches the observed spread ( p < 0.00005 p<0.00005 ). At the category level, mean grid side has essentially no linear association with mechanic solve rate ( r = − 0.022 r=-0.022 ), and panel set operations have the smallest mean side but the second-lowest solve rate. This aggregate correlation does not control for size variation within mechanics.

This profile is not an intrinsic ranking of the operations. Every stratified task was authored by GPT-5.6 from a prompt naming the target mechanic, so each rate also reflects how that generator instantiates the requested operation. Public and generated mechanic rates do not show a reliable rank correspondence ( ρ = 0.300 \rho=0.300 across 16 mechanics). Hand-written comparison families expose a substantial task-construction confound: hand-written gravity tasks score 80–100%, compared with 2.9% for generated gravity-and-stacking tasks, and hand-written counting tasks score 85–95%, compared with 18.6% for generated object counting. Touching-panel overlays remain difficult under both authoring modes, scoring 7.5% when hand-written and 10.4% when generated. The mechanic table therefore characterizes this evaluation distribution, while controlled generators are required to isolate properties of the operation itself.

### A.3 Controlled structural manipulations

The mechanic taxonomy describes which transformation a task requires, but does not encode how demonstrations select or parameterize that transformation. We construct ten deterministic Python-generated ladders to vary such properties directly. Each step holds the grid at 12 × 12 12\times 12 , uses three demonstrations, one test pair, and a fixed palette, and contains 40 tasks. The retained analysis contains 1,840 task evaluations. The selection results use a corrected 240-task rerun because the original generator omitted a required rule from some demonstration sets. Table 8 summarizes the largest observed changes.

For conditional selection, a corner marker determines which of two demonstrated rules applies. A control in which the marker varies but both values invoke the same rule is solved on 40/40 tasks. Pooling color, position, and count cues, genuine selection between two rules is solved on 68/120 tasks, or 56.7% ( 95 % 95\% CI [ 48 , 65 ] [48,65] ), a decrease of 43.3 points ( p = 9 × 10 − 9 p=9\times 10^{-9} , two-sided Fisher’s exact test). The control shows that the decrease attaches to using the cue to select a rule, rather than to the presence of a varying marker.

A second ladder uses marker count to specify a downward shift. When the test value appears among the demonstrations, the model solves 12/40 tasks. When the value is absent, it solves 0/120: 0/40 for an unseen value interpolated between demonstrated values and 0/80 for two extrapolation conditions ( p = 1.4 × 10 − 8 p=1.4\times 10^{-8} , two-sided Fisher’s exact test, pooled against the demonstrated-value condition). Within this family, the relevant boundary is whether the value is demonstrated, not whether it lies inside the demonstrated range. The demonstrated-value condition itself reaches only 30.0%, so the result may share the conditional-parameterization limitation observed in rule selection.

For panel union, two panels at opposite corners are solved on 26/40 tasks, whereas three panels are solved on 1/40. The generator holds answer color and expected union density fixed across panel counts. Separation matters even before a third panel is added: two touching panels score 3/40, while opposite-corner panels score 26/40. This pattern is consistent with a segmentation limitation, although the analysis does not directly test the model’s segmentation.

Other manipulations show no detectable cost over the tested range. Four axis-aligned operations in sequence are solved on 38/40 tasks, counting 10–12 objects reaches 34/40 compared with 36/40 for one to three objects, and three independently moving objects are solved on 40/40. Within the axis-aligned family, operation count carries no detectable cost over the tested range. Dependency between objects behaves differently: support chains decrease monotonically from 80.0% with one object resting on another to 67.5%, 52.5%, and 27.5% as the chain grows to three, five, and eight objects. The three-object independent control shows that multiplicity alone does not cause failure at that scale. Object count and dependency depth co-vary in the longer chains, so the gradient is consistent with a dependency cost but does not isolate it from object count at every level.

### A.4 Failure structure and scope

Finally, we compare incorrect predictions on the 400-task public and calibrated generated sets. The calibrated set contributes 251 failed tasks and the public set 282. As Table 9 shows, most failures preserve output dimensions and palette, and shape-correct failures often differ from the target in a small fraction of cells.

These output-level signatures characterize the final grids but do not identify the latent rule the model followed. The evaluator returns grids rather than a reasoning trace, so a correct rule applied incompletely is observationally indistinguishable from a narrower rule applied completely.

The generated analyses have additional scope limitations. Manual review found at least one task whose stated test output contradicts the rule expressed consistently by its demonstrations; the prevalence of this defect is unknown. Independent LLM mechanic labels agree with the requested label on 82.9% of the nonrandom 659-task subset that was labeled, and some mechanic definitions overlap. Deduplication removed 24% of the merged generated pool, concentrated in flood fill and panel set operation; additional batches produced few new distinct forms at these extremes. Each ladder condition contains only 40 tasks from one puzzle family. Consequently, the results establish strong within-family differences for conditional selection, unseen parameter values, and panel number and separation, together with a graded association for support-chain length; they do not yet establish that these differences generalize across visual operations.

Taken together, the evaluation set analyses separate surface appearance from transformation and structural demand. Coarse descriptors such as grid size weakly predict success; operation-based groups reveal large differences but remain confounded by task authoring; and deterministic ladders isolate sharper limitations in selecting and parameterizing demonstrated rules. Many incorrect outputs preserve output dimensions, and among shape-correct failures many differences are localized. These outputs are consistent with substantial partial recovery, although final grids alone cannot establish where the underlying reasoning failed.

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