{"id":"87eafedf-db9d-46bc-9020-c2f6469e76ff","arxiv_id":"2608.00326","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":7,"one_line_summary":"An LLM agent fine-tuned on algebra tasks and optimized with exact symbolic rewards reaches 78.96% verified success on synthetic weighted sum-of-squares certificate search, versus 44.73% for the base model with the same tools.","lead":"This paper trains an LLM agent to find sum-of-squares certificates for polynomials by combining algebra practice, reinforcement learning, and exact SymPy tools. On a synthetic benchmark, the full system succeeds on 79% of weighted SOS problems, far above the 45% of the base model with the same tools.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Aggregate 78.96% may be inflated by easy generator strata (m=1 perfect squares, low-degree cases); per-stratum success is needed to support the coordination claim.","rationale":"The reader correctly identifies the same-generator synthetic distribution as the key weakness. I go one step further: even within that same-generator setting, the aggregate acceptance rate may be dominated by easy slices (m=1, low degree) where no meaningful tool coordination is required. This is a load-bearing concern for the paper's qualitative claim about coordination, not for the numerical correctness of the four-way comparison. The paper's explicit scope caveats weaken the OOD-generalization version of the concern, but they do not address internal difficulty stratification. The proposed test is cheap and requires no new training: just record generator parameters and recompute success by stratum. I considered the missing SFT+Tools ablation, but the paper consistently frames its comparisons as complete systems and disclaims component attribution, so that gap is acknowledged rather than hidden. The exact terminal verifier and the honest failure taxonomy are real independent strengths. Keeping the CONDITIONAL verdict seems right: the central numbers are probably correct, but the coordination interpretation should be conditional on per-stratum evidence.","tokens_in":17323,"tokens_out":12085,"duration_ms":122935,"concrete_test":"Re-analyze the existing 10,000 weighted-SOS test instances with generator metadata attached: m, degree, number of variables, weight multiset, and whether factorize_polynomial returns a perfect-square factor. Report verified success for Full and Base+Tools stratified by m=1 / m=2 / m>=3 and by degree 2 / 4 / 6, with Wilson intervals per stratum. If the 78.96% and the Full-vs-B+T gap concentrate in m=1 or degree-2 strata, the coordination claim is unsupported for the hard multi-square regrouping cases; if the advantage persists across m>=3 and degree-6 strata, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central qualitative claim is that the full system learns to coordinate sequential symbolic transformations in weighted-SOS search (Introduction; Figure 2). But the headline 78.96% is a single average over a generator that samples m uniformly in {1,...,5}, degrees in {2,4,6}, and weights in {1,...,10} (Appendix, Table A2). No result is reported separately for the parameter slices where coordination is actually needed. In particular, m=1 instances are exactly f = c·q^2: one factorize_polynomial call exposes the square and the certificate is immediate. Degree-2 instances can often be solved by completing the square with very few regroup-and-check steps. If such easy cases are a large share of the 10,000-instance test set and drive the Full advantage over Base+Tools (78.96 vs 44.73), then the aggregate metric does not establish the paper's coordination story on the regroup-and-revise cases that motivate the environment. The paper honestly notes the same-generator scope (Experiments; Conclusion), but that limitation statement does not address the internal difficulty mix of the generator. The exact verifier makes the raw numbers credible; the unproven step is the inference from an aggregate acceptance rate to 'coordination' of symbolic tools.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a controlled synthetic environment in which an LLM agent must produce verified weighted sum-of-squares (SOS) certificates for polynomials. The pipeline begins with Phi-4-reasoning-plus, applies QLoRA SFT on 1.35M synthetic examples covering eight supporting algebraic tasks plus weighted SOS, and then applies GRPO with task-specific symbolic rewards. At evaluation, two of the four configurations receive a native four-function SymPy interface for SOS search, and every terminal certificate is checked by exact expansion and coefficient comparison. On 10,000 held-out, same-generator test polynomials, the full SFT+GRPO+tools system reaches 78.96% verified success, compared