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Beyond Action Entropy: Quotient-Space Exploration for Genome-Scale Metabolic Model Repair

Xuan Gong, Hanbo Huang, Wenbin Dai, Jing Wang, Lei Bai, Xiang Xiao, Weishu Zhao, Shiyu Liang

arXiv:2610.11627v1cs.LGq-bio.MN

TL;DR

For GEM repair, phenotype feedback may verify a solution without identifying its edit, while equivalent edits can make raw-action diversity differ from mechanism diversity. QuotientPO canonicalizes equivalent repairs and applies kernelized Rényi exploration; it raises Success@32 from 17.93% (Vanilla GRPO) to 20.10%.

  • Problem

    Verifier-guided GEM repair lacks gold edit targets, while equivalent reaction edits can represent the same mechanism and raw-action entropy rewards representation multiplicity.

    Source: §1 · p. 1, §4.1 · p. 4

  • Method

    QuotientPO canonicalizes verifier-preserving repairs into repair cores and uses a kernelized Rényi objective to estimate graded mechanistic crowding.

    Source: §1 · p. 2, §4.1 · p. 4, §4.2 · p. 5

  • Results

    Success@32 rose from 17.93% (Vanilla GRPO) to 20.10%, and core discovery rose from 0.50 to 0.57.

    Source: §5.2 · p. 7

  • Takeaways & Limitations

    The findings support mechanism-level rather than raw-edit diversity as the exploration target for verifier-guided GEM repair.

    Source: §4.1 · p. 4, §7 · p. 10

    Training supervision uses phenotypes only through verifier feedback, and at test time the policy receives only the initial GEM and organism representation.

    Source: §3 · p. 3

Abstract

from arXiv · show

Repairing scientific models from functional observations differs fundamentally from supervised prediction: feedback may certify a solution without revealing which structural correction is responsible. We study this setting for genome-scale metabolic model (GEM) repair, where multiple reaction edits can explain the same phenotypes and many apparently distinct edits correspond to the same biological mechanism. This many-to-one structure creates a hidden failure mode for conventional exploration: diversity in the output space need not translate into diversity of scientific hypotheses. We introduce QuotientPO, which collapses equivalent repairs into canonical mechanisms and optimizes exploration directly over the resulting quotient space. To make quotient exploration informative under finite rollouts, we derive a kernelized R\'enyi estimator that resolves graded crowding among distinct repair cores beyond coarse exact-match counts. On 2,212 held-out GEMs, QuotientPO improves Success@32 from 17.93% to 20.10% (+12.1% relative) while consistently increasing distinct successful-core discovery under the same sampling budget. These results establish quotient-space exploration as a principled approach to mechanism-level discovery under verifier-induced equivalence.

1 INTRODUCTION

Phenotype feedback can validate a GEM repair without uniquely identifying its underlying mechanism, while distinct reaction edits may encode the same mechanism. QuotientPO addresses this mismatch by canonicalizing equivalent repairs and exploring distinct mechanisms.

1 INTRODUCTION

  • Phenotype observations can validate repairs without uniquely identifying mechanisms, while many different edit sets can implement the same mechanism.Consequently, surface-level diversity need not correspond to distinct scientific hypotheses.Source: §1 · p. 1
  • Verifier-guided repair asks how to diversify exploration when scientific identity is defined up to verifier-preserving transformations.The verifier supplies functional feedback rather than a gold target for the structural correction.Source: §1 · pp. 1–2
  • QuotientPO canonicalizes verifier-preserving edits into equivalence classes, making scientific identity rather than syntactic multiplicity the unit of exploration.It pairs this quotient-space view with a kernelized Rényi objective to detect graded crowding among non-identical repairs.Source: §1 · p. 2, Figure 1 · p. 2
  • The paper formulates phenotype-supervised GEM repair, develops a theory and algorithm for quotient exploration, and evaluates recovery of distinct mechanisms.Its contributions include a large-scale amortized repair benchmark and gains in multi-budget repair and distinct mechanism recovery.Source: §1 · p. 2

2 BACKGROUND AND RELATED WORK

Prior GEM repair methods either address models independently or target missing-reaction recovery, while learned diversity methods do not resolve verifier-induced equivalence among repair edits. The paper instead learns a shared policy for functional bidirectional repair across organisms.

