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Blackboard Intelligence Can Surpass Autoregressive on Globally Constrained Problems

Woosang Jeon, Jaeyeon Kim, Sham Kakade, Yilun Du, Amrit Singh Bedi, Arun Kumar Chithanar, Chul Lee, Taehyeong Kim, Sitan Chen

arXiv:2609.38806v1cs.LGcs.CL

TL;DR

Next-token models can struggle when solutions must satisfy complex global constraints, raising the question of whether their inference interface contributes to that difficulty. The paper uses diffusion models as revisable blackboards, with mean confidence guiding search over partial states. Blackboard improves inference across three domains with the checkpoint fixed, including 90.4% accuracy on ZebraLogic-Hard.

  • Problem

    Next-token prediction can struggle on globally constrained problems, where maintaining coherence across interacting decisions and revising earlier choices is difficult.

    Source: §1 · p. 2, §3.1 · p. 4

  • Method

    Blackboard uses diffusion language models on a fixed, revisable canvas, with mean confidence guiding inference-time search and revision over partial solution states.

    Source: §1 · p. 2, §3.2 · p. 5, §4.1 · p. 7

  • Results

    Blackboard consistently improves inference across three domains and outperforms same-scale autoregressive baselines with the checkpoint fixed, reaching 90.4% accuracy on ZebraLogic-Hard.

    Source: §1 · p. 2, §5 · p. 9

  • Takeaways & Limitations

    The results suggest that globally constrained problem solving may depend on what intermediate state inference exposes and how that state can be evaluated and revised.

    Source: §5 · p. 9

    The study focuses on globally constrained problems; how far blackboard intelligence extends beyond this regime remains an open question.

    Source: §5 · p. 9

Abstract

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Next-token prediction has driven remarkable progress in large language models, yet a growing body of evidence suggests that they can struggle on problems governed by complex global constraints. In this work, we focus on this regime and ask whether some of these limitations arise from the inference interface induced by next-token prediction itself. We study this question through blackboard intelligence: an inference-time perspective in which a model works on a fixed, revisable canvas and searches over candidate solution states rather than committing to a causal, left-to-right trajectory. We instantiate this idea with diffusion language models, whose any-order prediction interface naturally exposes predictions over partially filled solution states. Our key observation is that mean confidence, a simple model-internal quantity available from the standard masked diffusion objective, provides a useful proxy for global coherence and can guide inference-time search and revision. Empirically, across ZebraLogic, Nurse Rostering, and Job-Shop Scheduling, Blackboard consistently improves inference while holding the fine-tuned LLaDA-8B-Instruct checkpoint fixed and substantially outperforms same-scale autoregressive baselines, reaching 90.4% accuracy on ZebraLogic-Hard, 76.4% exact feasibility on Nurse Rostering, and 80.2% optimality on JSSP. Stronger autoregressive search and refinement also fail to close the gap on ZebraLogic-Hard, while Blackboard surpasses tested frontier LLMs there and on JSSP despite their substantially greater scale and strong test-time reasoning. We open-source our codebase at https://github.com/jwoosang1/blackboard-intelligence.

1 Introduction

The paper asks whether next-token prediction’s inference interface contributes to difficulties on globally constrained problems. It introduces blackboard intelligence using diffusion language models and reports improved inference across three such domains.

1 Introduction

  • The work examines whether some difficulties with complex combinatorial or logical constraints arise from the next-token inference interface, beyond model scale or data.It frames revisable, partial solution states as an alternative to a causal left-to-right trajectory.Source: §1 · p. 2
  • Blackboard intelligence uses a fixed canvas for iteratively building and revising candidate solutions, with diffusion language models providing an any-order inference interface.The paper presents masked diffusion as a natural instantiation of this approach.Source: §1 · p. 2
  • Mean confidence provides a model-internal signal of state-level global coherence and guides inference-time search, revision, and adaptive intervention.The signal arises from the standard masked diffusion objective and predictions over arbitrary masked positions.Source: §1 · p. 2
  • With the fine-tuned LLaDA-8B-Instruct checkpoint fixed, Blackboard improves inference across three domains, reaching 90.4% accuracy on ZebraLogic-Hard, 76.4% exact feasibility on Nurse Rostering, and 80.2% optimality on JSSP.The domains are ZebraLogic, Nurse Rostering, and JSSP.Source: §1 · p. 2
  • LLaMA-3.1-8B Instruct reaches 35.0% with wider beam search and 36.6% with verifier-guided refinement on ZebraLogic-Hard, while tested frontier models remain below Blackboard on ZebraLogic-Hard and JSSP.The passage names GPT-5.4 and Gemini 2.5 Pro among the frontier models.Source: §1 · p. 2
  • The results point to an interface-level distinction: persistent partial solutions can be evaluated and revised during inference, unlike in a standard next-token decoding pass.The evaluated autoregressive search and refinement procedures do not recover comparable gains from additional inference-time computation.Source: §1 · p. 2