with 44.73% for the base model with the same tools; the eight-task direct macro-accuracy is 93.35%. The paper explicitly scopes the evidence to same-generator synthetic problems, one backbone, and one run per training stage, and distinguishes checkpoint-level ability from complete-system certificate search.","tokens_in":17685,"tokens_out":6018,"duration_ms":65570,"significance":"If the empirical claims are taken at face value, this is a useful, carefully instrumented case study of tool coordination for exactly checkable algebraic outputs. The main strengths are: (i) the terminal verifier is exact, so all accepted certificates are correct; (ii) the paper cleanly separates checkpoint-level competence from complete-system search; (iii) the training–deployment mismatch (no native tool calls during training, native calls at test time) is explicitly documented; and (iv) the authors are unusually candid about the scope and about what the four-configuration design can and cannot identify. The result is not a claim of state-of-the-art inequality proving, but rather a controlled benchmark for studying sequential symbolic-tool use. The central quantitative claim is credible as a measurement on this synthetic benchmark, but the paper has not yet supplied the per-stratum analysis needed to support the qualitative claim that the learned system is specifically coordinating multi-step symbolic search rather than exploiting easy generator strata.","major_comments":[{"comment":"The headline weighted-SOS success is a single average over a generator with m uniform in {1,...,5}, degree in {2,4,6}, and variables in {1,2,3}. Since every test polynomial is SOS by construction and m=1 is exactly a single square f = c·q^2, the aggregate cannot distinguish learned multi-step coordination from success on easy strata. Please report verified success for Full and Base+Tools stratified by m, degree, and number of variables, and ideally the mean number of native calls per stratum. Without such a breakdown, the Introduction's 'coordination' narrative — regroup, revise, retry — is not established for the cases that motivate the environment.","section":"Experiments, Table 6; Appendix Table A2"},{"comment":"Eq. (A21) defines the trajectory-shaping reward R_tool in terms of N_schema, N_exec, N_repeat, and T, which are properties of native tool-call trajectories. However, Table 3 and the deployment section state that GRPO optimizes chat-template responses with no native calls. The paper does not specify where R_tool is evaluated during training. If it is applied to simulated traces, the trace format and the exact computation of these counts must be described; if it is not part of GRPO, the reward definition should be corrected or moved. This matters because the GRPO reward is a central component of the proposed recipe and the reader must be able to reproduce the training signal.","section":"Verifier-grounded GRPO and Appendix Eqs. (A20)–(A21)"}],"minor_comments":[{"comment":"The nine-task macro combines checkpoint-level direct-task accuracies with a system-level SOS success rate. The text does acknowledge this heterogeneity, but the table caption should repeat the caveat to prevent misreading the number as a uniform checkpoint metric.","section":"Table 6"},{"comment":"The paper reports that successful Full episodes use 3.8 native calls on average. Reporting the distribution of call counts, and the call counts of failed episodes, would strengthen the coordination narrative and help the reader judge where search difficulty lies.","section":"Experiments, 'Verification separates search from proof'"},{"comment":"There are a few transcription/formatting artifacts in the text (e.g., 'macro- average' and 'on90,000separately generated' lack spacing). These should be cleaned up in the camera-ready version.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within scope and the exact verifier makes the measured numbers trustworthy. The main risk is that the central 'coordination' claim is not yet supported by per-stratum results. The missing clarification about where R_tool is applied during GRPO is also essential. I would not reject; both concerns are addressable within the current scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. It's a well-executed, honestly scoped study of whether you can post-train an LLM to coordinate exact symbolic tools on weighted SOS certificates. The training recipe is new as a combination: eight-task synthetic curriculum, SFT on simulated traces with no native tool calls, then GRPO with exact symbolic rewards, then deployment with a native SymPy interface. The terminal verifier is exact expansion and coefficient comparison, so every accepted certificate is genuinely correct. That is real evidence, and the paper deserves credit for it. The authors also separate checkpoint-level skill from complete-system search, report a terminal failure taxonomy, and give Wilson intervals. They explicitly concede the same-generator scope, one backbone, and one run per stage.