BDH-CQ: In-Context Learning with Recurrent Latent Reasoning — part 2/3

Cleaned-text code points [15992, 31907). Bibliography omitted here; source.md includes it.

The ConceptARC profile shows where performance differs across broad concept families, but not how an individual operation behaves as its demands increase. We therefore construct controlled tasks after freezing the model and vary one source of complexity at a time. The experiments carry forward boundary propagation as a relative strength, copying and ordering as pressure points, and nested containment as a direct test of increasing relational depth.

We generated fresh ARC-like tasks after freezing the model. Every task contains three demonstrations and two held-out inputs, and every target output is produced by a deterministic oracle. Four families vary one factor while holding the rule fixed: horizontal propagation distance, the number of independent motif copies, the number of bars that must be sorted, and the number of nested containment relations. Demonstrations use propagation distances 1–3, one or two copies, sequences of length 2–4, and nesting depths 1–3. Held-out instances extend beyond those ranges. We use six tasks per level for propagation and copying and, after an initial pilot, 18 per level around the ordering and nesting transitions (12 for the two longest ordering levels).

Figure 4 shows one held-out input and oracle output from each family at the upper end of the tested range. Propagation and copying enlarge a locally specified operation; ordering requires constructing a sequence from eight spatially separated objects; and nesting requires resolving five containment relations before recoloring the selected cells.

Figure 5 shows sharply different generalization patterns. Propagation remains correct on 48/48 held-out outputs at both pass@1 and pass@2 across distances 2–8, and copying remains correct on 48/48 as the number of target sites increases from one to four; neither reaches a ceiling in the tested range. Ordering is nearly saturated through five objects, then falls to 29/36 outputs at length six, 8/24 at seven, and 1/24 at eight at pass@2. Nesting remains nearly saturated through depth four but falls to 29/36 at depth five.

The two declines have different signatures. At ordering length eight, only 3/24 held-out outputs have the correct dimensions. By contrast, all 36 depth-five nesting outputs have the correct dimensions, and mean best-candidate cell accuracy exceeds 99.9%; errors typically differ from the target in a single containment decision. Ordering failures therefore affect construction of the output as a whole, whereas nesting failures usually preserve its structure and make a localized relational error.

To determine if the source of the model’s failure is its execution capacity or its extrapolation ability, we rerun byte-identical length-eight ordering and depth-five nesting inputs under contexts of varying complexity. A “short” context stops at length or depth three to four; a “supported” context includes one demonstration at the test complexity. As Table 3 shows, matched support raises depth-five nesting from 19/24 to 24/24 exact outputs at pass@2. For ordering, it recovers 13/24 outputs from a 0/24 baseline. The nesting cliff is therefore largely a failure to extrapolate the demonstrated relation depth. Long ordering also benefits from support, but retains an execution bottleneck.

### 6.3 Context binds dense mappings, but composition is operation-dependent

To probe contextual memory directly, each task defines a fresh color permutation through its demonstrations and requires elementwise application to two held-out sequences. The system solves all 96 held-out outputs at rank one as the number of simultaneous bindings increases from two to eight (24/24 at every level). The model therefore recovers and consistently applies a relatively dense task-specific mapping introduced entirely through context.

To determine the ability of the model to compose operations, we compare the model’s performance on a new experiment where the demonstration examples are either a single operation or a composition of operations. Each demonstration and test input board consists of a 3x3 grid of colors, which we call a motif, placed in the top left corner, as shown in Figure 6 . We test horizontal reflection, a demonstration- defined swap of two colors, and clockwise rotation alone. Additionally, we also test demonstration examples which consist of the composition of each one of these operations with relocation of the motif to a marked position. Three dense, asymmetric 3 × \times 3 motif families provide 72 held-out outputs per condition. One family, the original family, has a fixed color layout, where different motifs in the family differ only by a permutation of colors. There are also two shuffled families, where motifs do not follow a fixed color layout, in order to balance color multiplicities. Across families, we hold the canvas, upper-left source position, single-cell anchor, demonstrations per task, and semantic operations constant.

Relocation alone, atomic reflection, and atomic rotation are each solved on 72/72 held-out outputs. Rotation composed with relocation is solved on 72/72 tasks, while reflection composed with relocation is solved on 47/72 tasks. The model learned to compose reflection with relocation on most color layouts, but not all. Color swap is acquired atomically only in the original family (26/72 pooled) and the model never learns to compose it with relocation (0/72). The two shuffled families each reach only 1/24 in isolation, showing that the fixed layout in the original motif family made the swap substantially easier to infer; their composed failures cannot be attributed to composition alone. Figure 6 illustrates the task construction, and Table 4 reports the pooled results.

