Efficient Reasoning with Flow Language Models

• Discrete denoising passes only one categorical value per position, so alternative-token probabilities are not explicitly preserved between steps.Early token choices can affect predictions at other positions through bidirectional attention before sufficient information has been integrated.Source: §1 · p. 2
• FLMs evolve continuous sequence representations during denoising and decode discrete tokens only at the end, allowing alternatives to be refined before a discrete answer.The introduction frames this as a way to retain soft representations of multiple candidate tokens.Source: §1 · p. 2
• The study compares reasoning efficiency by solution accuracy under a fixed budget of denoiser evaluations, focusing on structured, verifiable tasks.Its stated goal is to assess the accuracy–computation trade-off, particularly in the few-step regime.Source: §1 · p. 2
• FLMs achieve stronger few-step reasoning efficiency than discrete diffusion across maze planning and Sudoku, with higher sequence accuracy at matched model sizes and small denoising steps.The reported advantage grows with maze difficulty, and FLMs reach matched maze accuracy targets with smaller models.Source: §1 · p. 2
• The paper uses a superposition account to explain how continuous states preserve useful intermediate information, supported by candidate-retention measurements, interventions, and decoder probes.The introduction positions iterative diffusion as an alternative to sequential thought generation when reasoning involves exploring or comparing candidate paths.Source: §1 · pp. 1–2

• Conditioning inputs and target outputs are serialized and tokenized as sequences, with each token represented by a vocabulary-sized one-hot vector.The setup denotes the tokenizer vocabulary by Vtok and its size by K.Source: §2 · p. 2
• During training, the conditional FLM keeps the conditioning state fixed and constructs its continuous answer state by interpolating Gaussian noise and the clean target.The state is denoted Zt and has shape Mx × K.Source: §2 · p. 2
• At each answer position, the model predicts a distribution over the tokenizer vocabulary and is trained to recover the clean answer tokens.Source: §2 · p. 2
• At inference, the model initializes the answer state with Gaussian noise, integrates the learned flow for N evaluations, then discretizes each position once by argmax.The decoded tokens are detokenized and parsed into a predicted path.Source: §2 · pp. 2–3

3 THEORETICAL ANALYSIS

• The section organizes its account around temporal candidate superposition, history-dependent prediction changes, and predictive information available beyond hard projection.Source: §3 · p. 3

3.1 TEMPORAL CANDIDATE-SEQUENCE SUPERPOSITION

• The FLM state contains a temporally accumulated convex superposition of candidate-sequence vertices, which cannot be represented by one hard sequence when multiple sequences have positive integrated weights.The paper connects this retained non-top-1 historical support to subsequent refinement and notes that hard-state-only solvers cannot distinguish such histories at a fixed time.Source: §3.1 · p. 3

3.2 HISTORICAL INFLUENCE WITHIN A FIXED HARD STATE

• At a fixed hard-token sequence, sufficiently small admissible changes to retained non-top-1 history can change the prediction margin when the stated sensitivity condition holds.The analysis separates attention reweighting through Q/K, message changes through V, and bypass responses; their signed effects may reinforce or cancel, and influence can reach other positions.Source: §3.2 · pp. 3–4

3.3 TASK-RELEVANT INFORMATION BEYOND HARD PROJECTION

• The analysis bounds task-relevant predictive information beyond hard projection, with a predictive advantage guaranteed when 2νt > εZt and the Bayes-risk gap vanishing as bt approaches zero.Matched decoders on intermediate FLM states and their hard projections are used to test whether retained information predicts the correct solution.Source: §3.3 · pp. 4–5

4.1 EXPERIMENTAL SETUP

• The evaluation covers maze planning and Sudoku, matches model sizes and denoising steps across methods, and uses complete-sequence accuracy as its primary metric.Maze difficulty varies with maze size; Standard Sudoku tests transfer from Sudoku-Extreme without fine-tuning.Source: §4 · p. 5, §4.1 · p. 6

4.2 MAZE PLANNING: REASONING EFFICIENCY ACROSS TASK DIFFICULTIES

• At 64 denoising steps on Maze15, the smallest tested models reaching 95% sequence accuracy have 88.25M parameters for FLM and 139.00M for MDLM.At comparable model sizes, FLM maintains higher accuracy in several settings as maze difficulty increases.Source: Figure 2 · p. 7, §4.2 · p. 7

