{"id":"19254f42-970a-4136-9216-f92d71cffdd9","arxiv_id":"2608.13129","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"LLM numeracy failures can be usefully organized as separate representational and procedural grounding problems, with procedural arithmetic improving more than number understanding when models are asked to reason.","lead":"This survey proposes a framework that splits LLM numeracy into two parts: recognizing what numbers mean and carrying out arithmetic correctly. It then uses that split to organize benchmarks, explain failure modes, and recommend fixes for current frontier models.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5.2's RG battery is internally inconsistent with Table 1: 'digit counting' and 'number length' are labeled RG although the taxonomy assigns digit manipulation to PG, so the §5.3 RG-PG gap may be a labeling artifact rather than a measured dissociation.","rationale":"The reader identified the unvalidated RG/PG battery as the weakest assumption, and I agree. The internal inconsistency between Section 5.2 and Table 1 makes the problem concrete rather than merely methodological: the RG battery appears to contain at least two task types the taxonomy itself assigns to PG. Because the empirical sections are framed as testing NGF's predictions, this labeling issue cuts directly to the strongest claim. I do not see fraud or even carelessness beyond an unvalidated experimental design; the survey and framework have independent value as an organizing synthesis, and the Pretrained-Model Constraint is a fair synthesis of prior work. But the paper's own empirical contribution cannot be treated as measurement until the battery split is either validated or corrected. The CONDITIONAL verdict remains appropriate; it should be conditioned on a reanalysis or explicit reframing of Sections 5.3-5.4 as exploratory rather than confirmatory.","tokens_in":15617,"tokens_out":6600,"duration_ms":60699,"concrete_test":"Recompute the Section 5.3 RG-PG gap after moving digit counting and number length from the RG battery to the PG battery, as required by Table 1's Digit Manipulation row, and report the per-model gaps with item-level counts and confidence intervals. If the positive gap does not survive this reclassification across all three model families, the dissociation is a labeling artifact rather than a psychological or architectural fact. As a secondary check, release item-level responses so a confirmatory factor analysis can test whether a one-factor difficulty model fits as well as the two-factor RG/PG model.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim (Section 7.1) is that numerical grounding is architecturally distinct from mathematical reasoning, and its main empirical support is the Section 5.3 dissociation result: every model configuration shows higher RG than PG, with an average RG-PG gap of approximately 0.19 in domain and 0.27 out of domain. That result depends entirely on the RG/PG battery split in Section 5.2. The split is not validated, and it is internally inconsistent: Section 5.2 assigns digit counting and number length to the RG battery, while Table 1 classifies the Structural row, whose example is a digit-access question ('What is the 3rd digit of 3.1415?'), as PG Digit Manipulation. Counting digits and judging number length are digit-manipulation tasks under that taxonomy. If those items belong in PG, the RG battery is inflated with PG-like tasks and the reported RG-PG gap is partly a labeling artifact. More broadly, no item-level validation, inter-rater agreement, or factor analysis is provided to show the two batteries measure distinct constructs rather than task difficulty, output length, or pretraining token frequency; Section 4.4 itself identifies operand frequency as a strong predictor. The dissociation is therefore not established, and the claims in Sections 5.4 and 7.1 that reasoning preferentially compensates PG and that RG/PG are architecturally distinct lose their empirical foundation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes the Numerical Grounding Framework (NGF), which splits LLM numeracy into Representational Grounding (RG) and Procedural Grounding (PG), and uses this framework to organize a survey of diagnostic benchmarks, failure modes, structural root causes, and mitigation strategies. It also reports a coordinated evaluation across Number Cookbook, NumericBench, and GSM-Symbolic, claiming a robust RG–PG dissociation, an asymmetric benefit of extended reasoning for PG, tokenizer-specific RG blind spots, and partial primitive–contextual transfer. The central conclusion is that numerical grounding is architecturally distinct from mathematical reasoning, with tokenization as the primary RG bottleneck and the Pretrained-Model Constraint shaping available interventions for deployed