### 6.4 Within-task consistency

The 18.5-point gap between semantic ConceptARC pair accuracy (77.92%) and strict task accuracy (59.38%) is not just a consequence of a stricter metric. Under pass@2, the system solved zero test pairs for 13 tasks, one for 15 tasks, two for 37 tasks, and all three for 95 tasks. Thus, 52/160 tasks have one or two correct test inputs but are not solved as tasks. Under a rule-induction account of ARC, a correctly induced rule should transfer to every test input of its task; this partial success is evidence that the observed transformation is not applied consistently across inputs.

### 6.5 Identifier and batch-context replication

The most direct request-side confound is that the system could exploit a semantic task_id or a concept-grouped batch context rather than the grids. The opaque replication replaced identifiers with cryptographically opaque labels and mixed concept areas within batches. Aggregate performance did not move: it solved 374/480 test pairs in both conditions and 96/160 opaque tasks versus 95/160 semantic tasks. On the paired test-pair outcome, there were exactly six semantic-only and six opaque-only pass@2 successes. This supplies no directional evidence that removing those request-side cues changed performance.

Aggregate stability did not imply identical outputs. Across the two ConceptARC executions, 442/480 first candidates, 276/480 complete ordered candidate lists, and 455/480 attempt counts agreed. Request context, call time, or an internal search path can therefore change the candidates returned without changing the aggregate score. The replication rules out this combined identifier-and-batch confound as an explanation for the observed ConceptARC score; it does not make ConceptARC a fresh benchmark, rule out exposure through training or checkpoint selection, or isolate the two interventions factorially.

### 6.6 Attempt delivery, repeatability, and effort

In the opaque ConceptARC replication, all 75 single-candidate records were already correct at rank one. Consequently, nominal pass@2 should not be interpreted as the result of two independently sampled attempts for every input.

On the full public ARC-AGI-1 evaluation, the MIN effort setting cost one third of standard ( $ 0.00088399 \$0.00088399 versus $ 0.00265246 \$0.00265246 per task) and scored 111/400 rather than 118/400 pass@2, a difference of − 1.75 -1.75 percentage points. The paired split was 105 tasks solved by both settings, 13 standard-only, 6 min-only, and 276 neither (two-sided exact McNemar p = 0.167 p=0.167 ). All four reported endpoints favor standard in point estimate, but this comparison is statistically unresolved.

Repeated identical requests were byte-identical at both effort tiers: the standard re-run matched all 419 ARC-AGI-1 test inputs from the earlier standard run, and repeated min runs also matched all 419.

## 7 Scaling reasoning effort increases pass@2

In this section we evaluate the impact of reasoning effort (latent thinking effort) on the pass@2 and cost.

For this task, we train the model changing the levels of latent reasoning during training. In doing this, we get a model that has been exposed to different reasoning efforts during training. For inference, we can choose the level of reasoning to apply. In table 5 and figure 7 , we show an increase in pass@2 for different levels of latent reasoning (three different reasoning effort levels). It is evident that adding more effort from LOW to MEDIUM and from MEDIUM to HIGH boosts the accuracy.

## 8 Related work

#### Verbalized reasoning.

Chain-of-thought prompting elicits intermediate natural-language steps and can substantially improve reasoning in sufficiently large language models ( Wei et al., 2022 ) . This mechanism combines in-context specification with an autoregressive computational scratchpad. BDH-CQ combines the same contextual flexibility with a continuous recurrent workspace, making intermediate language generation unnecessary.

#### Continuous thoughts in language models.

Coconut feeds a Transformer’s previous final hidden state back as the next input embedding and uses a curriculum that progressively replaces verbal CoT steps with continuous thoughts ( Hao et al., 2024 ) . It provided the seminal demonstration of continuous hidden-state feedback in an autoregressive language model and showed that one state can preserve multiple candidate continuations during planning. Its staged internalization curriculum is essential to the reported performance; curriculum controls in the same study recover much of the result on its logical reasoning tasks. BDH-CQ explores a different operating regime: demonstrations update recurrent task memory, and a structured multi-vector workspace solves previously unseen visual transformations. We evaluate that combination as a complete costed system and probe its binding, extrapolation, composition, and execution behavior.

