FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation

Qiufeng Wang; Ruoran Xu; Wending Gao; Xiaoqing Kang

FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper claims a closed-loop pipeline can generate multimodal analytic geometry problems at scale, producing over 7,000 verified problems with a median ground-truth measurement error of 0.70%.

desk verdict A genuinely useful pipeline and honest quality audit, but the 1%-tolerance benchmark is not supported by the paper's own ground-truth error statistics. read the letter →

arxiv 2607.12982 v3 pith:IMJQMJGB submitted 2026-07-14 cs.AI cs.MAcs.SC

Ruoran Xu , Wending Gao , Xiaoqing Kang , Qiufeng Wang This is my paper · ORCID

classification cs.AIcs.MAcs.SC

keywords analyticgeometrymultimodalLLMdatagenerationsigneddistancefieldconicsectionsclosed-loopverificationformallanguagegeometricbenchmark

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to prove that multimodal analytic geometry problems—questions about conic sections that require reading both an equation and a diagram—can be produced automatically, in volume, without human annotation. Its route is a formal intermediate language (CDL) that compiles each problem into a differentiable signed-distance-field scene, so the same representation both resolves geometric constraints and renders an exact diagram. A vision-based Measurer then reads the answer from the rendered image, while a Quality Verifier checks solvability, annotation completeness, and answer plausibility in a loop that retries failures. The result is AnalyticGeo7K, a dataset of 7,043 verified problems, and the paper reports that measured ground truth has a median relative error of 0.70% and that frontier multimodal models score up to 77.6% on it. If correct, this closes the data-scarcity gap that has kept analytic geometry out of multimodal math benchmarks.

What carries the argument

The load-bearing object is CDL, a declarative Condition Description Language in which every statement maps either to a visual element or to a differentiable constraint; it is compiled by a Signed Distance Field (SDF) engine into a scalar field whose zero level set is the rendered curve. This unification makes constraint solving and rendering one operation: underdetermined points are resolved by gradient descent, and because the diagram is generated from known geometry, the pixel-to-coordinate map is exact, which the Measurer exploits for visual ground-truth extraction. The closed loop is completed by a Quality Verifier that parses, completes, and cross-checks each stage and issues structured

What would settle it

Recompute exact symbolic answers for all 7,043 problems (or a much larger stratified sample) and diff them against the stored ground truth. If the fraction of stored answers with relative error above 1% is close to the observed 17.7% (the share beyond 5% in the 164-sample evaluation), then the 1%-tolerance benchmark scores are contaminated by ground-truth noise and need re-scoring or filtering.

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Extended reading notes

Core claim

The central claim is that conic-section problems can be generated end-to-end with no human annotation by decomposing the work into four LLM components joined by a formal language. The Generator drafts a problem with a clear split between text conditions and visual conditions; the Formalizer translates it into CDL; the SDF engine compiles CDL into diagrams with exact coordinate-to-pixel mappings; and the Measurer extracts the answer purely from the image, without seeing equations, using computer-vision tools. A Quality Verifier checks the output at three gates—symbolic solvability, CDL completeness, and cross-validation of measured against symbolic answers—and feeds structured errors back for

Load-bearing premise

The load-bearing premise is that the stored ground truth—obtained by pixel-level vision measurements on 512×512 rendered diagrams and checked by a verifier that tolerates up to 20% deviation—is accurate enough to score model answers at a 1% tolerance; the paper's own statistics show 17.7% of measured answers exceed 5% error.