4.3 SUDOKU: FEW-STEP CONSTRAINT REASONING AND GENERALIZATION

• At one sampling step, FLM reaches 29.5% sequence accuracy on Standard Sudoku, exceeding the strongest discrete result of 27.1%.Models trained on Sudoku-Extreme were evaluated on Standard Sudoku without additional fine-tuning.Source: Table 2 · p. 7, §4.3 · p. 8

4.2 MAZE PLANNING: REASONING EFFICIENCY ACROSS TASK DIFFICULTIES

• FLM leads one-step sequence accuracy on all three maze tasks and maintains a clear few-step advantage on Maze15 and Maze20.Source: §4.2 · p. 7

4.3 SUDOKU: FEW-STEP CONSTRAINT REASONING AND GENERALIZATION

• 21.0% and 36.7% sequence accuracy at one and two steps on Sudoku-Extreme exceed the strongest discrete results of 19.2% and 36.2%, respectively.Further sampling yields more limited FLM gains, while enhanced discrete samplers achieve higher accuracy from 16 steps onward.Source: Table 2 · p. 7, §4.3 · p. 8

4.4 SUPERPOSITION ANALYSIS

• At t = 0.80, 23.8% of cells in Et retain a second candidate with probability at least 0.05, compared with 4.9% in Pt.Et also has a larger effective candidate count, indicating broader mass across candidate digits.Source: Figure 3 · p. 8, §4.4 · p. 8
• Cell-level recovery decreases as candidate information is removed: RecoveryNatural ≥ RecoveryTop-2 > RecoveryTop-1, while Digit-soft causes little degradation.Digit-soft retains and renormalizes locally feasible digits; Top-2 and Top-1 retain only the two or one highest-scoring feasible digits.Source: Figure 4 · p. 8, §4.4 · pp. 8–9
• At k = 4, raw states reach F1-score 0.8080, versus 0.6614 for Top-2 and 0.5929 for Hard; by k = 7 and 8, Raw and Hard agree to four decimal places.The decoders predict solution-path membership from a frozen FLM's eight-step Maze15 trajectory.Source: Table 3 · p. 9, §4.4 · p. 9

• Continuous diffusion and flow-matching language models use embeddings, autoencoding latents, and other continuous or relaxed representations.Source: §5 · p. 9
• Continuous latent reasoning performs intermediate computation in hidden representations without expressing every step as text.Source: §5 · p. 9
• Coconut, HRM, and TRM exemplify iterative approaches, using hidden-state feedback or recurrent updates to refine reasoning representations.Source: §5 · p. 9

6 CONCLUSION

• FLMs can achieve higher sequence accuracy with few denoising steps on maze planning and Sudoku, while analysis and interventions support candidate information persisting for subsequent reasoning.The conclusion describes this as a possible mechanism supported by the theoretical analysis and intermediate-state experiments.Source: §6 · pp. 9–10
• The evaluation covers structured reasoning tasks, not general language modeling; whether the benefits extend to natural-language reasoning or larger model capacities remains open.The paper identifies text-based reasoning benchmarks and scaling to larger capacities as directions for further study.Source: §6 · p. 10

APPENDIX OVERVIEW

• The appendix collects training and sampling algorithms, theoretical proofs, and additional experimental details and results.Source: APPENDIX OVERVIEW · p. 13

A TRAINING AND SAMPLING ALGORITHMS

• Training interpolates between Gaussian noise and the clean tokenized answer, then updates the model to predict clean answer tokens.The algorithm samples a graph, query, and target before constructing the noisy answer state.Source: A TRAINING AND SAMPLING ALGORITHMS · p. 14
• At inference, the conditional FLM evolves a continuous answer state for N denoising evaluations, then discretizes it into tokens and a path.Sampling begins from Gaussian noise and uses the learned flow.Source: A TRAINING AND SAMPLING ALGORITHMS · p. 14

B ADDITIONAL RELATED WORK

• Discrete diffusion methods use categorical intermediate states; examples include masked denoising, remasking and resampling, and confidence- or entropy-guided token selection.The passage describes D3PM, SEDD, MDLM, LLaDA, Dream, and ReMDM.Source: B ADDITIONAL RELATED WORK · p. 14