2 Preliminaries and Framework

The section contrasts causal next-token generation with masked diffusion’s predictions over unrevealed positions. It outlines how diffusion inference selects positions to unmask, including confidence-based greedy decoding.

2 Preliminaries and Framework

  • The notation defines pdata as the target distribution over discrete sequences and x<i as the prefix before position i.Sequences have vocabulary V and length L.Source: §2 · p. 2
  • Autoregressive models approximate the next-token posterior from the preceding prefix and generate by repeatedly sampling and appending the next token.The section writes the data distribution as a causal factorization over sequence positions.Source: §2 · p. 3
  • Masked diffusion models learn to predict clean tokens at all unrevealed positions from partially observed sequences, using cross-entropy over masked positions.The model outputs a categorical distribution for each masked position.Source: §2 · p. 3
  • Diffusion training masks a uniformly selected subset of sequence positions and trains predictions for those masked coordinates.The number of masked positions is sampled from 1 to the sequence length.Source: §2 · p. 3
  • At inference, diffusion starts from a fully masked sequence or a prompt with masked positions, then chooses a subset to unmask at each step.The process follows a decreasing time grid, and the choice of subset is flexible.Source: §2 · p. 3
  • Greedy decoding selects masked positions with the highest confidence scores, which can use maximum probability, top-two probability margin, or negative entropy.The section describes this as a common, practically effective dLLM decoding strategy.Source: §2 · p. 3

3 Mean Confidence as a Probe of Global Coherence

Across globally constrained tasks, next-token generation faces difficulty with maintaining and revising coherent partial solutions. The section develops mean confidence as a state-level coherence signal and finds it tracks solution quality on ZebraLogic and JSSP, with the same pattern reported for Nurse Rostering.

3.1 Next-token prediction under global constraints

  • ZebraLogic performance deteriorates as search spaces grow from S to XL, while Nurse Rostering and JSSP difficulty also rises with denser conflicts and larger problems.The section frames serialized generation as cumbersome for backtracking and branching because changing an earlier decision requires regenerating what follows, and local next-token training gives no explicit signal of global coherence.Source: §3.1 · pp. 3–4

3.2 Deriving mean confidence

  • Under a finite-temperature Gibbs abstraction, mean confidence averages posterior concentration across masked positions, and higher confidence tightens a bound on expected completion potential toward the best attainable potential.The Gibbs potential represents constraint violations in ZebraLogic and Nurse Rostering, and feasibility and suboptimality in JSSP; finite temperature smooths the zero-temperature solution distribution.Source: §3.2 · pp. 4–5
  • Masked diffusion predicts arbitrary unrevealed positions, making a state-level reliability signal available, whereas autoregressive next-token predictions do not natively expose an analogous signal.The model's coordinate-wise predictions approximate unmasking posteriors, which support empirical mean confidence for a partially filled state.Source: §3.2 · pp. 4–5

3.3 Empirically verifying mean confidence

  • Solved ZebraLogic puzzles show higher, more stable confidence trajectories than failed puzzles, and optimal JSSP runs show higher confidence and increasing optimal-solution rates across confidence quintiles.The same confidence pattern is reported for Nurse Rostering.Source: Figure 4 · p. 5, §3.3 · p. 6, Figure 5 · p. 6

4 Confidence-Driven Blackboard Inference

Blackboard uses mean confidence to control search and revision over partially specified solutions, improving the same fine-tuned LLaDA checkpoint across three globally constrained tasks. Autoregressive search and refinement show limited accuracy scaling on ZebraLogic, while confidence-triggered correction and exploration adapt to task structure.