Genome-scale metabolic reconstruction and model repair

  • Classical gap-filling and phenotype-reconciliation methods repair GEMs independently, without sharing repair knowledge across organisms.Learning-based work largely targets synthetically removed reactions rather than functional bidirectional repair.Source: §2 · p. 2
  • A shared repair policy uses phenotypes as training supervision to predict functionally validated bidirectional edits without test-time phenotype access.The same phenotypic constraints can admit multiple valid repairs.Source: §2 · p. 2

Diversity-aware reinforcement learning for structured prediction

  • Existing diversity methods include entropy policies, skill discovery, quality-diversity optimization, GFlowNets, and semantic or kernelized approaches, but do not resolve verifier-induced many-to-one repair mappings.Raw-output diversity can reward representation multiplicity without expanding distinct repair hypotheses.Source: §2 · pp. 2–3

3 PRELIMINARIES AND PROBLEM FORMULATION

The paper frames GEM repair as constrained reaction editing evaluated through phenotype predictions from flux balance analysis. It learns a stochastic repair policy from phenotype consistency while withholding phenotypes as model inputs.

3 PRELIMINARIES AND PROBLEM FORMULATION

  • A GEM is represented as a directed bipartite graph between metabolites and reactions, with a stoichiometric matrix specifying reaction consumption and production coefficients.Source: §3 · p. 3
  • A repair adds candidate reactions and removes existing ones, and flux balance analysis predicts growth for tested phenotypes in the repaired model.The verifier compares these predictions with observed phenotypes to measure consistency.Source: §3 · p. 3
  • The learned stochastic policy maps an initial model and protein representation to valid repairs, using phenotype consistency only as training supervision.Corrected GEMs and gold reaction-edit sets are unavailable, and observed phenotypes generally do not uniquely identify the repair.Source: §3 · p. 3
  • At test time, the policy receives only the initial model and protein representation, and evaluation uses held-out phenotype consistency.Source: §3 · p. 3

4 METHODOLOGY

The method addresses reward-equivalent repairs by exploring canonical repair mechanisms rather than raw edits. It combines quotient entropy, a graded kernelized Rényi estimate, and group-relative policy optimization to guide finite-rollout exploration.

4 METHODOLOGY

  • Policy-gradient reinforcement learning optimizes non-differentiable verifier rewards, while explicit exploration addresses multiple reward-equivalent repairs that can leave a collapsed policy without a signal to discover alternatives.Group-relative policy optimization supplies rollout groups for quotient estimation.Source: §4 · p. 3

4.1 REPRESENTATION-INVARIANT EXPLORATION WITH QUOTIENT ENTROPY

  • Quotient exploration canonicalizes equivalent raw repairs into repair cores and evaluates diversity using their aggregated probability distribution, removing within-core representation multiplicity from the exploration signal.Theorem 4.1 characterizes this invariance: exploration depends on the induced distribution over cores, not how probability is distributed among equivalent raw realizations.Source: §4.1 · p. 4

4.2 GRADED QUOTIENT ESTIMATION WITH R´ENYI KERNELS

  • Exact core matching gives only binary crowding information and becomes uninformative when sampled cores rarely repeat.A kernelized quadratic Rényi estimate uses Jaccard similarity to assign graded crowding to overlapping repair cores; Figure 2 contrasts isolated and clustered candidates.Source: §4.2 · pp. 4–5, Figure 2 · p. 4

4.3 QUOTIENTPO: QUOTIENT-SPACE POLICY OPTIMIZATION

  • QuotientPO combines verifier feedback and quotient diversity in candidate-level credit, with a shaped reward that ranks fully verified repairs above partial improvements.A leave-one-out group-entropy linearization yields crowding costs, where positive values indicate crowded cores and negative values indicate expanded quotient coverage.Source: §4.3 · p. 5
  • The complete objective combines group-relative verifier advantage with a quotient term that shifts probability away from crowded cores toward distinct hypotheses.An auxiliary term adds credit for newly discovered successful cores and can penalize edits outside the canonical core.Source: §4.3 · p. 5

5 EXPERIMENTS

Experiments evaluate phenotype-guided GEM repair on a held-out test set and compare QuotientPO with verifier-guided, imitation, and diversity-oriented baselines. QuotientPO improves repair success and successful-core discovery, with analyses tracing the gains to quotient-space coverage and testing robustness across objectives, architectures, and design choices.