C.1 PROOF OF THEOREM 1

• Theorem 1 strengthens the instantaneous result: the FLM ODE integrates candidate-sequence superpositions over time and retains them in its persistent continuous state.The proof proceeds from candidate-sequence vertices and their superpositions to the FLM ODE and its integral representation.Source: §C.1 · p. 15

C.1.1 CANDIDATE SEQUENCES AS DISCRETE VERTICES

• A complete hard-token sequence is a vertex of the product simplex, whereas the FLM state lies in the ambient continuous space and its denoised component lies in the simplex.Each hard-token position is represented by a one-hot row.Source: §C.1.1 · p. 16

C.1.2 GEOMETRY OF MULTI-CANDIDATE SUPERPOSITION

• A positive-weight mixture of candidate vertices that disagree at any token position is in their convex hull but is not a hard-token vertex.The proof notes that the mixture need not lie in the relative interior when the vertices are not affinely independent.Source: §C.1.2 · p. 16

C.1.3 ACCESSIBILITY OF THE COMPLETE GRAPH

• With finite attention logits, no inference-time attention dropout, and only padding masked, every answer position has an attention route to every graph node; this does not ensure the model uses every node or that candidates are valid paths.Source: §C.1.3 · pp. 16–17

C.1.4 PROOF OF INSTANTANEOUS CANDIDATE-SEQUENCE SUPERPOSITION

• The position-wise clean-token prediction is a superposition of candidate sequences, and differing positive-weight candidates make it non-one-hot at a position where they disagree.The induced factorized distribution does not establish that candidate sequences are valid graph paths or capture general joint path dependencies.Source: §C.1.4 · p. 17

C.1.5 DERIVATION OF THE FLM ODE

• At sampling time, the model substitutes its clean-state prediction for the unknown clean answer to define the learned velocity field.Under an ideal clean-state predictor, the prediction approximates the conditional mean of the clean state.Source: §C.1.5 · p. 18
• The stated dynamics and integral identities apply for t < 1 because the velocity-field denominator vanishes at t = 1.Source: §C.1.5 · p. 18

C.1.6 INTEGRAL REPRESENTATION AND PROOF OF THE MAIN THEOREM

• The current state equals a residual contribution from initial noise plus accumulated clean predictions, as an exact identity along the ODE trajectory.The identity remains exact even though the predictions depend on the evolving state.Source: §C.1.6 · p. 19
• When distinct candidate sequences have positive integrated weights, their contributions accumulate into a nontrivial geometric superposition along the same FLM ODE trajectory.The proof shows the denoised component is not one-hot when distinct sequences contribute.Source: §C.1.6 · p. 19

C.2 PROOF OF THEOREM 2

• Theorem 2’s proof establishes that a non-top-1 candidate can be perturbed without changing the hard state, while the prediction changes under the stated nonzero-sensitivity condition.The derivative decomposes into attention-weight and value contributions, with no sign restriction on individual contributions; the local argument also covers admissible one-sided perturbations at interval endpoints.Source: §C.2 · pp. 20–21
• The proof shows that the response range is nonzero exactly when the sensitivity is not identically zero, and that the first lower bound is tight.A scalar readout using only context, time, and hard state is constant along the intervention path.Source: §C.2 · pp. 21–22

C.3 HISTORICAL PERSISTENCE AND CONTINUOUS REFINEMENT

• Under the stated winner-gap and step-size conditions, historical support can persist while an update narrows a candidate’s gap to the current winner without changing the hard-token sequence.The proof separates direct persistence from the model’s prediction response; conditioning coordinates stay fixed.Source: §C.3 · pp. 22–23

C.4 PROOF OF THEOREM 3

• Theorem 3’s proof bounds the conditional entropies by χ_t = hbin(min{1/2, 2b^2_t}), and notes that matching decoder architecture and training budget does not eliminate excess-risk terms.The proof does not assume either trained decoder attains its Bayes risk.Source: §C.4 · p. 24

D.1.1 MAZE DATASETS

• Maze10, Maze15, and Maze20 each use 10,000 training examples and 1,000 test examples, with corresponding logical lattices of 10 × 10, 15 × 15, and 20 × 20.Mazes are deduplicated by layout, split without layout overlap, rasterized, and serialized as character-level input/output sequences.Source: §D.1.1 · pp. 24–25

