Flow Reasoning Models: Turning Flows Into Efficient Recurrent Reasoners — part 4/4

Cleaned-text code points [45833, 48137). Bibliography omitted here; source.md includes it.

Method Inference-time control Charged computation
FMLM / FLM Flow-integration steps All denoiser evaluations
MDLM Denoising steps All denoiser evaluations
ReMDM Denoising and remasking steps All denoiser and remasking evaluations
Adaptive MDLM Update cap and confidence rule Realized adaptive evaluations
HRM / TRM / FPRM Recurrent depth All recurrent model evaluations
EqR Equilibrium depth and breadth All iterations and evaluated branches
FRM (ours) Recurrent depth and sampler All recurrent denoiser evaluations
Table 4:Inference controls used to construct accuracy–compute frontiers.Every learned-model
invocation is included in the reported computation. Exact evaluated grids are checkpoint- and dataset-
specific.
Adaptive computation.Token-adaptive methods can incur a nonzero minimum cost even at their
lowest nominal setting. We therefore plot measured realized FLOPs rather than the maximum-step
label.
B.2 Training-Recipe Results
Table 2 reports three-seed means and standard deviations for every dataset. We report aggregate
statistics consistently and omit individual-seed values.
C Additional Dynamics Analysis
We distinguish productive refinement from convergence to an incorrect fixed point using the following
depth-dependent diagnostics.
Depth-dependent performance and distance to the solution.At each depth, we record exact
solve rate and token-averaged cross-entropy to the one-hot solution on the same pinned examples.
Thus changes isolate additional recurrent computation. In Fig. 6, FPF improves accuracy while
driving ground-truth loss downward; conventional self-conditioning becomes increasingly confident
in incorrect states.
Convergence residual and correctness prediction.For adjacent distributions pk andpk−1, we
measure the token-averaged symmetric-KL residual rk =D SKL(pk,pk−1) and pair it with the
decoded state’s exact-solve label. A near-zero residual indicates convergence, not necessarily
correctness. Conventional self-conditioning yields chance-level discrimination (AUROC 0.50); under
FPF, convergence is strongly predictive of correctness (AUROC1.00) on Sudoku-Extreme.
Interpretation.Exact solve rate remains primary; loss-to-gold and adjacent-state residuals reveal
whether depth corrects the solution or settles into a confidently incorrect fixed point.
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