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SanSi: A Looped Typed Decision Model for System 1.5 Thinking

Shuyu Gan, Young-Jun Lee, Dongyeop Kang

arXiv:2610.07730v1cs.CLcs.AIcs.LG

TL;DR

Typed decision models provide option probabilities in one pass, but some decisions require several dependent reasoning steps. SanSi loops a pre-trained language-model stack and trains a readout at each loop; it improves benchmark accuracy over single-pass comparators and extends solvable depth on two controlled tasks.

  • Problem

    A single pass gives fixed computation, while multi-step decisions may require following chains or composing rules; generated reasoning adds tokens and latency and gives up the typed contract.

    Source: §1 · p. 1

  • Method

    SanSi applies a pre-trained looped model repeatedly, reading and training option probabilities after every loop so one model serves budgets from one to eight.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    Looping adds 13.5 points over a same-shape non-looped model, and on two depth-controlled tasks it solves depths beyond training where the larger single-pass model fails.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Three loops provide 88% of the loop-1-to-8 gain, while most answers settle by loop four and harder items settle later.

    Source: §1 · p. 2, §9 · p. 10

    The findings rely on Ouro-1.4B and Ouro-2.6B backbones pre-trained to loop, and how well they transfer to other or larger looped backbones remains open.

    Source: Limitations · pp. 10–11

Abstract

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Typed decision models answer a declared question without generating text: a decision head returns a probability for each of the declared options in a single forward pass. A single pass is fast, intuitive System 1 thinking. We study what lies between one pass and generated reasoning: looping, in which the same layers are recursively applied several times before one typed readout. Each loop lets the model revise its hidden state before it commits to an answer, without generating a token; we call this System 1.5 thinking. We propose SanSi, which turns a pre-trained looped language model into a typed decision model. The option probabilities are read after every loop, and every loop is trained with a proper scoring rule, so that one model serves every budget from one loop to eight in a single run. On 10,027 test decisions from 59 sources, SanSi reaches 72.0% accuracy: 13.5 points above a non-looped model of the same shape trained with the same recipe, 5.3 points above a newer non-looped model of its size, and 1.8 points below one with three times the parameters. On two depth-controlled tasks, loops extend the solvable depth beyond the depths seen in training, where the larger single-pass model fails. Used as the judge for policy optimization with reinforcement learning, without gold answers, SanSi raises the generator's F1 by 7.7 points.

1 Introduction

The paper studies looping as a way to spend additional computation on typed decisions without adding parameters or generating text. SanSi applies a pre-trained looped model to this task, and its evaluation reports accuracy gains, deeper problem-solving, and a verifier use case.

1 Introduction

  • Looping adds 13.5 points over same-shape SmolLM2-1.7B and brings SanSi within 1.8 points of Qwen3.5-4B, which has three times its parameters.Figure 1 compares accuracy with backbone size for models trained on the authors’ data.Source: Figure 1 · p. 1, §1 · p. 2
  • Multi-step reasoning gains +15.1 points and classification +3.1, while long documents gain +16.7 and knowledge questions +17.6.The gains are smallest on classification and largest on knowledge questions.Source: §1 · p. 2
  • Three loops deliver 88% of the loop-1-to-8 gain, most answers settle by loop four, and harder items settle later.Looping costs 7.7 times the computation of a single pass, and calibration improves only up to loop three.Source: §1 · p. 2
  • On liar chains of 9–16 steps, beyond the training maximum of 8, SanSi is right on 72.6% of items while Qwen3.5-4B is at chance with 50.0%.The comparison is on a depth-controlled task where the larger single-pass model fails.Source: §1 · p. 2
  • As the only reinforcement-learning reward, SanSi raises a generator’s F1 from 39.5 to 47.3 without gold answers.The paper presents this as a use case for SanSi as a judge for policy learning.Source: §1 · p. 2

2 Background and Related Work

Prior work established typed decision models and explored looping in language models, while SanSi builds on a backbone pre-trained to loop. The paper relates multi-depth readouts to adaptive computation and early exit, without proposing a halting rule.