\n\nThe soft spots are real but proportionate. First, the stress-test critique is fair: the generator samples m uniformly from 1 to 5, and m=1 instances are just f = c q^2, where a single factorize call exposes the square. Low-degree cases can often be solved by completing the square in a few steps. The paper reports only the aggregate 78.96%, with no breakdown by m or degree. Without that, the aggregate doesn't establish the 'coordination' claim on the regroup-and-revise cases that motivate the environment. It's not fatal—successful Full episodes average 3.8 calls, which suggests real multi-step search—but it's a gap. Second, the four-configuration design is missing SFT+Tools and GRPO-without-tools, so the recipe's components aren't isolated; the paper says this itself, so it's a limitation rather than a hidden flaw. Third, single-run-per-stage means no training variance estimate, and no public code/data makes reproduction harder.\n\nAll of these are addressable in revision. The central measurement is likely correct, and the paper is honest about what it does and doesn't show. It's worth a serious referee; I'd want the per-stratum breakdown and the two missing ablations before accepting. This isn't a paper that changes practice in AI for math, but it's a useful controlled case study for anyone working on tool-calling agents with exact verifiers.","headline":"A solid, honestly scoped controlled case study of tool-coordination training for SOS certificates, with a credible exact-verified headline result but an aggregate metric that doesn't yet isolate the coordination claim.","tokens_in":18160,"tokens_out":2758,"would_cite":true,"duration_ms":28678,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A language model can be post-trained to coordinate exact symbolic tools and find verified sum-of-squares certificates: the full system succeeds on 78.96% of held-out synthetic problems, versus 44.73% when the same base model is given the sa","keywords":["weighted sum-of-squares certificates","LLM tool calling","symbolic computation agents","verifier feedback","GRPO","synthetic curriculum","polynomial nonnegativity","exact verification"],"falsifier":"Stratify the reported 78.96% by the hidden number of squares m: if verified success is near 100% on m=1 and drops sharply for m=4–5, or if retraining on an out-of-generator sampler (more variables, degree 8, non-SOS distractors) makes the Full-minus-Base+Tools gap collapse, the coordination claim is an artifact of the generator.","tokens_in":17241,"feed_emoji":"🧮","tokens_out":9688,"duration_ms":84988,"temperature":0.7,"pith_summary":"This paper argues that giving a language model exact symbolic tools is not enough: the agent must also learn which operations to request, in what order, and when to stop. The authors make this concrete with weighted sum-of-squares (SOS) decomposition, a certificate of the form f = sum_j c_j p_j^2 with positive rational weights, where checking a candidate is exact polynomial expansion but finding one requires strategic regrouping and multi-step search. They build a 1.35-million-example synthetic curriculum over nine polynomial tasks, first fine-tune on direct algebra and simulated symbolic traces, then optimize the policy with verifier-grounded reinforcement learning (GRPO). On held-out same-generator problems, the complete system reaches 78.96% verified SOS success versus 44.73% for the same base model given the same native tools, and 91.75% macro accuracy across the nine tasks. The authors stress that this is a complete-system comparison and that success measures certificate-search yield inside a controlled environment, not natural-problem generalization.","feed_headline":"Algebraic post-training lifts verified SOS success to 79%","feed_subtitle":"Nine-task algebra training plus verifier rewards beats the same tools on an untrained model: 78.96% vs 44.73%.","key_machinery":"The load-bearing object is the exact terminal verifier: a candidate is accepted only if it parses as sum_j c_j p_j^2 with positive rational c_j, each p_j a polynomial, and expand(f - sum) is identically zero, making every accepted certificate a machine-checkable proof of nonnegativity. Around this verifier, the training recipe combines a nine-task synthetic curriculum (four local operations, four structural-algebra tasks, weighted SOS), supervised fine-tuning on direct algebra and simulated symbolic traces, and GRPO with rewards computed from exact algebraic