systems.","tokens_in":15795,"tokens_out":4645,"duration_ms":39275,"significance":"If the dissociation were empirically established, NGF would be a useful organizing device: it maps failure modes to structural causes, separates scaling gains from architectural fixes, and makes concrete predictions about intervention efficacy. The paper's strengths are its systematic survey of the 2024–2025 diagnostic benchmark generation, the failure-mode taxonomy, the root-cause analysis linking BPE tokenization, positional encodings, embedding geometry, and pretraining data to RG/PG, and the explicit Pretrained-Model Constraint. These contributions are valuable independent of the empirical section. However, the current empirical validation falls short of the claims made in Sections 5 and 7, so the central theoretical claim is not yet established to the standard the paper asserts.","major_comments":[{"comment":"The RG battery in Section 5.2 includes digit counting and number length, yet Table 1 classifies digit manipulation as PG with the example 'What is the 3rd digit of 3.1415?' Counting digits and judging number length are digit-manipulation tasks under that taxonomy, so the RG battery is inflated with PG-like items. The reported RG–PG gap of about 0.19 in-domain and 0.27 out-of-domain in Section 5.3 may therefore be partly a labeling artifact rather than a measured dissociation. No item-level validation, inter-rater agreement, or factor analysis is provided to show that the two batteries measure distinct constructs rather than task difficulty, output length, or pretraining token frequency—a confound the paper itself identifies in Section 4.4. This inconsistency must be resolved before the dissociation claim can be accepted.","section":"Section 5.2, Table 1"},{"comment":"The 'robust' dissociation result is not supported by the reported statistics. No sample sizes, item counts, error bars, confidence intervals, or significance tests are given. Table 2's layout is confusing: the Gemini rows omit the digit-match columns without a stated N/A convention, and the Claude row can be misread as having '4.6' as a value rather than a model version. The 0.19/0.27 RG–PG gap is presented as an average without variance or an inferential test, so the result cannot be distinguished from noise or from a small number of outlier items.","section":"Section 5.3, Table 2"},{"comment":"The claim that extended reasoning improves PG more than RG is based on a single model family (Gemini 3, MINIMAL versus HIGH) and a single configuration at temperature 0. The 21.3× token-cost comparison and the in-domain regression rest on one pair of runs with no repeated sampling or confidence intervals. To support the framework's prediction, the asymmetry should be replicated across at least one additional model family and with multiple seeds or prompt variants.","section":"Section 5.4, Table 3"},{"comment":"The tokenizer-specific RG blind-spot claim is asserted from cross-model accuracy differences, but the paper reports no tokenizer analysis (for example, actual BPE segmentation of the test numerals), no controlled comparison of models with identical training data and different tokenizers, and no release of prompts or exact API versions. Differences between GPT-5.4, Claude Opus 4.6, and Gemini 3 could be due to pretraining data, alignment, or instruction-following rather than tokenization. The claim therefore overreaches the evidence presented.","section":"Section 5.5"},{"comment":"The framework's predictions partially follow from its definitions: RG is defined as mapping numerals to value and PG as executing procedures, and tasks are assigned to batteries accordingly. Finding RG > PG may then reflect task difficulty or exposure rather than a genuine cognitive dissociation. Section 4.4 itself identifies operand frequency as a strong predictor, and no attempt is made to regress out difficulty, output length, or token frequency. A dissociation claim needs item-level controls or at least a demonstration that the RG and PG items are matched on difficulty and frequency.","section":"Sections 3.6 and 5.1"}],"minor_comments":[{"comment":"Format the table so that model names and values are unambiguous, and state explicitly that digit-match columns are not available for Gemini 3 configurations.","section":"Table 2"},{"comment":"The NumericBench reference (Li et al., 2025) contains the placeholder text 'Verify authors and arXiv ID before final submission' and must be completed.","section":"References"},{"comment":"Table 6 lists CoT/reasoning as costing approximately 18× tokens, while Table 3 reports a measured 21.3× token cost for Gemini HIGH; these numbers should be reconciled.","section":"Section 6.4.1, Table 6"},{"comment":"The experimental setup does not specify how many items are in each RG and PG battery, how in-domain and out-of-domain splits were constructed, or the exact prompt templates; an appendix with these details is needed for reproducibility.","section":"Section 5.2"},{"comment":"The y-axis label 'maximum digit length at which a model maintains at least 90% exact-match accuracy' is not defined; the interpolation method used to obtain the threshold should be stated.","section":"Figure 8"},{"comment":"There are several typos and spacing errors, for example 'theNumerical' in Section 3.1, 'awiderangeof' in Section 2.1, and 'improvesome' in Section 2.3; a careful proofread is needed.