2 Background and Related Work

  • Jev and its open reimplementation Kev return probabilities over declared options in one pass; this work instead starts from a backbone pre-trained to loop and reads every loop.The authors consider one question per call, unlike Kev’s option to carry several questions about the same state.Source: §2 · p. 2
  • Looped language-model research covers loop use, stability, architectures, adding loops to non-looped models, and tool calling [Dau et al., 2026; Yang et al., 2026; Jeddi et al., 2026; McLeish et al., 2025; Popescu et al., 2026b].The paper also notes repeated fixed operations at the agent level [Kim et al., 2026].Source: §2 · p. 2
  • Reading predictions at several depths relates to adaptive computation and early exit, but the paper does not propose a halting rule [Graves, 2016; Xin et al., 2020].Prior work finds later exits can reverse correct predictions and that settling depth measures example difficulty; the authors report analogous loop behavior [Kaya et al., 2019; Baldock et al., 2021].Source: §2 · p. 3

3 SanSi: A Looped Typed Decision Model

SanSi repeatedly applies a pre-trained transformer stack and reads option probabilities after every loop. It trains each readout toward a target distribution, allowing one model to provide decisions at multiple loop budgets.

3 SanSi: A Looped Typed Decision Model

  • Each typed decision comprises a state, a question, and at least two declared options; the output is a probability distribution, with no text.Targets are one-hot for answerable items, uniform for unanswerable items, and the annotators’ label distribution for crowd-labelled items.Source: §3 · p. 3
  • Ouro-1.4B applies one stack of 24 transformer layers repeatedly, with each normalized hidden state serving as the next loop’s input and no re-injection of input tokens.The stack was pre-trained with four loops; SanSi is trained with eight.Source: §3 · pp. 3–4
  • At every loop, the model reads option probabilities from the final prompt-token state, so a single run provides decisions for budgets 1 through T.The readout is restricted to declared options and uses a trained rank-16 correction and scale for each loop.Source: §3 · p. 3
  • Every loop is trained with cross-entropy plus Brier score, and each loop’s loss is backpropagated through all preceding loops while the backbone remains frozen.The trained parameters are LoRA adapters and readouts; their total is about 61 million parameters.Source: §3 · p. 3
  • The decision suite table organizes item counts by type and distance from training data; Type C covers unanswerable and crowd-labelled items.The test set includes 231 public JevBench items within 10,027 total test items.Source: Table 1 · p. 4

4 Experimental Setup

The experiments use a typed-decision suite assembled from public sources and compare SanSi with single-pass controls and reference models. The setup defines accuracy, calibration, evidence ranking, and GPU-time measures.

4 Experimental Setup

  • The suite combines public datasets, Kev training and transfer suites, and public JevBench items, with 10,027 test items and 2,471 development items.Tests are grouped by distance from training data; knowledge questions appear only in the test set.Source: §4 · p. 4
  • Comparators include same-shape Ouro-1.4B and SmolLM2-1.7B controls, Qwen3.5-2B and Qwen3.5-4B references, and a Kev-4B model trained with Kev’s recipe.Fine-tuned models use the same training items, LoRA rank, readout, and loss, with three seeds each.Source: §4 · p. 4
  • Accuracy, confidence, ECE, evidence AUROC, and test-set GPU time measure prediction quality, calibration, evidence sensitivity, and cost.Evidence AUROC is measured on 786 items from sources containing both evidence-present and evidence-absent examples; intervals are 95% bootstrap intervals over related-item groups.Source: §4 · p. 4

5 Results

Across the mixed test suite, looping improves typed decisions at a computation cost. Gains arrive early, and on depth-controlled tasks looping extends performance beyond the training depths.