5.2 MAIN RESULTS

  • 20.10% Success@32 for QuotientPO versus 17.93% for Vanilla GRPO, with core discovery rising from 0.50 to 0.57.The phenotype-informed MILP oracle reaches 15.01% full repair within its fixed 600 s search budget.Source: Table 1 · p. 6, §5.2 · p. 7

5.1 EXPERIMENTAL SETUP

  • The sealed test set contains 2,212 species-disjoint GEMs, and the terminal verifier rewards repairs reproducing observed phenotypes across 18 FBA assays.Each tested nutrient is the sole elemental source against an element-depleted background.Source: §5.1 · p. 6

5.2 MAIN RESULTS

  • At k = 32, QuotientPO’s margin over Vanilla GRPO reaches +2.17 Success@k points and +0.068 successful cores per GEM.Both repair success and successful-core discovery improve as the sampling budget increases.Source: Figure 4 · p. 7, §5.2 · p. 7

5.3 EMPIRICAL ANALYSIS OF QUOTIENTPO’S MECHANISM

  • Successful QuotientPO repairs are farther from the MILP teacher than SFT-Soft repairs: mean distance is 0.961 versus 0.780.This supports the view that verifier feedback reveals successful alternatives beyond the supervised mode.Source: §5.3 · pp. 7–8
  • Over 80% of MaxEnt and GFlowNet sampled representations collapse after canonicalization, while QuotientPO recovers more distinct successful cores even with equal successful-sample budgets.At r = 2, 4, and 8, QuotientPO recovers 1.46 vs. 1.43, 2.05 vs. 1.96, and 2.78 vs. 2.66 cores, respectively, compared with Vanilla GRPO.Source: §5.3 · p. 8
  • Structural pretraining and ESM-2 conditioning improve single-edit imitation, while set-level SFT provides rollout reward variance for subsequent RL.Without structural pretraining, operation-level Hit@1 drops from 20.7% to 11.4%; without ESM-2 conditioning, it drops to 14.3%.Source: Figure 7 · p. 8, §5.3 · p. 8

5.4 ROBUSTNESS AND ABLATION ANALYSIS

  • Under RLOO, QuotientPO reaches 17.21 Success@32 versus 15.75 for Vanilla; under GRPO, it improves both Soft and Hard architectures.For GRPO, Soft results are 20.10 vs. 17.93 and Hard results are 6.65 vs. 5.92.Source: §5.4 · p. 9, Table 3 · p. 9
  • Removing the quotient term drops S@32 from 20.10 to 17.54, and quotient-space exploration beats the raw-space counterpart with the same kernel and budget.The comparison is 20.10 vs. 18.16 on Success@32 and 0.57 vs. 0.52 on successful-core discovery; larger bandwidth over-smooths distant cores.Source: §5.4 · p. 9, Table 3 · p. 9

6 CASE STUDY

The case studies contrast missed viable mechanisms with surface variation around the same outcome-determining repair core. QuotientPO finds functionally distinct repairs in both examples.

6 CASE STUDY

  • On S. mashuensis, Vanilla GRPO collapses onto boundary entry blockade, while QuotientPO finds an intracellular reductase-knockout mechanism.The alternative jointly knocks out NITR and NTR3B.Source: §6 · p. 9, Figure 8 · p. 9
  • On S. palmae, varied Vanilla proposals retain the SUCRe bypass, while QuotientPO recovers clean successful cores for sucrose utilization.Counterfactual replays show that toggling SUCRe directly flips repair outcomes.Source: §6 · p. 9, Figure 8 · p. 9
  • The cases expose two failures of raw exploration: missing viable mechanisms and mistaking surface variation around one outcome-determining core for useful diversity.QuotientPO directs exploration toward functionally distinct mechanisms.Source: §6 · p. 9

7 CONCLUSION

The conclusion frames GEM repair as a mismatch between edit-level exploration and mechanism-level identity. QuotientPO canonicalizes verifier-preserving edits and explores the resulting quotient space.