Theory gives a more precise motivation than a generic appeal to higher-dimensional states. For directed graph reachability, Zhu et al. (2025) show that continuous thoughts encode multiple search frontiers simultaneously and can expand them in parallel. Their construction demonstrates the central computational opportunity: one continuous state can carry several active hypotheses. Xu and Sato (2026) characterize a broader trade-off: latent iteration can exploit parallel structure in computation graphs, whereas stochastic token decoding supplies natural mechanisms for approximate counting and sampling.

#### Compressed discrete and communicated latent states.

Abstract-CoT replaces verbal rationales with a short autoregressive sequence from a learned reserved vocabulary and reports up to 11.6-fold fewer reasoning tokens while approaching verbal-CoT post-training performance ( Ramji et al., 2026 ) . It occupies the discrete end of the nonlinguistic-reasoning spectrum: compressed and abstract, yet still serial and externally decoded. Separately, latent communication transfers embeddings, hidden states, or inference caches between agents rather than between reasoning steps within one model ( Liu, 2026 ) . This adjacent literature reinforces the need to distinguish continuous latent thought, compact abstract-token languages, and inter-agent latent channels.

#### Recurrent depth and looped models.

Recurrent-depth language models repeatedly apply a shared block to a sequence-wide latent state and can improve some reasoning benchmarks with additional iterations ( Geiping et al., 2025 ) . Looped Transformers provide empirical and theoretical evidence that many iterative problems require effective depth without requiring distinct parameters at every depth, and can simulate bounded tokenized CoT computations under stated constructions ( Saunshi et al., 2025 ; Zhu et al., 2026 ; Zhang et al., 2025 ) . These works primarily treat recurrent-depth as a way to design parameter-efficient architectures and to scale test-time compute. Our focus is demonstration-dependent learning and expression of visual transformations.

#### Task-trained recursive solvers.

HRM and TRM recursively update latent and candidate-answer states and report strong results on Sudoku, mazes, and ARC ( Wang et al., 2025 ; Jolicoeur-Martineau, 2025 ) . Their ARC pipeline is transductive: demonstration pairs from evaluation tasks are augmented and used in optimization. Each augmented puzzle receives a learned identity embedding, and predictions are voted over augmentations ( ARC Prize Foundation, 2025 ) . A previously unseen hidden task therefore requires backward-pass adaptation before it can be evaluated. BDH-CQ targets that specific conjunction: task information is written into recurrent memory through context, and latent computation applies it without puzzle-specific optimization. ARC Prize reports costs of $1.48 per task for HRM and $1.76 per task for TRM, reflecting their coupling of task-specific optimization with inference ( ARC Prize Foundation, 2025 ; ARC Prize Foundation, 2026 ) . HRM-Text subsequently demonstrated that hierarchical recurrence can also support efficient conditional language modeling at one billion parameters, with strong results on knowledge, reading, and mathematical benchmarks ( Wang et al., 2026 ) .

#### ARC and concept-organized evaluation.

ARC was proposed to study skill-acquisition efficiency under sparse evidence ( Chollet, 2019 ) . ConceptARC groups ARC-like tasks around a designed ontology of spatial and semantic concepts, enabling analyses hidden by one aggregate score ( Moskvichev et al., 2023 ) . We use this ontology as the organizing map for capability analysis.

## 9 Discussion

The system-level result is direct: a 150M-parameter BDH-CQ reaches 29.5% pass@2 on the public ARC-AGI-1 evaluation set at a computed $0.00070 per task. This breaks through the reported cost–accuracy Pareto frontier and establishes a new state of the art in benchmark cost efficiency. The unusually low starting cost also creates room to increase model capacity and reasoning compute while remaining competitive with existing systems.

The controlled experiments show what that combination can express. BDH-CQ applies dense task-specific color mappings to all 96 held-out outputs, demonstrating substantial contextual binding capacity. Boundary propagation and copying extrapolate throughout the tested ranges, while ordering and nested containment expose distinct limits. Matched demonstrations eliminate the observed depth-five nesting failures and substantially improve length-eight ordering, showing that coverage in the in-context examples can determine whether a learned operation extrapolates. Composition is more representation-dependent: rotation composes with relocation on all 72 held-out outputs, whereas reflection succeeds on 47/72 and varies sharply across motif families; color swapping is not acquired reliably outside the original fixed layout. The resulting picture is not a single scalar notion of reasoning difficulty, but a capability profile shaped by contextual binding, demonstrated support, output construction, and the representation through which operations are composed.