5.1 RQ1: Does looping help, and at what cost?

  • 72.0% vs 58.4% accuracy (+13.5 points) for matched-shape SanSi and SmolLM2; SanSi is 5.3 points above Qwen3.5-2B and 1.8 below Qwen3.5-4B.The main comparison uses the same training data and recipe for SanSi and SmolLM2; Qwen3.5 models are reference comparisons.Source: §5.1 · pp. 4–5, Table 2 · p. 5
  • Looping adds 17.6 points on test-only knowledge questions and 15.1 on multi-step reasoning, versus 3.1 on classification.Gains are also 16.7 points on long documents; by training distance, they are 10.5 in-distribution, 10.0 near-transfer, and 15.8 far-transfer.Source: §5.1 · p. 5, Figure 3 · p. 6
  • Looping costs 7.7× the inference computation and 9.5× the training GPU time; three loops reach 70.4% accuracy at three eighths of the full-pass cost.SanSi-2.6B with three loops exceeds Qwen3.5-4B by 0.9 points, at 5.6× versus 2.4× the one-loop computation cost.Source: §5.1 · p. 5, Figure 3 · p. 6

5.3 RQ3: What do the loops do to the probabilities?

  • Mean confidence rises on both right answers (0.769 to 0.869) and wrong answers (0.567 to 0.660), while ECE reaches its minimum of 0.082 at loop 3.By loop 8, ECE rises to 0.093 as confidence continues increasing after answers settle.Source: Figure 4 · p. 7, §5.3 · p. 7
  • On items with removed key evidence, hard answers fall from 27.1% at loop 1 to 17.5% at loop 8, while evidence AUROC rises from 0.835 to 0.935.The single-loop-trained backbone reaches 0.837 AUROC; only Qwen3.5-4B and Kev-4B (our data), both three times larger, score higher among single-pass models.Source: Figure 4 · p. 7, §5.3 · p. 7

5.2 RQ2: How do the answers change from loop to loop?

  • Loops 2 and 3 deliver 88% of the loop-1-to-8 accuracy gain; after loop 4, 90.2% of answers are final, and fixes and breaks balance after loop 5.At most 11.7% of items are right at some loop but wrong at loop 8.Source: §5.2 · p. 6, Figure 5 · p. 7

5.4 RQ4: Does looping buy reasoning depth?

  • On unseen liar-chain depths 9–16, eight-loop SanSi reaches 72.6% accuracy versus 49.9% at one loop; Qwen3.5-4B is at chance on unseen depths.After eight loops, SanSi holds depth 9 on object swaps; on liar chains, it holds depth 11, three steps beyond the longest training chain.Source: Figure 6 · p. 8, §5.4 · p. 8

6 Examples and Errors

Examples show loops can revise surface-cue answers or reduce confidence when decisive evidence is missing, but they can also overturn a correct answer. Errors remain common, including confident errors.

6 Examples and Errors

  • In an item with its deciding sentence removed, SanSi shifts from “yes” at 0.97 to below the hard-answer threshold from loop 2 onward, unlike Qwen3.5-2B at 0.98.A separate example shows loops changing a surface-cue answer to “unknown” by loop 2.Source: §6 · p. 8
  • At loop 8, SanSi is wrong on 27.2% of items with one gold option; 19.7% of those errors have confidence of at least 0.9.Qwen3.5-4B also fails on 68.0% of these errors, and 60.9% are wrong at every loop.Source: §6 · p. 8

7 Ablations

Ablations show that per-loop supervision makes early reads usable, while extending beyond trained depth hurts accuracy. Loop pre-training is necessary for the gains, and averaging loop probabilities improves calibration with little accuracy change.