7 CONCLUSION

  • Combinatorial candidate edits can collapse to the same underlying mechanism, leading standard exploration to mistake syntactic multiplicity for scientific diversity.The conclusion presents this as a structural mismatch in GEM repair from indirect functional feedback.Source: §7 · p. 10
  • QuotientPO canonicalizes verifier-preserving edits and uses kernelized R'enyi relaxation to replace coarse exact-match statistics with graded finite-sample crowding signals.The exploration unit changes from raw reaction-level actions to invariant repair mechanisms.Source: §7 · p. 10
  • The conclusion reports improved repair success and coverage of distinct valid mechanisms, and suggests defining exploration over equivalence classes when solutions have equivalent representations.Source: §7 · p. 10

AI USE STATEMENT

The authors used generative AI to assist with writing refinement and code generation, then reviewed the AI-assisted work.

AI USE STATEMENT

  • Generative AI assisted with refining author-written text and generating or revising code, which the authors manually reviewed, tested, and verified or edited.The authors take full responsibility for the work's correctness and final content.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

This material includes an ethics statement alongside implementation details for QuotientPO and background on GEM formulation. The technical descriptions cover policy outputs, canonical repair cores, exploration credits, and the optimization objective.

ETHICS STATEMENT

  • The work uses public microbial genomic and phenotype data and computational resources, with experiments conducted in silico and no human participants, animal experiments, or living-organism manipulation.Potential downstream metabolic engineering or synthetic biology uses should follow applicable biosafety and regulatory practices.Source: ETHICS STATEMENT · p. 10

A ALGORITHM DETAILS OF QUOTIENTPO

  • QuotientPO fine-tunes a supervised one-shot repair policy using verifier reward, kernelized quotient crowding, successful-core discovery credit, and a penalty for edits outside the canonical core.The policy conditions on the initial GEM and genomic representation; phenotype labels are used exclusively by the verifier.Source: A ALGORITHM DETAILS OF QUOTIENTPO · p. 16

A.1 ONE-SHOT REPAIR POLICY

  • The policy predicts the complete unordered repair set in one shot, treating ADD and REMOVE of a reaction as distinct legal signed edits.Source: §A.1 · p. 16

A.2 VERIFIER-PRESERVING CANONICALIZATION

  • Canonicalization deterministically removes redundant edits while preserving verifier outcomes, yielding a deletion-irreducible core and a well-defined partition of repairs.The redundancy cost is the edit-set size minus the canonical core size.Source: §A.2 · p. 17

A.3 KERNELIZED QUOTIENT CROWDING (QUOTIENT TERM)

  • Kernelized crowding compares signed repair cores; positive crowding cost suppresses locally redundant cores, while negative cost favors isolated ones.The implemented score is clipped to [-8, 8] for finite-sample numerical stability.Source: §A.3 · p. 17

A.4 FINITE-PANEL SUCCESSFUL-CORE DISCOVERY CREDIT (QPI TERM)

  • Finite-panel QPI supplements geometric diversity with credit for successful-core discovery, sharing credit among successful candidates and rewarding unique or newly archived cores.The reported setting repartitions each group of 32 rollouts four times into panels of size 8.Source: §A.4 · p. 17

A.5 COMPLETE QUOTIENTPO OBJECTIVE

  • The objective combines verifier-guided GRPO, kernelized quotient diversity, finite-panel QPI, and a penalty for surface edits removed by canonicalization.The auxiliary outcomes and costs are detached from the actor graph; gradients flow through the complete edit-set likelihood.Source: §A.5 · p. 18

B.1 FORMULATION OF GEM MODELING

  • A GEM is a directed, weighted bipartite graph of metabolites and reactions, with stoichiometric edges; reaction edits add or remove reaction nodes and their matrix columns.FBA imposes steady-state mass balance and reaction directionality bounds; the background also describes a biomass-supporting flux threshold.Source: §B.1 · p. 19

B.2 THEORY OF REPRESENTATION-INVARIANT EXPLORATION

This section develops a theory of representation-invariant exploration, explaining how quotient-space objectives address the bias of action-space entropy and how kernelized crowding relates to quotient diversity and mechanism coverage. It also describes the construction of assay-conditioned GEM cohorts and leakage-controlled splits.