4.1 Blackboard as an inference interface

  • For feasibility tasks, confidence triggers additional inference when a trajectory appears unreliable, then guides search and backtracking; for JSSP, it triggers exploration while the objective selects among completed candidates.The corrective action is adapted to task structure: search and revision for feasibility, and objective-guided candidate exploration for optimization.Source: §4.1 · p. 7

4.2 Comparison to Autoregressive Inference

  • Accuracy rises from 78.4% to 90.4% on ZebraLogic, exact feasibility from 73.0% to 76.4% on Nurse Rostering, and optimality from 70.0% to 80.2% on JSSP with Blackboard over fine-tuned LLaDA.The checkpoint is held fixed across the comparisons.Source: §4.2 · p. 8
  • Widening LLaMA beam search from B = 8 to B = 64 raises ZebraLogic accuracy only from 34.8% to 35.0%; perfect-verifier refinement reaches 36.6%, versus 34.6% under greedy decoding.Self-critique reaches 26.2%, and the tested autoregressive methods do not match Blackboard’s gains over the same LLaDA checkpoint.Source: §4.2 · p. 8
  • Blackboard reaches 90.4% on ZebraLogic and 80.2% optimality on JSSP, versus 73.2% and 77.0% for GPT-5.4 Thinking, while GPT-5.4 Thinking remains stronger on Nurse Rostering.The comparison includes substantially larger frontier models with strong test-time reasoning.Source: §4.2 · p. 8

4.3 Comparison to Existing dLLM Methods and Ablations

  • Selective confidence triggering reaches 90.4% accuracy while reducing average inference effort from 280 to 129 NFE on ZebraLogic, compared with 86.8% accuracy for always-on correction.On Nurse Rostering, selective triggering improves over greedy inference and always-on correction (76.4% versus 73.0% and 57.4%); on JSSP it reduces average NFE from 217 to 149 versus untriggered BoN while retaining comparable solution quality.Source: §4.3 · p. 9
  • PRISM improves ZebraLogic accuracy only modestly over greedy decoding (79.2% versus 78.4%), while LoPA reduces inference effort from 35 to 6.74 NFE.The methods use confidence differently: LoPA supports parallel commitment to reduce inference cost, whereas Blackboard evaluates candidate states and decides when more inference is needed.Source: §4.3 · p. 9

5 Conclusion

The conclusion frames blackboard intelligence as an alternative inference interface for globally constrained problems: mean confidence tracks partial-state coherence and controls search or revision without changing the LLaDA checkpoint. The study leaves open how far this approach extends beyond the tested regime.

5 Conclusion

  • Across ZebraLogic, Nurse Rostering, and JSSP, mean confidence tracks the global coherence of partial solutions, and Blackboard improves inference with the fine-tuned LLaDA checkpoint fixed.Stronger autoregressive inference does not consistently recover these gains with additional test-time computation.Source: §5 · p. 9
  • The study focuses on globally constrained problems; whether blackboard intelligence extends beyond this regime, and which problems benefit most, remains an open question..Source: §5 · p. 9

A Related Works

  • Prior work studies globally structured problems and confidence-driven diffusion inference; this work uses mean confidence as a proxy for global coherence to decide when to trust, search, or backtrack without changing architecture or training objective.Source: A Related Works · p. 13

B Mean Confidence under the Idealized Gibbs Model

  • Under the idealized Gibbs model, higher true mean confidence tightens an upper bound on conditional excess potential; the derivation does not directly bound the model-derived confidence used in experiments.Source: B Mean Confidence under the Idealized… · p. 14

C.1.1 ZebraLogic

  • ZebraLogic uses single-token attribute-value cells and unique-solution puzzles; its held-out hard variant has comparable search-space scale but higher Z3 conflict counts than the official benchmark.Source: §C.1.1 · pp. 14–15

C.1.2 Nurse Rostering

  • Nurse Rostering assigns staff to shifts under coupled coverage, workload, and temporal constraints, with held-out instances scored by exact agreement with the unique satisfying roster.Source: §C.1.2 · p. 15

C.1.3 JSSP

  • JSSP represents schedules as machine-by-job grids; evaluation uses CP-SAT-certified optima and counts invalid outputs as failures for optimality rates.Source: §C.1.3 · p. 16

C.2 SFT Recipe, Hyperparameters, and Compute

  • LLaMA and LLaDA are fine-tuned with matched configurations within each task, while task-specific sequence lengths and training durations vary.Training wall-clock time is not treated as a cross-task efficiency comparison.Source: §C.2 · p. 16

C.3 Pre-Fine-Tuning Task Adaptation

The section reports near-zero task success before task-specific fine-tuning, then examines confidence diagnostics, correction and refinement, and a remasking-head baseline.