Effect of the training objective

  • 35.6% at loop 1 versus 58.4% shows that last-loop-only supervision sharply reduces early-loop accuracy; loop-8 accuracy falls 1.1 points to 70.9%.Replacing supervision with outcome-only reinforcement learning leaves accuracy nearly unchanged at 71.7% versus 72.0%, while removing the Brier term gives 71.6%.Source: §7 · p. 8

Illustrative decisions

  • Figure 7 compares option probabilities across SanSi’s loops and three single-pass models on four test items, including one with its deciding sentence removed.Correct options are starred and outlined, and column maxima are bold.Source: Figure 7 · p. 9

Effect of the number of loops

  • 72.0% at loop 8 falls to 70.1% at loop 16 when SanSi runs beyond its trained depth; the four-loop model also declines after loop 4.Training four loops yields 70.8% at loop 4, just 0.8 points below SanSi at loop 4.Source: §7 · p. 9, Figure 8 · p. 10

Effect of the backbone

  • Adding loops to non-loop-pretrained SmolLM2-1.7B gives 33.5% or 58.3% accuracy, versus 58.4% without a loop; SanSi-2.6B reaches 75.8%.SanSi-2.6B is 2.0 points above Qwen3.5-4B despite using 63% of its parameters.Source: §7 · p. 9

Effect of averaging the loops

  • Averaging eight loops preserves accuracy at 71.8% versus 72.0% and halves ECE from 0.093 to 0.044 without extra computation.The loops are already computed, so averaging requires no labels or training.Source: §7 · p. 9

8 Use Case: SanSi as a Verifier

The paper tests SanSi as the sole reward for training a generator without gold answers. Reading the verifier after four or eight loops improves F1, while a one-loop reward damages it, with reward quality tracking the difference.

Setup

  • The experiment trains SmolLM2-1.7B with LoRA and GRPO on 2WikiMultiHopQA, using frozen SanSi’s probability over sampled answers as reward without gold answers.Only the verifier readout loop varies: 1, 2, 4, or 8, with three seeds each.Source: §8 · p. 10

Result

  • Reading SanSi at loop 8 raises generator F1 from 39.5 to 47.3 (+7.7), and reading it at loop 4 raises F1 to 45.8 (+6.3).At loop 2 F1 is unchanged, while loop 1 lowers it to 29.1 (−10.5).Source: §8 · p. 10

Why reward depth matters

  • Loop-1 rewards distinguish gold-matching answers less well than loop-4 or loop-8 rewards early in training: AUROC 0.78 versus 0.91 and 0.90.The weaker reward more often ranks a wrong answer above a right one.Source: §8 · p. 10

9 Conclusion

The conclusion presents looping as a way for a small typed decision model to spend computation instead of adding parameters. It reports accuracy gains, depth generalization, and verifier benefits, while noting limits from calibration, trained depth, and loop pre-training.

Accuracy and compute

  • A looped 1.4B model is 13.5 points more accurate than a same-shape non-looped model and 5.3 points above a newer non-looped model of its size, within 1.8 points of a model with three times its parameters.The looped model uses 3.3 times that larger model’s GPU time.Source: §9 · p. 10

Loop budgets

  • Three loops provide most of the gain, most answers settle by loop four, and harder items settle later.A loop-adaptive rule is a proposed next step because 11.7% of items are right at some loop but wrong at the last.Source: §9 · p. 10

Depth generalization

  • On two depth-controlled tasks, looping solves problems beyond training depths where a single-pass model with three times the parameters fails.The conclusion also cautions that accuracy falls when running more loops than were trained.Source: §9 · p. 10

Calibration and scope

  • Calibration improves only through loop three because confidence keeps rising after answers settle; these effects depend on a backbone pre-trained to loop.Source: §9 · p. 10

Verifier use

  • As the sole reward for generator training, SanSi helps when read after four or eight loops but harms when read after one.Source: §9 · p. 10

Limitations

SanSi’s evidence is bounded by its reliance on Ouro backbones pretrained to loop, with most analyses using Ouro-1.4B. The evaluation also has fixed-loop compute costs and several task, dataset, and seed constraints.