B.2 THEORY OF REPRESENTATION-INVARIANT EXPLORATION

  • Action-entropy optimization assigns quotient mechanisms probability proportional to nm exp(Uy(m)/β), adding an implicit β log nm utility for representation multiplicity.The multiplicity factor arises when the probabilities of syntactically distinct representatives are aggregated at the mechanism level.Source: §B.2 · pp. 20–21
  • The quotient marginalization map is surjective, and exploration functionals invariant to raw-policy redistributions at fixed quotient law factor through the quotient.Continuity and convexity or concavity are preserved through a linear representative lift.Source: §B.2 · pp. 21–22
  • Quotient Shannon entropy removes the representation-volume factor nm from the unrestricted mechanism-level optimum.The result contrasts with action entropy, whose induced quotient distribution includes the multiplicity factor.Source: §B.2 · p. 22

B.3 COLLISION SPARSITY AND KERNEL R´ENYI ESTIMATION

  • When rollout groups are collision-free, exact-match leave-one-out statistics give every sampled candidate identical exploration credit, namely zero.The collision probability vanishes along sequences where L^2 times the quotient collision concentration tends to zero.Source: §B.3 · p. 23
  • The signed-Jaccard kernel is positive semidefinite and converges uniformly to the exact-match kernel as bandwidth h tends to zero.Its order-two kernel concentration estimator is unbiased, and the associated expected kernelized Rényi estimate converges uniformly at rate O(L^-1) for fixed h.Source: §B.3 · pp. 24–25
  • For fixed bandwidth, the kernel R´enyi estimator is uniformly O(L^-1) consistent over quotient laws, but its bound is conservative and signed-Jaccard similarity measures structural overlap, not biological equivalence.The convergence of the kernel functional to H2(q) is a population result.Source: §B.3 · p. 26

B.4 QUOTIENTPO OPTIMIZATION THEORY

  • The leave-one-out crowding cost is positive for above-average kernel similarity and becomes a locally consistent score surrogate for population quotient diversity as rollout groups grow.The local-consistency result assumes a differentiable policy with locally fixed support and a fixed kernel during the policy update.Source: §B.4 · pp. 26–28
  • Success@K depends only on successful-core probability mass, whereas distinct-core coverage also depends on its allocation and is maximized by uniform allocation at fixed mass.Equalizing probability across successful cores weakly increases expected distinct-core coverage.Source: §B.4 · pp. 28, 30, 31

C.1 CONSTRUCTION PIPELINE

  • The construction pipeline yielded 36,855 draft GEMs and 21,615 phenotype-error-eligible post-training GEMs from BacDive-linked proteomes and standardized nutrient assays.The candidate GEMs were built under four CarveMe settings, and eligible models required at least one decidable target and one initial phenotype error.Source: §C.1 · p. 31
  • Identity-component and genus-held-out partitioning produced a 2,212-GEM final test set with zero biological-identity overlap across pre-training, SFT, RL, validation, and test.The held-out genera were Bacillus, Clostridium, Lactobacillus, Mycobacterium, and Streptomyces.Source: §C.1 · p. 32

C.2 DATASET CHARACTERIZATION AND QUALITY CONTROL

The appendices characterize the phenotype-supervised GEM dataset and sealed benchmark, then describe the MILP and GFlowNet baselines, including the GFlowNet anchor selection and the three-stage training pipeline.

C.2 DATASET CHARACTERIZATION AND QUALITY CONTROL

  • The linked dataset contains 28,842 retained GEMs and 153,869 model-level BacDive target groups, with broad phenotype-partition consistency and expected shifts in the genus-held-out test set.Counts measure phenotype-target coverage, not repair errors; organisms can contribute through multiple GEM variants and compound-level annotations remain independent targets.Source: §C.2 · pp. 32–33

C.3 BENCHMARK VALIDATION

  • The sealed benchmark covers 2,212 GEMs across five held-out genera, four reconstruction templates, and 16 core metabolites, with both false-negative and false-positive errors.These errors require policies to coordinate reaction additions and deletions.Source: §C.3 · p. 33