Editorial extensions

If this is right

A 7K-item multimodal analytic-geometry benchmark can be built without human labor, and the same pipeline can be rerun to scale it further.
The explicit information partition—text conditions and diagram conditions are disjoint—means the dataset tests whether models truly use the image, not just the text.
The image-versus-text accuracy gap reported for the best model (about 41.6 percentage points) shows current multimodal models rely on diagrams substantially for conic-section problems.
Because the viewport metadata is known exactly, the same data can evaluate visual measurement tools independently of math reasoning.
Extending CDL with new primitives would let the framework cover locus problems and parametric equations without a redesign.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A stricter release could filter the 17.7% of ground-truth answers that deviate more than 5% from exact values by recomputing symbolic answers, making the 1%-tolerance benchmark scores cleaner.
The same CDL/SDF machinery could be extended to locus problems, parametric curves, and region-shaded areas once signed-region SDF composition is added—the paper flags region rendering as future work.
The dataset's exact viewport metadata makes it a natural testbed for separating visual measurement skill from algebraic reasoning; a model that solves text-only versions may not actually read diagrams.

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Signed reviews

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is the first pipeline I've seen that actually produces multimodal analytic-geometry problems at scale, and CDL plus SDF-as-solver-and-renderer is a real design win. But the benchmark claims are undermined by the paper's own reported ground-truth error: a median of 0.70% doesn't make a 1% accuracy threshold safe when the mean is 2.84% and 17.7% of the manual-eval sample sits above 5%. That's a load-bearing soft spot, not a cosmetic one.

What's new: CDL extends Assertional Logic with construction primitives (Focus, Intersection, TangentOfPoint, RightPart) so a formal annotation can drive rendering, not just parse equations. SDF as a unified constraint solver and renderer is clever, and the exact viewport mapping genuinely removes the axis-reading problem. The 164-problem manual symbolic evaluation is real external evidence; they don't hide the heavy tail.

Where it gets shaky: Gate 3's "independent" solver is a same-family LLM whose tolerance is PASS/WARN at 5–20%, and the design claims I_v∩I_t=∅ while simultaneously using a text-only solver to verify diagram-dependent answers. Either the solver sees the visual conditions (violating the partition) or it can't verify those problems and falls back to a range check. The paper doesn't say which. On top of that, the dataset and code aren't shipped, so the central artifact can't be checked.

The good news: these are fixable. Report the within-1% GT-error fraction and a misclassification analysis; state plainly what fraction of problems Gate 3 actually verified with a symbolic pass versus range-checked; release the data and code. Then the benchmark is defensible.

Who it's for: anyone building multimodal math data or evaluating MLLMs on analytic geometry. As it stands it deserves a serious referee — the framework is real engineering — but the 1% tolerance claims need to survive a referee who does the same arithmetic I just did.

Referee Report

3 major / 5 minor

Summary. The paper introduces FormalAnalyticGeo, a fully automatic pipeline that generates multimodal analytic-geometry problems. Natural-language problems are formalized in CDL, rendered through an SDF-based engine, and visually measured to produce ground-truth answers. A Quality Verifier applies three gates with feedback-driven retries, yielding a dataset of 7,043 problems with aligned text, diagram, CDL annotation, and measured answer. On 164 manually solved problems the authors report median ground-truth relative error 0.70% and 82.3% within 5%. They then evaluate eight MLLMs on the dataset at a 1% relative-error tolerance, report ablations showing each framework component contributes, and plan to release the framework and dataset.

Significance. If the framework and dataset are sound, this is a useful contribution: it addresses a genuine gap in multimodal analytic-geometry data, and the combination of a formal intermediate language (CDL), differentiable SDF rendering, and closed-loop verification is a plausible design. The 164-problem manual symbolic evaluation is a real strength, as are the ablations showing that the quality gates and CV tools matter. However, the two central validity claims — that the stored ground truth is accurate enough for a 1%-tolerance benchmark and that Gate 3 provides independent cross-validation — are not established by the reported evidence. The paper should be published only after these are resolved.

major comments (3)