Limitations

  • SanSi uses only Ouro-1.4B and Ouro-2.6B backbones pretrained to loop, and the gain relies on that pretraining.How findings transfer to other and larger looped backbones remains open.Source: Limitations · p. 10
  • Eight loops take about 3.3 times Qwen3.5-4B’s GPU time, four loops take 1.6 times, and SanSi-2.6B takes about 6.3 times.The model uses a fixed number of loops; early stopping on easy items is left to future work.Source: Limitations · p. 11
  • The reasoning-depth analysis uses two program-generated tasks, while the verifier case study uses one dataset and one generator and is illustrative, not a general reward-model evaluation.Source: Limitations · p. 11
  • The suite is English and assembled from existing datasets with an author-chosen item mix, and possible pretraining exposure to public datasets cannot be excluded.Source: Limitations · p. 11
  • Main comparisons use three seeds, so one- or two-point differences on smaller test groups should be read alongside their intervals.Source: Limitations · p. 11

Ethical Considerations

The paper reports overconfidence and evidence-free answers because typed decisions may be automated, while noting that its data come from public benchmarks and include no newly collected human data. Its suite spans six item types, specifies distinct targets for unanswerable and crowd-labelled items, and uses controls for related-item splits and prompt formatting.

Ethical Considerations

  • Typed-decision probabilities may be used for automation, so the paper reports SanSi’s overconfidence and how often it answers without evidence; all data are public benchmarks, with no new human data collected.Source: Ethical Considerations · p. 11

A Data

  • The suite covers six item types, from classification and multi-step reasoning to uncertain evidence, long documents, sentence pairs, and test-only knowledge questions.It draws on 20 training sources and 59 test sources.Source: A Data · pp. 14–15
  • Unanswerable items have their key evidence removed and use a uniform target; crowd-labelled items use label distributions as targets, with accuracy based on the most probable label.Source: A Data · p. 15
  • Related items are kept on one side of development/test splits, and test candidates sharing a state or at least 80% of sentences with training items are excluded.Source: A Data · p. 15
  • Prompts place the state, question, options, and answer cue in sequence; options use one-token letters, allowing at most 26 choices.Most test items have two to four options; the largest sets have 14, 18, and 26 options.Source: A Data · p. 15

B Training and Implementation Details

This appendix documents SanSi’s training setup and evaluates its performance across models, test sources, item types, and task settings. It also examines how its answers and calibration change across loops, including on missing-evidence and depth-controlled tasks.

D.1 All models; B Training and Implementation Details

D.3 The comparison with the model of the same shape

  • SanSi at loop 8 is 13.4 points above Ouro-1.4B run once, and its gap over SmolLM2-1.7B remains above 11 points after longer training.Doubling training to 2,000 steps adds 1.9 points to SmolLM2-1.7B and 0.9 to SanSi; it worsens both models’ calibration.Source: §D.3 · p. 18, §D.2 · p. 18, Table 10 · p. 21

D.6 The Qwen3.5 models

  • SanSi at loop 8 is 5.3 points above Qwen3.5-2B and 1.8 points below Qwen3.5-4B.Qwen3.5-2B reaches 66.7% and Qwen3.5-4B reaches 73.8%; SanSi draws level with Qwen3.5-2B by loop 2.Source: Table 7 · p. 19, §D.6 · p. 19, Table 8 · p. 20

D.4 Every test source

  • SanSi at loop 8 is more accurate than SmolLM2-1.7B on 55 of 59 test sources, with four exceptions.One exception is natural-language ProofWriter, where SanSi scores 67.2% against 74.4% for SmolLM2-1.7B.Source: §D.4 · p. 19, Table 11 · p. 22

D.5 Item types

  • On multi-step reasoning, SanSi at loop 8 is level with Qwen3.5-4B; it trails on knowledge questions and uncertain-evidence items.The differences are 0.0 points on multi-step reasoning, 4.9 points behind on knowledge questions, and 3.5 points behind on uncertain-evidence items.Source: §D.5 · p. 19, Table 12 · p. 23