D.1 MILP

  • The phenotype-informed MILP oracle uses test-time labels to minimize phenotype errors before edit count, generating phenotype-guided cuts through FBA separation.Candidates are independently verified under the original assay conditions; the oracle retains the best lexicographic candidate found within a 600-second budget.Source: §D.1 · pp. 33–34

D.2 GFLOWNET

  • The GFlowNet baseline samples complete raw edit sets, so distinct edits remain distinct objects even when they map to the same repair core.Its one-shot policy uses independent Bernoulli decoders and a log-partition variance loss to match reward-tilted target probabilities.Source: §D.2 · p. 34
  • Validation selected anchor strength γ = 0.6, achieving Success@8 of 9.38% with 10.98 ADD edits; the identity-uniform diagnostic yielded 0.276% validation success.The unanchored run degenerated into thousands of spurious ADD edits, while the identity-uniform diagnostic collapsed toward nearly empty repairs.Source: §D.2 · pp. 34–35, Table 5 · p. 35
  • The baseline combines verifier-guided GRPO with raw-reward LPV distribution matching, while repair cores are computed only after training for evaluation.It shares the backbone, verifier, action masks, and frozen SFT initialization with the other RL methods.Source: §D.2 · p. 35

E EXPERIMENTAL DETAILS

  • Training proceeds through pretraining, phenotype-supervised SFT, and verifier-grounded RL to learn GEM structure, a repair prior, and joint consequence optimization.Source: E EXPERIMENTAL DETAILS · p. 35

E.1 BASE MODEL ARCHITECTURE.

This section describes a shared graph-and-proteome model, staged supervised initialization, and verifier-guided one-shot policy optimization. It also defines finite-sample credit-resolution measures and evaluates quotient-exploration objectives in terms of reconstruction-template performance, mechanism diversity, graded credit, and downstream performance.

E.1 BASE MODEL ARCHITECTURE.

  • At five million pretraining examples, the 768×12 model reaches 63.49% ADD recall and 80.73% REMOVE recall.It is the shared backbone: a 12-layer edge-aware GINE encodes GEM graphs, a universal reaction encoder represents candidate ADDs, and frozen ESM-2 protein embeddings are fused with the global GEM representation.Source: §E.1 · p. 35, §E.2 · p. 37, Figure 12 · p. 37

E.3.1 SFT-A: PARTIAL-LABEL REACTION-EDIT RANKING

  • Pretrained + ESM2 raises operation-macro Hit@1 from 0.016 on pretrained initialization to 0.207 and achieves development loss 3.248.SFT-A ranks improving and required-neutral edits as multi-positive labels while masking regressive actions, retaining alternative valid repair pathways; removing pretraining or ESM2 lowers Hit@1 to 0.114 or 0.143, respectively.Source: §E.3.1 · pp. 37–38, Table 6 · p. 38

E.3.2 SFT-B: COMPLETE SET-LEVEL REACTION-EDIT SUPERVISION

  • Soft-SFT exposes nonzero within-group reward variance for 51.6% ± 4.9% of organism groups, versus 32.0% ± 5.1% for Hard-SFT.Soft-SFT uses a state-conditioned inclusion threshold so repair size emerges from continuous candidate scores, whereas Hard-SFT predicts a discrete edit count; separate certified completions remain distinct training examples.Source: §E.3.2 · pp. 38–39

E.4 REINFORCEMENT LEARNING DETAILS

  • QuotientPO, Vanilla GRPO, and MaxEnt share one-shot Bernoulli repair sampling initialized from Soft-SFT; MaxEnt uses a fixed entropy coefficient of 2e−4 after larger values inflated ADD cardinality.Evaluation covers 2,212 held-out initial GEMs with 32 stochastic repairs per GEM; QuotientPO adds 8.9% per-update runtime over Vanilla GRPO, mainly from verifier-side canonicalization and kernel scoring.Source: §E.4 · p. 40, §E.5 · p. 41

F.4 SUCCESS-CONTROLLED MECHANISM COVERAGE

  • Under equalized successful-draw budgets, QuotientPO's distinct-core advantage over Vanilla GRPO widens from +0.035 at r = 2 to +0.246 at r = 16.The comparison uses paired GEM cohorts where both methods achieve at least r successes among 32 rollouts; r is capped at 16 because larger budgets leave few paired GEMs.Source: Table 10 · p. 42, §F.4 · p. 42