[§4.3, Table 3, Appendix C.4] The 1% benchmark tolerance is not compatible with the reported ground-truth error distribution. On the 164 sampled problems, the paper reports median relative error 0.70%, mean 2.84%, and 82.3% within 5%; equivalently, at least 17.7% of sampled ground-truth answers are more than 5% from the exact symbolic solution. Since every such item also has error >1%, at least 17.7% of benchmark items can be misclassified at the 1% scoring threshold: a correct model answer can be marked wrong, or a compensating-error answer marked right. The paper never reports the within-1% rate, and the statement that the median GT error being below 1% 'remains reliable' is invalid — a median does not bound the per-item error tail. Further, Gate 3 explicitly retains answers whose error is 5–20% (WARN interval, Appendix C.4). The paper must report the full GT error CDF, the within-1% quantile, and either filter/cor
[§3.7, Appendix C.4, §3.4] Gate 3's cross-validation is not an independent check for the problems this pipeline is designed to generate. All four components and the Quality Verifier use Gemini-3.1-Pro-Preview. The Generator is explicitly required to place measurable numeric information in the diagram, not the text (I_v ∩ I_t = ∅, §3.4), so a text-only solver such as solve_text_only lacks the quantities needed to compute an exact reference answer for a well-formed diagram-dependent problem. The stored ground truth is the Measurer's own output, filtered by a verifier of the same model family, whose PASS band is 5% and whose WARN band is 5–20%. The manual 164-problem evaluation is a post-hoc accuracy study, not a pipeline gate. Consequently the 'verified' label overstates what Gate 3 guarantees. The authors should either use an independent symbolic solver (not an LLM of the same family) or clearly state the residual
[§4.3, Figure 4] The quality analysis is based on 164 manually solved problems spread over 20 curve-type × goal-type cells, i.e., roughly 8 per cell. The paper gives aggregate and per-category means but no per-category sample sizes, confidence intervals, or per-category error quantiles (e.g., the within-1% rate for parabola–area, the hardest cell). Given that the benchmark's scoring tolerance is 1%, the paper should report the within-1% rate per cell, not only the mean-based accuracy proxy 1−mean error, which can be misleading when errors are skewed. Please provide the full error distribution and uncertainty quantification for the 164-problem sample.

minor comments (5)

[Abstract] Minor typo: 'ground truth.Experiments' is missing a space. Also 'AnalyticGeo7k' and 'AnalyticGeo7K' are used inconsistently.
[Table 3] The table lists a model named 'Qwen-VL-Max', while reference [3] is the Qwen2.5-VL technical report. Please clarify the model naming and ensure the reference matches.
[Figure 4] The caption describes accuracy as 1−mean relative error. This conflates two different measures; it should be labeled as '1 − mean relative error' and discussed separately from within-5% accuracy.
[Appendix B.4] The rendering resolution is described as 'typically 512×512'. Since pixel quantization directly affects the measured answers and the 1% benchmark tolerance, the actual resolution and visibility threshold used for the released dataset should be stated exactly.
[§3.4] The notation I_v ∩ I_t = ∅ is intuitive but not formally defined. A short definition of what counts as text-condition versus visual-condition information would help readers assess the information-partition claim.

Circularity Check

1 steps flagged · score 6.0 of 10

Dataset-wide 'verified' status reduces to same-model text-only self-agreement because Gate 3 cross-checks against a solution that, by the paper's own information-partition design, cannot be computed from text alone.

self definitional [§3.4 (Information partition), §3.7 (Quality Gates), §4.1, Appendix C.1/C.4]
"let 𝐼𝑡 denote the conditions stated in the problem text and 𝐼𝑣 those that must be read from the diagram. The constraint 𝐼𝑣∩𝐼𝑡 =∅ ensures that solving the problem requires both modalities. ... The problemMUSTrequire reading the diagram to solve. ... The problem text says 'As shown in the figure' and the diagram shows the specific numeric configuration. ... Gate 3 (after visual measurement): checks the answer range for plausibility and cross-validates against a text-only symbolic solution. ... All four components (Generator, Formalizer, Measurer, and Quality Verifier) use Gemini-3.1-Pro-Preview."