C.3 Pre-Fine-Tuning Task Adaptation

  • Before task-specific fine-tuning, exact task success is near zero across all three domains; on JSSP, LLaMA yields 55/400 valid schedules (13.8%) and no optimal schedules, while LLaDA yields none.The pre-fine-tuning table reports exact feasibility for ZebraLogic-Hard and Nurse Rostering, and exact optimality for JSSP.Source: §C.3 · p. 17, Table 4 · p. 17

C.4 Prompts

  • The task prompts use Markdown-table outputs; chain-of-thought versions add a worked example and reasoning instructions, with clue-by-clue self-checking omitted for JSSP.The Nurse Rostering chain-of-thought prompt uses a reasoning–self-check–final-table structure.Source: §C.4 · p. 17, §C.4.1 · p. 17, §C.4.2 · p. 18

D.1 Main Evaluation Results

  • Table 5 covers ZebraLogic-Hard comparisons with frontier LLMs, same-scale autoregressive baselines, standard LLaDA inference, and existing diffusion inference-time methods.The supplied passage identifies the comparison scope but does not provide table values.Source: §D.1 · p. 20, Table 5 · p. 20

D.2 Separation and Grid-Size Controls

  • Solved ZebraLogic-Hard runs maintain higher mean confidence across fill intervals and grid sizes, while prediction changes are largest at direct clue links and decay with graph distance.The graph-distance pattern indicates that predictive changes are localized along the puzzle constraint graph.Source: §D.2 · p. 20, Table 6 · p. 20, Table 7 · p. 20

D.4 Sparse Correction Recovery on Greedy Failures

  • Revealing 15% of ground-truth cells solves 87.0% of greedy-failed ZebraLogic-Hard puzzles, and revealing 25% solves all failures.Most recovered puzzles require only one or two revealed cells.Source: §D.4 · p. 21, Table 8 · p. 21, Figure 8 · p. 21

D.5.1 Refinement Protocol

  • The refinement diagnostic starts from the same deterministic LLaMA solution; perfect-verifier feedback identifies violated clues without revealing correct assignments or repairs.The study uses the same fine-tuned LLaMA-3.1-8B checkpoint and does not add revision-specific supervision.Source: §D.5 · p. 21, §D.5.1 · p. 22

D.5.2 Tiered Refinement Results

  • Perfect-verifier refinement raises ZebraLogic-Hard accuracy from 34.6% to 36.6%, while self-critique lowers accuracy and stronger autoregressive inference remains far from comparable Blackboard gains.Widening beam search from B = 8 to B = 64 adds only 0.2 pp while increasing FLOPs by 4.9×.Source: §D.5.2 · p. 22, Table 9 · p. 22, §D.5.3 · p. 23

D.6.1 PRISM

  • PRISM adds a remasking head trained jointly with the unmasking objective; at inference, tokens are remasked when the head output exceeds a threshold, and the best tested configuration reaches 79.2% on ZebraLogic-Hard.The reported best configuration uses threshold 0.8 and temperature 1.0.Source: §D.6.1 · p. 23

D.6.2 Blackboard Search and Trigger Ablations

The ablations test puzzle-level triggering and lookahead depth, finding that triggering improves accuracy while reducing inference effort.

D.6.2 Blackboard Search and Trigger Ablations

  • Triggered full Blackboard reaches 90.4% accuracy versus 86.8% untriggered, while reducing average NFE from 280 to 129; it matches triggered Blackboard Search at 90.2% with less than half the NFE.The trigger improves Blackboard Search from 84.6% to 90.2% while reducing average NFE from 866 to 273.Source: §D.6.2 · pp. 23–24, Table 11 · p. 24

D.6.3 Lookahead Depth

  • Depth d = 3 performs best overall; d = 4 slightly helps X-Large but lowers Medium and Large performance, while d = 5 is lower overall.The depth sweep holds the selected trigger, branching factor, and anchor threshold fixed.Source: §D.6.3 · p. 24, Table 12 · p. 24

D.7 Official ZebraLogic Benchmark

The supplementary evaluations test Blackboard on an adapted official ZebraLogic benchmark and examine confidence and performance patterns across Nurse Rostering difficulty bands.