F.2 SOLUTION SPACE GEOMETRY: SFT VERSUS POLICY OPTIMIZATION

  • QuotientPO spreads successful repairs across disconnected edit-space clusters, while SFT-Soft rollouts cluster near the teacher and standard RL baselines often collapse onto redundant mechanisms.Across six representative GEMs, QuotientPO discovers more distinct successful canonical repair cores than Vanilla GRPO, MaxEnt, and GFlowNet.Source: §F.2 · p. 42, §F.3 · p. 42

F.5 FINITE-SAMPLE CREDIT RESOLUTION ANALYSIS

  • Exact R´enyi-2 tie rate rises from 5.67% to 10.23%, while kernel credit resolves 78.20% of its exact-count ties and 84.86% of Vanilla GRPO's at step 864.TieRate measures indistinguishable multiplicities under exact counting; ResolvedRate measures the fraction of those ties distinguished by graded kernel credit.Source: §F.5 · pp. 42–43
  • Exact-count tie rates measure distinct-core pairs indistinguishable under multiplicity credit within rollout groups, while kernel-resolved rates measure the fraction of these ties separated by graded kernel credit.Table 11 presents the full training trajectory for this analysis.Source: Table 11 · p. 43
  • Table 12 reports downstream performance of quotient exploration objectives.Source: Table 12 · p. 43

F.6 MULTI-SEED ROBUSTNESS AND ABLATION RESULTS

Multi-seed tests and ablations assess QuotientPO’s robustness, while biological case studies distinguish where its focal-mechanism coverage gains arise.

F.6 MULTI-SEED ROBUSTNESS AND ABLATION RESULTS

  • 17.21 ±0.81 Success@32 for QuotientPO exceeds Vanilla’s 15.75 ±1.20 across three seeds.Table 13 reports Success@32 as mean ± standard deviation over three seeds.Source: §F.6 · p. 44, Table 13 · p. 44
  • Removing the quotient term drops Success@32 from 20.10 to 17.54, the steepest ablation decrease.The raw-space kernel reaches 18.16, compared with 20.10 for the default configuration.Source: §F.6 · p. 44, Table 13 · p. 44

G BIOLOGICAL ANALYSIS AND CASE STUDIES

  • Vanilla’s focal alternatives are lost at three stages: during canonicalization, in unsuccessful repairs, or before proposal; QuotientPO retains or discovers them.These case-study tiers correspond respectively to retention, viability, and discovery gaps.Source: G BIOLOGICAL ANALYSIS AND CASE STUDIES · p. 44

H LIMITATIONS AND FUTURE WORK

The study’s conclusions are bounded by its reconstruction, media, and verifier choices; proposed extensions include broader pipelines, richer conditions, and additional biological constraints.

H LIMITATIONS AND FUTURE WORK

  • All GEMs use CarveMe reconstructions in the BiGG reaction system, so transfer across reconstruction pipelines remains untested.The shared vocabulary and modeling convention provide consistency across organisms.Source: H LIMITATIONS AND FUTURE WORK · p. 44
  • Standardized category-specific media proxy BacDive conditions but omit some strain-specific environmental details.The abstraction addresses heterogeneous or incomplete culture descriptions in many BacDive assays.Source: H LIMITATIONS AND FUTURE WORK · p. 44
  • The verifier assesses steady-state metabolic feasibility, not regulatory or kinetic effects.Source: H LIMITATIONS AND FUTURE WORK · p. 44
  • Future work proposes other reconstruction pipelines, richer or uncertainty-aware culture models, additional biological constraints, and targeted experimental validation.The passage also suggests extending quotient-space exploration to other scientific domains.Source: H LIMITATIONS AND FUTURE WORK · p. 44
  • The case-study tables distinguish coverage gains at retention, successful-combination, and raw-module stages.They describe cases where Vanilla loses focal alternatives during canonicalization, fails to include proposed modules in successful repairs, or never proposes the module.Source: Table 14 · p. 47, Table 15 · p. 48, Table 16 · p. 49

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