By the paper's own design, the numeric visual configuration (I_v) is deliberately withheld from the text, so a text-only solver has no basis to compute the exact answer. Gate 3's 'independent analytical answer' is therefore either underdetermined or an ungrounded LLM completion. Comparing the Measurer's answer to that answer—using the same Gemini-3.1-Pro-Preview family for both—certifies only internal agreement, not agreement with the exact symbolic solution. The 'verified' label for all 7,043 problems is thus defined by the framework's own outputs rather than by an external ground truth.

full rationale

The paper does contain one independent anchor: §4.3 manually solves 164 sampled problems and reports median relative error 0.70% and 82.3% within 5%. That external computation prevents the whole derivation from being circular. However, the framework's central claim—that the 7,043 released problems are 'verified' with no human annotation—rests on Gate 3, whose text-only cross-check is impossible under the paper's own I_v∩I_t=∅ rule and whose judge is the same model family that produced the candidate answers. Separately, the benchmark-noise issue (at least 17.7% of sampled GTs deviate by more than 5%, and hence more than the 1% scoring tolerance, while Table 3 scores with a 1% threshold) is a serious validity concern, but it is a statistical flaw rather than a circularity. Score 6 reflects the one load-bearing self-referential verification step; the independent 164-item manual evaluation and the deterministic SDF/CDL pipeline keep the framework from being entirely equivalent to its inputs.

Assumptions & free parameters 5 free parameters · 7 assumptions · 3 invented entities

The main unpaid-for premises: (i) hyperbola SDF approximation and threshold-rendering precision suffice for ~1% measurement; (ii) non-convex SDF optimization converges for retained problems; (iii) Gate 3's text-only solver is independent despite the visual-information partition; (iv) LLM judges of the same model family as the generators are reliable verifiers. The quality claim has one external anchor: 164 manually computed symbolic answers, which is real but small relative to the 7,043-problem dataset.

free parameters (5)

Gate 3 verification tolerances = PASS ≤5%; WARN 5–20%; FAIL >20% relative error
Appendix C.4: hand-set constants that determine which measured answers enter the dataset; they shape the reported 82.3%-within-5% figure.
Benchmark scoring tolerance = 1% relative error
Table 3 caption: hand-set accuracy threshold applied to noisy ground truths; its reliability argument uses the median GT error (0.70%), which does not bound the tail.
Rendering resolution and visibility threshold = 512×512 grid; τ_vis = 1.5Δ
Appendix B.4: fixed pixel budget and thresholding band set the precision floor for all measurements (~pixel scale).
SDF optimization hyperparameters = λ=0.1; τ=0.2; LR 0.1→1e-6; iters 10,000; batch ≤2048
Appendix B.3/Table 9: hand-set engineering constants for the constraint-satisfaction optimizer; no sensitivity analysis reported.
Retry budget = up to 2 retries (3 attempts) per gate
§3.7: bounds the discard rate (780/7,823 candidates) and therefore which problems are retained as 'verified'.

assumptions (7)