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The empirical section reads as an early preprint that has not undergone rigorous methodological review. The model names and some benchmark citations raise verification concerns, and the taxonomy inconsistency plus missing statistics are load-bearing for the central claim. The issues are fixable within the manuscript's scope: the authors should either supply a much stronger empirical case with item-level validation and replicated reasoning-budget comparisons, or explicitly reframe the paper as a survey with illustrative pilot data. I would not accept the paper in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's real contribution is organizational, not empirical. The RG/PG split is a clean way to sort a messy literature, even if it borrows heavily from Harnad, Dehaene, and existing representational-vs-procedural distinctions in the LLM arithmetic literature. What is new is applying that split systematically across the 2024-2025 diagnostic benchmarks, failure modes, root causes, and mitigations. The single most useful thing in the paper is the Pretrained-Model Constraint: noting that tokenizer and embedding fixes help scratch-trained models but are largely unavailable to people using pretrained systems. That is a practical synthesis worth having.\n\nThe survey side is solid. The benchmark landscape section is organized and current, the mapping of BPE tokenization to RG failures is coherent, and the mitigation table is genuinely useful for practitioners.\n\nThe empirical section, though, cannot be accepted as evidence. There are no sample sizes, no error bars, no significance tests, no released prompts, and no API model versions. One reasoning-budget comparison comes from a single model family. Table 2 is malformed, and the NumericBench citation is an explicit placeholder. That alone would require major revision before the measurements could be relied on.\n\nThe deeper problem is the one the stress-test note caught. Table 1 classifies digit access and digit manipulation as PG (\"What is the 3rd digit of 3.1415?\"), but Section 5.2 puts digit counting and number length in the RG battery. Counting digits and judging number length are structurally the same kind of task as digit access. If those items belong in PG, then the RG battery is inflated with procedural items and the reported RG-PG gap of 0.19/0.27 is partly a labeling artifact. The paper provides no item-level validation, no inter-rater agreement, and no factor analysis to show the two batteries measure distinct constructs rather than task difficulty or token frequency—which Section 4.4 itself identifies as a strong predictor. The dissociation claim in Section 5.3 and the architectural-distinctness claim in Section 7.1 lose their empirical footing without that validation.\n\nThe \"fundamental limitations\" framing also overclaims. The paper demonstrates observed fragility, not a proven lower bound. That is a rhetorical mismatch, not a fatal one, but it should be fixed.\n\nWho is this for? Practitioners and LLM reliability researchers who want a map of the numeracy literature and a practical sense of which interventions work on pretrained models. They will get value from the survey even if the empirical centerpiece fails. It deserves a serious referee, but the referee should demand data, corrected tables, and a reanalysis of the dissociation with a defensible battery split before publication.","headline":"Useful survey and a genuinely practical constraint, but the RG–PG dissociation result is not established: the task labeling does part of the work, and the empirical section lacks the artifacts to back its claims.","tokens_in":16403,"tokens_out":1662,"would_cite":false,"duration_ms":17408,"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":"This survey claims that LLM numeracy failures are grounding failures, not reasoning failures: the mapping from numeral to value fails separately from arithmetic execution, so the two need different fixes.","keywords":["numeracy","large language models","representational grounding","procedural grounding","tokenization","length generalization","symbol grounding","arithmetic reasoning"],"falsifier":"Re-score the Number Cookbook items after matching RG and PG