D.7 Kev’s recipe on our data

  • With Kev’s recipe on the same 12,800 items, Kev-4B reaches 74.3%, while SanSi-2.6B is 1.5 points ahead of it.SanSi is 2.4 points behind Kev-4B (our data) at loop 8; the two training recipes give the Qwen3.5-4B backbone similar accuracy.Source: §D.7 · p. 20

D.8 Released models: Kev-4B and the Jev API

  • On 6,040 items neither model was trained on, SanSi-2.6B is level with released Kev-4B, while the Jev API scores 83.5%.The Jev API reaches 78.9% overall but gives hard answers to 62.8% of unanswerable items; comparisons with released systems are limited by training-data overlap.Source: §D.8 · pp. 20–21, Table 13 · p. 23

E.1 Every loop; E.2 Changes between consecutive loops

  • After fine-tuning, accuracy rises steeply through loop 3 and is flat from loop 4; changes between consecutive answers roughly halve early and fall to 2.3% by loops 7–8.The loop-1-to-8 gain is largest on far transfer (50.8% to 68.0%) and smallest on near transfer (61.1% to 67.7%); up to loop 5, each step fixes more answers than it breaks.Source: §E.1 · p. 23, Figure 10 · p. 24, §E.2 · p. 24

E.3 Changes seen from the first and from the last loop

  • 83.3% of answerable items could be correct if each used its best loop, versus 71.6% at loop 8, but this gold-answer upper bound is not proposed as a selection rule.Across loops, 46.7% of items are right at every loop, 20.0% are fixed once and remain right, 6.5% are broken once and remain wrong, 10.1% switch more than once, and 16.7% are always wrong.Source: §E.3 · p. 25

E.4 When answers settle

  • 56.7% of answers never change and are correct in 82.2% of cases; SanSi’s largest gains occur on items answered correctly by two single-pass models, rising from 52.3% to 77.4%.Items answered correctly by no single-pass model settle at 2.8 loops versus 3.0 for items answered correctly by at least one; unchanged wrong answers end with mean confidence 0.782, above 0.763 for right answers reached after a change.Source: §E.4 · pp. 25–26, §E.5 · p. 26

F.2 Confidence of right and wrong answers

  • 0.795 is the loop-8 confidence AUROC for distinguishing right from wrong answers, above 0.685 for the settling loop; SanSi’s ECE is below the one-loop backbone and Qwen3.5 models but above SmolLM2-1.7B.Fitting a development-set temperature reduces the four models’ ECE to 0.022–0.029 and their hard-answer rates to 11.0–14.7%, without changing evidence AUROC. On far transfer, ECE falls from 0.156 to 0.084 by loop 4; on near transfer, it rises from 0.053 to 0.149.Source: §F.2 · pp. 26–27, §F.3 · pp. 27–28

F.4 Missing evidence

  • At loop 8, SanSi gives hard answers to 17.5% of unanswerable items and reaches evidence AUROC 0.935; Qwen3.5-4B has AUROC 0.948 and an 18.2% hard-answer rate.SanSi’s loop-1 values are 27.1% and 0.835, and its loop-4 values are 18.1% and 0.928; the Jev API gives hard answers to 62.8% of unanswerable items.Source: §F.4 · p. 28

G.1 Tasks, training and checks

  • SanSi is trained for eight loops on liar chains and object swaps with depths 1–8, then tested through 16 loops on depths 1–16; deeper items settle later, with mean settling loops reaching 4.2 and 4.5 at depth 16.The tasks use generated items with no test examples in training; liar-chain examples include an irrelevant chain, while object-swap examples include additional swaps not involving the queried object. The probability-measure differences are reported with 95% bootstrap intervals.Source: §F.1 · p. 26, §G.1 · pp. 28–29, §G.4 · p. 31

G.2 Results in summary

  • 72.7% on liar chains and 64.0% on object swaps are SanSi’s unseen-depth accuracies after 16 loops, matching its eight-loop results; it surpasses Qwen3.5-4B from loop 2 onward.At one loop, SanSi trails Qwen3.5-4B by 5.3 points on trained liar-chain depths and 10.1 points on trained object-swap depths.Source: §G.2 · pp. 29–30