D.7 Official ZebraLogic Benchmark

  • On the adapted official ZebraLogic benchmark, Blackboard improves over greedy by +4.7 pp (80.9% → 85.6%) and uses fewer NFEs than its untriggered variant, 145 versus 223.The evaluation adapts all 1,000 official puzzles to a single-token grid-filling format while preserving solvable constraints.Source: §D.7 · p. 24, Table 13 · p. 25

E.1 Confidence Separation by Difficulty

  • Mean confidence separates solved from failed Nurse Rostering trajectories in every difficulty band, with moderately smaller separation as conflict density increases but a large gap remaining on d3.The bands are defined by Z3 conflicts: d0 has 0–1, d1 2–4, d2 5–9, and d3 10–31.Source: §E.1 · p. 25, Figure 9 · p. 25

E.2 Performance Across Conflict Bands

  • On Nurse Rostering, Blackboard improves over LLaDA greedy by +1.6 pp on d1, +5.6 pp on d2, and +6.4 pp on d3; d0 remains unchanged at 95.2%.GPT-5.4 Thinking has the strongest frontier result at 86.0% overall.Source: §E.2 · p. 25, Table 15 · p. 26

E.3 Why Confidence Triggering Matters

The Nurse Rostering analysis finds that confidence-triggered correction concentrates intervention on selected instances, while the JSSP table supplies the full evaluation comparison.

E.3 Why Confidence Triggering Matters

  • The trigger fires on 98/500 rosters: the cascade changes 4 greedy-correct rosters to 21 correct outcomes, yielding 19 repairs and 2 regressions, while 402 non-triggered rosters retain greedy outputs.The trigger uses mean confidence over the final 10% of filling steps and fires below τ = 0.95; always applying the cascade can overwrite already-correct rosters.Source: §E.3 · p. 26, Table 16 · p. 26

F.1 Main Evaluation Results

  • Table 17 reports JSSP evaluation results, including frontier LLMs and matched 8B autoregressive and diffusion models.The supplied passage identifies the comparison scope but does not provide the table values.Source: §F.1 · p. 26, Table 17 · p. 26

F.2 Per-Size Results

Blackboard uses confidence-triggered inference to reduce JSSP search effort while preserving near-comparable solution quality. The supplied results also examine performance by grid size, trajectory quality, trigger selection, and ZebraLogic correction sensitivity.

F.2 Per-Size Results

  • Blackboard improves most on larger JSSP grids, where greedy decoding is more often suboptimal, and ties for the highest within-5% rate on 8×8 grids.Table 18 reports % optimal, % within 5%, and mean ms_ratio by grid size.Source: §F.2 · p. 27, Table 18 · p. 27

F.3 Blackboard vs. Untriggered BoN

  • Blackboard reduces average JSSP NFEs by 31% relative to untriggered BoN, with a 1.0 pp drop in within-5% rate.The comparison is reported in Table 19.Source: §F.3 · p. 27, Table 19 · p. 27

F.4 Per-Trajectory Quality Before BoN Selection

  • Before Best-of-N selection, LLaDA produces higher-quality individual JSSP trajectories while retaining comparable schedule diversity.The comparison uses 10 samples per model on puzzles where the Blackboard trigger fires.Source: §F.4 · p. 27, Table 20 · p. 27

G.1 Confidence Trigger Selection

  • The selected JSSP trigger recalls 79.6% of non-optimal greedy schedules, although confidence separates optimal from suboptimal schedules less sharply than feasibility failures in the other tasks.Trigger settings are selected on held-out training trajectories and frozen before test evaluation.Source: §G.1 · p. 28

G.2 Corrective-Operator Sensitivity on ZebraLogic

  • ZebraLogic recovery is unchanged across anchor thresholds α = 0.85, 0.90, and 0.95, while wider candidate sets yield a gradual accuracy–compute trade-off.The main experiments use α = 0.90 and k = 5.Source: §G.2 · p. 28, Table 22 · p. 28

G.3 Evaluation Uncertainty and Paired Comparisons

The authors quantify uncertainty for selected headline comparisons using held-out evaluation intervals and a paired bootstrap for the ZebraLogic-Hard greedy gap. The LLaDA–LLaMA accuracy gap is positive across all four difficulty tiers.