tasks for difficulty, length, and tokenization exposure, for example by pairing magnitude comparison with digit manipulation on identical numbers; if the RG-PG gap shrinks to zero once difficulty is matched, the claimed dissociation collapses, and finding a model with high PG but low RG on the same tokenizer would violate the framework's predicted ordering.","tokens_in":15296,"feed_emoji":"🔢","tokens_out":14414,"duration_ms":111043,"temperature":0.7,"pith_summary":"This survey proposes that large language models' high mathematical reasoning scores hide a separate, weaker ability: basic numeracy, the reading and computing of numbers themselves. The paper introduces the Numerical Grounding Framework (NGF), which splits numeracy into Representational Grounding (RG)—mapping numeral forms like $9.11$ to their value and magnitude—and Procedural Grounding (PG)—executing arithmetic correctly. Its coordinated evaluation of three frontier model families across Number Cookbook, NumericBench, and GSM-Symbolic finds that every configuration scores higher on RG than PG, with an average gap of about 0.19 in-domain and 0.27 out-of-domain. If this dissociation is right, then reasoning strategies such as Chain-of-Thought mostly patch PG, tokenizer changes shape RG, and neither alone fixes the other; the paper argues the durable fixes must be built at pretraining time.","feed_headline":"Why LLMs ace math but flub 9.11 vs 9.9","feed_subtitle":"A survey frames numeracy as two separate skills—representation and procedure—failing in different ways.","key_machinery":"The central object is the Numerical Grounding Framework (NGF), a two-part construct that defines numerical competence as the conjunction of Representational Grounding (RG)—mapping numeral surface forms to value, magnitude, and format equivalence—and Procedural Grounding (PG)—executing arithmetic procedures consistently with their mathematical definitions. NGF does the argumentative work by assigning every failure mode, root cause, and mitigation to one or both dimensions: BPE tokenization and embedding discontinuity damage RG; positional encodings and training-data sparsity damage PG; fragility and tokenization artifacts are RG failures, while length-generalization and algorithmic-asymmetry failures are PG failures. The framework is operationalized by grouping Number Cookbook's atomic tasks into RG and PG batteries and testing them alongside NumericBench and GSM-Symbolic across three model families.","core_discovery":"On the paper's own terms, the central discovery is that numerical grounding is architecturally distinct from mathematical reasoning: a model can reason well and still fail to anchor numerals to values. Numeracy decomposes into Representational Grounding (RG), the faithful mapping of a numeral's surface form to ordering, magnitude, and format equivalence, and Procedural Grounding (PG), the faithful execution of arithmetic procedures. The empirical claim is a dissociation: every evaluated model configuration shows higher RG than PG, by about 0.19 in-domain and 0.27 out-of-domain, and the gap widens under length extension. Extended reasoning improves out-of-domain PG by roughly three times its improvement of RG, while tokenizer choice creates model-specific RG blind spots (for instance, one model keeps float comparison accurate to longer lengths, another leads on digit counting). The paper also finds that atomic primitives only partially predict contextual numeracy, and it states the Pretrained-Model Constraint: the most structural fixes, such as digit-aware tokenization or Abacus Embeddings, must be made before pretraining, leaving pretrained-model users with fine-tuning, process rewards, and external tools.","pith_inferences":["The manuscript's own reference list marks the NumericBench citation as needing verification; because NumericBench anchors the contextual-transfer half of the dissociation evidence, readers should treat that part of the empirical picture as provisional until the benchmark source is confirmed.","A testable extension the paper does not run: hold tokenization fixed and swap only task labels between RG and PG batteries; if the dissociation survives matched difficulty, the construct split is real rather than a difficulty artifact.","If NGF is right, the same two-channel decomposition should apply to other non-monotonic symbol systems, such as dates, units, chemical formulas, or hexadecimal numbers, where surface form does not carry semantics monotonically and RG failures should show the same tokenizer sensitivity.","The $9.11 > 9.9$ error should disappear under any input representation whose embedding distance is monotone in magnitude, implying that post-hoc embedding alignment might partially rescue pretrained models even without full retraining."],"forward_implications":["Benchmark accuracy on