G.3 Accuracy at every depth

  • For unseen-depth object swaps, SanSi accuracy improves from 57.3% after four loops to 63.7% after eight; for liar chains, it rises from 70.9% to 72.6%.At loop 8 on unseen-depth object swaps, SanSi seeds range from a low of 60.2%, above Qwen3.5-4B’s high of 34.9%. At k = 16 on liar chains, SanSi is correct on 60.3% with mean confidence 0.73.Source: §G.3 · p. 30

G.5 Single-pass models of SanSi’s size

  • In liar-chain training, none of the SanSi-sized single-pass models—SmolLM2-1.7B under two recipes and one-loop Ouro-1.4B—learned the task across two seeds per model.With the recipe above, SmolLM2-1.7B and the one-loop model gave the same option for every item, with 0.51–0.52 confidence and 50.0% accuracy at every depth. With the second recipe—half the learning rate and twice the steps—SmolLM2-1.7B’s accuracy was 45.0–59.2% at every depth.Source: §G.5 · p. 31

G.6 The Jev API on the two tasks

The Jev API was tested as offered on two depth-controlled tasks, but it was not trained on them, so its scores are not comparable with those of fine-tuned models. Its accuracy falls toward chance as the number of dependent steps grows.

G.6 The Jev API on the two tasks

  • Table 27 defines held depth as at least 75% accuracy at every shallower depth; Tables 28–30 report per-depth accuracy, settling, and single-pass controls.Table 28 compares SanSi with Qwen3.5-4B; Table 30 gives results for single-pass models of SanSi’s size on liar chains.Source: Table 27 · p. 32, Table 28 · p. 32, Table 29 · p. 33
  • 62.5% accuracy at liar-chain depth 3 falls to 42.5–53.3% from depth 6; on object swaps, accuracy is 18.6–31.1% from depth 3, near 20% chance.The API was not trained on these tasks, so these results show its performance as offered rather than a comparison with fine-tuned models.Source: §G.6 · pp. 31–32
  • Across depths 6–16, confidence is 0.61 on liar chains and 0.33 on object swaps, while accuracy is 47.6% and 23.6%, respectively.From depth 3 onward, confidence exceeds accuracy at every depth but one.Source: §G.6 · pp. 32–33

H Ablations: Details

  • Appendix H covers loop training, loop count, backbone choice, and loop averaging.The training discussion is in H.1–H.2, followed by loop count in H.3, backbone in H.4, and loop averaging in H.5.Source: H Ablations: Details · p. 33

H.1 Which loops carry the loss, and the Brier term

The appendix tests how loop supervision, reward-based training, loop budgets, backbones, and probability averaging affect SanSi. Its results show that loops work best within the trained budget, depend on looped pre-training, and can be averaged to improve calibration without a measurable accuracy loss.

H.1 Which loops carry the loss, and the Brier term

  • Training every loop mainly strengthens early readouts: last-loop-only supervision reaches 70.9% at loop 8, 1.1 points below SanSi, while sparse supervision costs little but buys nothing.Cross-entropy alone keeps accuracy close but worsens ECE at loops 3 and 8 and raises hard answers on unanswerable items.Source: §H.1 · pp. 33–34

H.2 Reinforcement learning instead of the supervised loss

  • After a 200-step supervised warm-up, sampled-answer rewards train from outcomes alone; loop-8 accuracy is 71.7% against 72.0%, with lower ECE but weaker evidence AUROC.The reward subtracts the probability stated for each sampled answer, making confidence costly and yielding a policy-gradient objective that follows a proper scoring rule.Source: §H.2 · pp. 34–35