domain assumption Hyperbola SDF may be approximated by |f(x)|/‖∇f(x)‖ (first-order implicit approximation) with 'sub-pixel accuracy near the curve'
Appendix B.1 and §3.3: no closed-form closest-point exists for hyperbolas, so the approximation is used for rendering and constraints. If inaccurate away from the boundary, rendered hyperbola geometry and subsequent measurements inherit the error.
domain assumption AdamW/cosine-annealing optimization over B random initializations finds configurations satisfying all CDL constraints
§3.3 Eq. (2): under-determined layouts are resolved via non-convex gradient descent; there is no feasibility guarantee, and problems that fail to converge are silently discarded rather than reported.
domain assumption Threshold rendering at τ_vis = 1.5Δ on a 512×512 grid conveys the visual conditions I_v accurately enough for vision-based measurement
Appendix B.4/§3.3: lines and curves render as ~3-pixel-wide bands with exact coordinate-to-pixel mapping, but positions remain pixel-quantized, bounding achievable measurement precision at roughly pixel scale.
ad hoc to paper A text-only symbolic solver (solve_text_only) can produce an independent analytical answer for problems designed to be diagram-dependent (I_v ∩ I_t = ∅)
Gate 3 protocol (Appendix C.4) invokes solve_text_only as independent verification, but §3.4 requires problems to be unsolvable from text alone; the premise is unstated and internally inconsistent for the visual-conditions lane.
ad hoc to paper The 1% evaluation threshold is reliable when the median GT relative error is 0.70%
§4.3: 'the median GT error (0.70%) is well below the threshold' — a median below threshold does not bound the tail; 17.7% of answers exceed 5% error, so per-problem classification at the 1% margin is substantially contaminated.
standard math Standard conic-section geometry and root-finding background
Appendix A/B: focus-directrix-asymptote formulas, discriminant analysis of general quadrics, Cardano/trigonometric cubic solvers, Shoelace formula — assumed textbook background.
domain assumption The LLM components (Gemini-3.1-Pro-Preview) follow their ReAct loops and tool-calling protocols faithfully enough that the quality gates are meaningful
§3.4–3.7: generation, formalization, measurement, and verification are all LLM-driven; pipeline reliability inherits from instruction-following of a single model family, with no independent audit of the gates except the 164-problem manual check.

invented entities (3)

CDL (Condition Description Language) independent evidence
purpose: Formal intermediate representation that bridges free-form problem text and SDF rendering; 35 operators across derived objects, predicates, metric constraints
Appendix A gives the full operator catalog and worked programs, so the language is independently implementable and its adequacy is falsifiable.
SDF-based rendering engine (constraint solver + renderer) independent evidence
purpose: Compiles CDL into exact diagrams with known coordinate-to-pixel mapping; resolves under-determined layouts by gradient descent
§3.3 and Appendix B provide SDF expressions, loss functions, and hyperparameters; independently re-implementable.
AnalyticGeo7K dataset
purpose: 7,043 verified multimodal analytic-geometry problems with text, diagram, CDL, and measured ground truth; also used to benchmark eight MLLMs
Release promised but not shipped; Table 3 scores and 0.70% median error cannot be independently checked until the dataset is public.

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Cite this review

Pith. "Pith review of FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation." pith.science (2026). https://pith.science/paper/IMJQMJGB

@misc{pith2026260712982,
  author       = {Pith},
  title        = {Pith review of: FormalAnalyticGeo: A Neural-Symbolic Based Framework for Multimodal Analytic Geometry Problem Generation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IMJQMJGB}},
  note         = {Machine review of arXiv:2607.12982}
}

read the original abstract

Math reasoning has achieved significant progress with the rapid advancement of Multimodal Large Language Models (MLLMs), however analytic geometry remains largely underexplored, primarily due to the scarcity of annotated samples. Existing diagram generation approaches struggle with analytic geometry: template methods cannot handle constraint-driven layouts, and generative models lack the geometric precision to render annotated conic curves correctly. We present FormalAnalyticGeo, a scalable framework for fully automatic generation of multimodal analytic geometry problems. Leveraging the rigor of formal languages, we design the framework around CDL (Condition Description Language), a formal intermediate representation that bridges free-form problem text with precise diagram rendering via a Signed Distance Field (SDF) engine. The framework employs four specialized LLM components in sequence: a Generator that produces diverse analytic geometry problems, a Formalizer that converts each problem into CDL for SDF-based rendering, a Measurer that extracts ground-truth answers through vision-based measurement on the rendered diagrams, and a Quality Verifier that checks outputs at three stages. Structured feedback from the Quality Verifier drives automatic retry, forming a closed loop that eliminates any need for human annotation. Applying FormalAnalyticGeo at scale yields AnalyticGeo7K, a dataset of over 7K verified multimodal problems, each with aligned text, diagram, formal annotation, and ground truth.Experiments show that the generated problems achieve a median ground-truth relative error of 0.70\%, with 82.3\% of answers falling within 5\% of the exact symbolic solution. Our framework and dataset will be publicly released.