GSM8K, MATH, and similar reasoning sets can remain high while elementary numeracy fails, so deployment decisions should be based on grounding tests, not reasoning benchmarks.","Chain-of-Thought and reasoning-model training will continue to close PG gaps faster than RG gaps, especially out-of-domain, and cannot repair tokenizer-induced errors that happen before reasoning begins.","Different tokenizers will keep producing qualitatively different numeracy strengths, so a single global numeracy score is misleading and evaluations should report per-task RG profiles.","The Pretrained-Model Constraint means practitioners who cannot retrain should rely on supervised fine-tuning on diverse numerical examples, process-reward training, self-consistency, and tool use for high-reliability arithmetic.","Tool use is the near-term route to reliable PG because it externalizes computation, but it does not remove RG requirements: the model must still extract quantities and formulate correct equations."],"supporting_citations":[{"why":"Provides the symbol grounding problem that motivates the claim that numeral tokens must be linked to value to be meaningful.","marker":"Harnad (1990)"},{"why":"Supplies the dual-system cognitive account of number sense that NGF maps onto RG and PG.","marker":"Dehaene (2011)"},{"why":"Documents tokenization sensitivity and length-generalization failures in transformer arithmetic, the BPE bottleneck evidence for RG damage.","marker":"Nogueira et al. (2021)"},{"why":"Supplies xVal and PCA evidence that number embeddings are discontinuous in magnitude, supporting the RG embedding-geometry bottleneck.","marker":"Golkar et al. (2023)"},{"why":"Provides Number Cookbook's 44 task-representation combinations, which the paper groups into RG and PG batteries for the dissociation test.","marker":"Wang et al. (2024)"},{"why":"Provides NumericBench's in-context numerical tasks, used to test whether primitive numeracy transfers to contextual reasoning.","marker":"Li et al. (2025)"},{"why":"Supplies GSM-Symbolic's numeric substitution and NoOp distractors, the fragility evidence the paper classifies as an RG failure.","marker":"Mirzadeh et al. (2024)"},{"why":"Introduces LEFT and Abacus Embeddings, the scratch-training-only PG interventions that ground the Pretrained-Model Constraint.","marker":"McLeish et al. (2024)"},{"why":"Documents Llama 3's fixed three-digit number tokenization, the primary example that RG-beneficial tokenization must be set at pretraining.","marker":"Dubey et al. (2024)"},{"why":"Introduces Chain-of-Thought prompting, the reasoning intervention whose preferentially PG improvement the paper measures.","marker":"Wei et al. (2022b)"}],"fun_headline_variants":["Numeracy gap: LLMs can reason but can't ground numbers","LLMs: Math reasoning strong, number grounding weak","Representation vs procedure: core of LLM numeracy gap","LLMs' numeracy: representation beats procedure","Why LLMs are better at representing numbers than calculating"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument's load-bearing assumption is that the task batteries labeled RG and PG really measure two independent constructs rather than two levels of difficulty: if representation tasks are simply easier or more frequent in training than procedure tasks, the observed RG-PG gap would be a relabeled easy-versus-hard effect, not a dissociation.","fun_headline_variants_meta":{"raw":{"variants":["Numeracy gap: LLMs can reason but can't ground numbers","LLMs: Math reasoning strong, number grounding weak","Representation vs procedure: core of LLM numeracy gap","LLMs' numeracy: representation beats procedure","Why LLMs are better at representing numbers than calculating"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001045,"raw_usage":{"total_tokens":4419,"prompt_tokens":1000,"completion_tokens":3419,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":3339}},"tokens_in":616,"tokens_out":3419,"duration_ms":22248,"temperature":1.0,"reasoning_tokens":3339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:12:02.987648+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-score the Number Cookbook items after matching RG and PG tasks for difficulty, length, and tokenization exposure, for example by pairing magnitude comparison with digit manipulation on identical numbers; if the RG-PG gap shrinks to zero once difficulty is matched, the claimed dissociation collapses, and finding a model with high PG but low RG on the same tokenizer would violate the framework's predicted ordering.","supporting_citations":[{"cited_title":"xVal: A continuous number encoding for large language models","cited_arxiv_id":null,"evidence_quote":"Supplies xVal and PCA evidence that number embeddings are discontinuous in magnitude, supporting the RG embedding-geometry bottleneck."}],"review_version":1}