H.3 The number of loops

  • 70.8%: a four-loop-trained model is 0.8 points below SanSi at loop 4, and running beyond trained loops does not extend depth on the controlled tasks.SanSi run to 16 loops falls from 72.0% at loop 8 to 70.1% at loop 16; its unseen-depth accuracy at 16 loops matches that at 8.Source: §H.3 · p. 35, Table 33 · p. 36, Table 34 · p. 37

H.4 The backbone

  • 75.8%: SanSi-2.6B exceeds Qwen3.5-4B by 1.8 points, while adding loops to SmolLM2 after pre-training adds no accuracy at this fine-tuning scale.The authors conclude that Ouro’s pre-training prepared the useful loops; their recipe turns those loops into a decision model but does not create them.Source: §H.4 · pp. 35–37, Table 36 · p. 38

H.5 Averaging the loops

  • 71.8%: averaging the eight loops is as accurate as loop 8 at 72.0%, while halving ECE from 0.093 to 0.044 without labelled data or further training.When calibration data come only from training sources, the average also has lower ECE than temperature scaling on other test items.Source: §H.5 · pp. 37–38, Table 37 · p. 39

I.1 Settings

  • The study uses 8,000 training, 500 development, and 3,000 test 2WikiMultiHopQA questions; its generator has 18.1M trained LoRA-adapter parameters, and evaluation reports greedy-answer token F1, exact match, and gold-answer containment.Source: §I.1 · pp. 38–39

I.2 Results

The verifier case study finds its largest F1 gain on inference questions, which require combining two facts. Exact match does not improve because the generator produces longer answers, and the reward does not prefer shorter variants.

I.2 Results

  • Across three seeds, eight-loop reward F1 ranges from 46.5 to 48.5; four-loop runs vary from 41.0 to 48.3, so four and eight loops cannot be separated.The one-loop reward gives the lowest F1 in every seed, while four- and eight-loop rewards change places.Source: §I.2 · p. 39

I.3 Exact match and the form of the answers

  • Exact match is 32.2 with the eight-loop reward versus 32.5 before training, as generated answers grow from 2.6 to 3.6 words on average.With eight loops, 17.5% of answers contain the gold answer plus further words; the reward treats some longer variants as the same option and cannot prefer the shorter form.Source: §I.3 · pp. 39–40, Table 39 · p. 40

I.2 Results

  • F1 on inference questions rises from 28.6 to 49.7 with the eight-loop reward, the largest gain across question types.The comparison is shown by question type in Figure 14.Source: §I.2 · p. 39, Figure 14 · p. 40

J Error Analysis: Details

The error analysis compares SanSi’s mistakes with single-pass models, traces how errors change across loops, and illustrates fixes, regressions, and persistent errors on unseen-source examples.

J Error Analysis: Details

  • Qwen3.5-4B is wrong on 68.0% of SanSi’s errors and chooses the same wrong option on 51.7% of them; SanSi is wrong on 72.0% of Qwen3.5-4B’s errors.SanSi’s error rate is 27.2%, compared with 25.7% for Qwen3.5-4B and 41.1% for SmolLM2-1.7B.Source: J Error Analysis: Details · p. 41
  • Between loops 1 and 8, SanSi fixes 20.9% of items and breaks 6.4%; 60.9% of its loop-8 errors were wrong at every loop.Errors are counted on test items with one gold option across the three seeds.Source: J Error Analysis: Details · pp. 40–41
  • 19.7% of SanSi’s errors have confidence of at least 0.9, compared with 24.0% for Qwen3.5-4B; SanSi’s error rate ranges from 13.2% in-distribution to 31.2% on far transfer.The rate on near transfer is 30.2%, and on JevBench items it is 27.7%.Source: J Error Analysis: Details · p. 41

K Examples

  • In unseen-source examples, loops correct early mistakes such as a word-overlap error and a true-to-unknown reasoning error, but can also overturn a correct answer or retain a confident error.For an unanswerable item created by removing the applicant’s age, loop 1 answers yes at 0.97; later loops bring the top probability below the 0.75 hard-answer threshold.Source: K Examples · pp. 42–43

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