G.3 Evaluation Uncertainty and Paired Comparisons

  • The ZebraLogic-Hard greedy comparison uses a paired bootstrap with 20,000 resamples, while marginal rates use 95% Wilson intervals.The intervals describe finite held-out evaluation uncertainty, not variation across independently retrained checkpoints.Source: §G.3 · p. 29, Table 23 · p. 29
  • LLaDA’s ZebraLogic-Hard greedy accuracy gap over LLaMA is positive in every tier: +19.2, +46.4, +59.2, and +50.4 pp on S, M, L, and XL.The 95% paired-bootstrap interval excludes zero in all four tiers.Source: §G.3 · p. 29
  • Across the full evaluation set, LLaDA alone solves 224 instances versus 5 for LLaMA; the continuity-corrected McNemar test gives χ2 = 207.5 (p < 10^-40).The marginal intervals indicate that the smaller Nurse Rostering gain is less separated.Source: §G.3 · p. 29

H Frontier API References

Frontier API results exceed Blackboard on Nurse Rostering but not on ZebraLogic-Hard or JSSP, and they are not compute-normalized comparisons. Supplementary studies test confidence-guided inference on rotating-shift rostering and asymmetric TSP.

H Frontier API References

  • Frontier API scale, serving configuration, training data, reasoning budgets, and hidden computation are not exposed, so latency and output-token counts are practical descriptors rather than comparable compute measurements.Nurse Rostering API results are reported as accuracy references only.Source: H Frontier API References · p. 29
  • Blackboard reaches 90.4% on ZebraLogic-Hard versus GPT-5.4 Thinking’s 73.2%, and 80.2% JSSP optimality versus its 77.0%; GPT-5.4 Thinking reaches 86.0% exact feasibility on Nurse Rostering versus Blackboard’s 76.4%.The frontier API models are black-box references, not compute-normalized comparisons.Source: H Frontier API References · pp. 29–30

I Supplementary Transfer Studies

  • Supplementary transfer studies test whether confidence supports task-appropriate inference under different constraint structures and output representations; the corrective action varies by task.Rotating-shift rosters use trajectory-level confidence for verifier-free candidate ranking, while ATSP confidence gates objective-ranked Best-of-N search.Source: I Supplementary Transfer Studies · p. 30

I.1.1 Task and Setup

  • The rotating-shift task is a 7×7 staff–day grid with a unique feasible completion; evaluation uses 320 held-out instances, and triggered cases rank ten sampled rosters by final-five-step mean confidence.No solver, task objective, or feasibility verifier is used for candidate ranking.Source: §I.1.1 · pp. 30–31

I.1.2 Difficulty-Controlled Results

In supplementary transfer studies, confidence-triggered inference reduces rotating-shift roster errors and supports objective-ranked ATSP search. The results cover roster difficulty and trigger diagnostics, plus ATSP performance by selection rule, optimality-gap difficulty, and compute allocation.

I.1.2 Difficulty-Controlled Results

  • On rotating-shift rostering, Blackboard reduces greedy errors from 15 to 7, a 53% reduction, and improves exact feasibility by 7.6 pp on the hardest d3 band.The largest gain occurs on search-hard d3 instances.Source: §I.1.2 · p. 31, Table 28 · p. 31

I.1.3 Trigger and Candidate-Selection Diagnostic

  • The roster trigger fires on exactly the 15 greedy-failure instances, with 100% precision and recall; LAST5 confidence selects a correct roster in 8 of 15 triggered pools.At least one correct roster appears in 13 of the 15 candidate pools.Source: §I.1.3 · p. 31, Table 29 · p. 31

I.2.1 Task and Setup

  • ATSP evaluation uses 300 held-out instances with unique certified optima; difficulty bands run from largest optimality gap (g0) to smallest (g3), and candidates are ranked by minimum valid tour length.The model outputs a permutation matrix decoded as a tour beginning and ending at city 0.Source: §I.2.1 · pp. 31–32

I.2.2 Matched Objective-Selection Results

  • Under objective-based Best-of-N selection, the autoregressive model rises from 68.7% to 89.0% exact optimality, while Blackboard reaches 92.7%.Table 30 compares matched 8B autoregressive and diffusion models under the same objective-selection rule.Source: §I.2.2 · p. 32, Table 30 · p. 32

I.2.3 Results by Optimality-Gap Difficulty

  • Blackboard’s ATSP gain over the baseline grows from +6.7 pp on g0 to +43.4 pp on the hardest g3 band.The paper relates this pattern to objective-based selection being especially useful when tours have similar likelihood but differ in global optimality.Source: §I.2.3 · p. 32, Table 31 · p. 32

I.2.4 Confidence-Triggered Objective BoN

  • The ATSP trigger fires more often from g0 to g3 (38%, 64%, 81%, and 95%) and preserves unconditional objective-BoN exact optimality while reducing average decodes by 27%.Confidence allocates search budget; it is not the candidate-ranking objective.Source: §I.2.4 · p. 32, Table 32 · p. 32

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