D.1.2 SUDOKU DATASETS

• Sudoku-Extreme uses 1,000,000 training puzzles, 1,000 validation puzzles, and 1,000 test puzzles; sequence accuracy requires exact agreement across all 81 solution cells.Models trained on Sudoku-Extreme are also evaluated on Standard Sudoku without fine-tuning.Source: §D.1.2 · p. 25

D.2.1 ARCHITECTURE AND OPTIMIZATION HYPERPARAMETERS

• FLM and discrete baselines share GPT-2-style backbones at each model size, but use different training objectives: clean-token cross-entropy for FLM and their respective discrete diffusion objectives.Maze training uses 1,000 epochs for FLM and 2,000 for discrete baselines; all Sudoku methods use 300 epochs.Source: §D.2.1 · pp. 25–26

D.2.2 INFERENCE DETAILS

• Inference evaluates FLM exactly K times on a uniform integration grid, then applies tokenwise argmax to its final continuous state; discrete baselines use reverse-time updates from an absorbing mask state.The evaluated budgets range from K = 1 to K = 512.Source: §D.2.2 · p. 27
• Discrete implementations use K reverse updates plus one noise-removal evaluation (K+1 total); input conditions remain fixed and all methods use EMA weights.Source: §D.2.2 · p. 27

D.2.3 FLM DECODER EXPERIMENTS’ IMPLEMENTATION DETAILS

• A frozen Maze15 FLM supplies cached states from one N=8 trajectory, which separate local CNN decoders use to predict solution-path masks.The eight-layer model has eight attention heads and width 768; decoders share a 9 × 9 receptive field, 38,177 parameters, and 480-update budget. Training uses 512/128/128 disjoint train/validation/test mazes and three matched initialization seeds; evaluation reports unweighted BCE and path-F1 at threshold 0.5.Source: §D.2.3 · p. 27

D.3 FULL-STATE INTERVENTIONS ON RETAINED HISTORICAL SUPPORT

• The intervention removes a strict non-top-1 candidate’s historical contribution while holding context, time, other coordinates, and initial noise fixed, then measures its effect on the next prediction margin.The cohort comprises 898 fixed Sudoku and Maze20 events with N ∈ {2, 4}, including 261 correct-candidate events; this is a retrospective mechanism analysis.Source: §D.3 · p. 27
• 232/261 original correct-candidate events showed increased margins when history was retained; all four groups had positive means with confidence intervals excluding zero.In a validation cohort, 47 of 48 correct-candidate events had positive endpoint effects; the additional Maze20 N = 4 group included only five such events.Source: §D.3 · p. 27, Table 7 · p. 28
• Q has the largest mean contribution in each group, V is positive in all four groups, and the bypass contribution is positive in Sudoku but near zero in Maze20.K intervals include zero, which does not establish that K is unused.Source: Table 8 · p. 28, §D.3 · p. 28
• V-other is positive with intervals excluding zero in every group, showing that history at one position can affect its prediction through other receiving positions and subsequent layers.In Sudoku, V-other exceeds total V because the signed contribution at the original receiver cancels part of it.Source: §D.3 · p. 28, Table 8 · p. 28
• The intervention results concern next-prediction margins, not final puzzle accuracy, and do not establish uniformly beneficial effects or positive derivatives throughout a continuous path.The validation inputs are not claimed to be unseen during checkpoint development, and some initial-noise keys are shared with the original cohort.Source: §D.3 · pp. 27–28

D.4 COMPLETE RESULTS ACROSS MODEL CAPACITIES

• The complete capacity sweep evaluates FLM and MDLM on each maze task at three matched model sizes across ten denoising-step budgets.Table 9 covers sequence accuracy for Maze10, Maze15, and Maze20; the supplied passages do not give its cell values.Source: §D.4 · p. 29, Table 9 · p. 29

D.5 WALL-CLOCK INFERENCE EFFICIENCY

• FLM has the lowest latency at N = 1 on all three tasks; at N = 64, its cost is comparable to standard discrete samplers, while confidence remasking approximately doubles latency.The latency table reports mean seconds per instance on one A100 at batch size one; budgets do not enforce matched accuracy.Source: §D.5 · p. 29, Table 10 · p. 30