contracts rather than string matches. At evaluation, the agent drives four native symbolic functions—expand, collect, reorder, factoriz","core_discovery":"The central claim is that a tool-using language model performs exact multi-step symbolic search far better when post-trained on the underlying algebra and optimized against an exact verifier than when the same tools are simply added to the base model. Concretely, the full configuration verifies 78.96% of held-out weighted-SOS instances, versus 44.73% for the base model with the same four native functions. Acceptance requires positive rational weights, polynomial squares, and exact coefficient-by-coefficient equality after expansion, so every counted success is a checked identity. The authors frame the object of study as tool coordination: exact operations do not eliminate errors in choosing,","pith_inferences":["If the result holds up, it implies tool-use evaluation should report the terminal state reached (the verified certificate) alongside call validity, since a schema-valid call can still be part of a failed search.","A natural stress test the paper does not run is stratifying success by hidden generator difficulty—number of squares m, degree, variable count; a concentration of failures at high m would indicate the coordination advantage is partly an easy-instance artifact.","The recipe may transfer to other domains with cheap exact verifiers—such as term rewriting, equality saturation, or proof-assistant checkable obligations—because the core idea is to train on subtasks whose outputs are exactly checkable before RL on the composite task.","Since every evaluation input is SOS by construction, the system is never asked to distinguish SOS from non-SOS; an out-of-distribution control with non-SOS polynomials would clarify whether the model learns to search or to exploit guaranteed solvability."],"forward_implications":["The full system's 78.96% verified SOS success, against 44.73% for the base model with the same tools, supports the claim that algebraic post-training complements interface access rather than substituting for it.","Full is best on all eight direct tasks (93.35% macro), so the trained checkpoint carries the algebraic skills even when native tools are disabled.","The gap over SFT grows from 12–18 points on local/structural tasks to 27 points on weighted SOS, consistent with the claim that the hardest part is global search and stopping, not single operations.","Exact verification separates failures into abstention, structural invalidity, and identity mismatch; because only the exact identity counts, the reported rate is a search-yield measure, not a recognition or syntax score.","Retention checks show no obvious loss on general-mathematics problems in a single 16-sample pass, though the authors explicitly label these as retention evidence, not transfer."],"fun_headline_variants":["Post-trained LLM agent verifies SOS: 79% vs 45% with same tools","Verifier-rewarded training lifts LLM SOS proofs to 79% from 45%","Tool-using LLM after algebra training: 79% exact SOS certificates","SFT+GRPO on algebra tasks boosts LLM SOS verification to 79%","LLM agents coordinate symbolic tools: 79% checked SOS success"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The claim's load-bearing premise is that success on same-generator synthetic polynomials (1–3 variables, degrees 2/4/6, weights 1–10, m≤5) measures learned tool coordination rather than exploitation of generator regularities, especially that every input is SOS by construction and m=1 instances are perfect squares.","fun_headline_variants_meta":{"raw":{"variants":["Post-trained LLM agent verifies SOS: 79% vs 45% with same tools","Verifier-rewarded training lifts LLM SOS proofs to 79% from 45%","Tool-using LLM after algebra training: 79% exact SOS certificates","SFT+GRPO on algebra tasks boosts LLM SOS verification to 79%","LLM agents coordinate symbolic tools: 79% checked SOS success"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000578,"raw_usage":{"total_tokens":2609,"prompt_tokens":838,"completion_tokens":1771,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":1662}},"tokens_in":582,"tokens_out":1771,"duration_ms":10364,"temperature":1.0,"reasoning_tokens":1662,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T00:42:53.427590+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Stratify the reported 78.96% by the hidden number of squares m: if verified success is near 100% on m=1 and drops sharply for m=4–5, or if retraining on an out-of-generator sampler (more variables, degree 8, non-SOS distractors) makes the Full-minus-Base+Tools gap collapse, the coordination claim is an artifact of the generator.","supporting_citations":[],"review_version":1}