NUMINA: Precision in Multimodal AI Research

• NUMINA is a multi-context AI label defining artifacts that enable precise numerical alignment in video, formal theorem proving, and 3D scene reasoning.
• The text-to-video framework uses an identify-then-guide pipeline that improves CountAcc from 42.3% to 59.1% on larger models, demonstrating robust numerical control.
• NuminaMath and related infrastructures facilitate automated theorem proving with over 750K validated formal pairs, while the 3D benchmark highlights TA@5 scores below 3% on challenging inference tasks.

NUMINA is a recurrent designation in recent arXiv literature for several technically distinct research artifacts spanning controllable video generation, formal mathematics, automated theorem proving, and 3D multimodal reasoning. In one line of work, NUMINA is a training-free identify-then-guide framework for aligning textual numerals with object multiplicities in text-to-video diffusion models; in another, Numina denotes a broader mathematical-AI ecosystem centered on NuminaMath, formalization datasets, Lean tooling, and agentic theorem proving; in a third, NUMINA is a ScanNet-based benchmark for multi-dimensional intelligence and numerical reasoning in indoor 3D scenes (Sun et al., 9 Apr 2026, Liu et al., 20 Jan 2026, Zeng et al., 20 Sep 2025).

1. Terminological scope and disambiguation

The name appears with at least three clearly separated meanings. A common source of confusion is that these meanings share a label while differing in modality, objective, and evaluation protocol. In the text-to-video literature, NUMINA is a numerical-alignment controller for diffusion sampling. In formal mathematics, Numina refers to datasets, infrastructure, and agents used for autoformalization and theorem proving. In 3D multimodal reasoning, NUMINA is a benchmark rather than a generative model (Sun et al., 9 Apr 2026, Guo et al., 8 Oct 2025, Zeng et al., 20 Sep 2025).

Context	Artifact	Brief characterization
Text-to-video generation	NUMINA	Training-free identify-then-guide framework for count-accurate video synthesis
Mathematical data and proving	Numina / NuminaMath / Numina-ATF	Competition-math corpus, synthetic Lean 4 formalizations, and downstream prover infrastructure
Formal geometry and agents	LeanGeo / Numina-Lean-Agent	Lean 4 geometry system, benchmark, and MCP-based agentic prover
3D multimodal evaluation	NUMINA	ScanNet-based benchmark for quantity, distance, and volume reasoning

Capitalization also tracks usage. The all-caps form is used in the video-generation framework and the 3D benchmark, whereas the capitalized form “Numina” is used for the mathematical datasets, Project Numina infrastructure, and theorem-proving agents. This suggests a family resemblance at the naming level, but not a single unified system.

2. NUMINA in text-to-video diffusion

In "When Numbers Speak: Aligning Textual Numerals and Visual Instances in Text-to-Video Diffusion Models" (Sun et al., 9 Apr 2026), NUMINA is a training-free framework for numerical alignment in text-to-video diffusion models. The target failure mode is that prompts such as “three red cars driving on a highway” are rendered with the wrong count, with under-counting, over-counting, and temporal drift across frames. The method is explicitly formulated as an identify-then-guide pipeline: it runs an early partial denoising pass, derives a countable latent layout from selected self- and cross-attention heads, conservatively refines that layout to match prompt numerals, and then restarts generation from the same noise while modulating cross-attention.

The identify stage exploits the observation that numeral tokens tend to have diffuse, low-contrast cross-attention, whereas a small subset of attention heads is instance-separable. For self-attention head selection, the method scores a 2D projection $\mathbf{SA}^h$ using foreground-background separation, structural richness, and edge clarity,

$S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$

and selects the best head as an instance-discriminative scaffold. For each noun token $T$, cross-attention head selection uses the peak activation

$C_T^h = \max_{x,y} \mathbf{CA}^h_T(x,y),$

to obtain a noun-specific semantic focus map. Self-attention regions $\{\mathbf{r}_i\}$ and a thresholded focus mask $\mathbf{F}$ are fused by the overlap score

$$S_o(\mathbf{r}_i, \mathbf{F}) = \frac{|\mathbf{r}_i \cap \mathbf{F}|}{|\mathbf{r}_i|},$$

yielding a layout map $\mathbf{M}_T$ whose connected foreground regions define the model’s implicit count.

Layout refinement is deliberately conservative. If the inferred count $m_{T,f}$ exceeds the requested count $k_T$, the smallest region is removed. If $S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$0, a missing instance is inserted using either the smallest existing region as a template or, when no reference exists, a circular geometric prior. Candidate placements are scored by overlap, centering, and temporal consistency:

$S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$1

The guide stage then perturbs pre-softmax cross-attention scores through a bias matrix $S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$2 in

$S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$3

using large negative bias for removed regions and positive or reference-copied bias for added ones, with time-dependent strength $S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$4.

Evaluation is performed on CountBench, a 210-prompt benchmark covering counts from 1 to 8 and 1–3 object categories per prompt. On Wan2.1-1.3B, NUMINA improves CountAcc / TC / CLIP from 42.3 / 81.2 / 33.9 to 49.7 / 83.4 / 35.6; on Wan2.2-5B, from 47.8 / 85.0 / 34.3 to 52.7 / 85.0 / 34.7; and on Wan2.1-14B, from 53.6 / 83.3 / 34.2 to 59.1 / 84.0 / 34.4 (Sun et al., 9 Apr 2026). Supplementary experiments report a CountAcc gain from 40.2% to 44.4% on CogVideoX-5B. The paper also reports that attention-derived layout construction outperforms detector-derived layout construction, that top-1 head selection is superior to head averaging or random head choice, and that temporal-consistency terms in refinement are especially important. The stated limitations include failure on very large counts, complex relational constraints, and cases where attention localizes only salient object parts, such as bird heads rather than full bodies.

3. Numina as a mathematical data substrate

In formal mathematics, Numina primarily denotes a large source corpus of competition-style problems and the datasets derived from it. "Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving" describes Numina as including 860K high-quality question and answer pairs sourced from MATH, GSM8K, AMC, AIME, the AoPS Forum, Chinese K-12 Exams, World Olympiads, and synthetic data (Lin et al., 11 Feb 2025). "Autoformalizer with Tool Feedback" describes NuminaMath-1.5 as a large collection of approximately 900k competition-style math problems and states that it had mostly been used as informal math training data prior to the introduction of a large formalization layer (Guo et al., 8 Oct 2025). The difference in counts reflects different descriptions and versions in the supplied works rather than a contradiction resolvable from a single source.

Numina-ATF is that formalization layer. The ATF paper states that ATF-32B was run on the complete Numina-V1.5 dataset, yielding 752,553 informal-formal pairs validated for both syntax and consistency. The formal side consists of Lean 4 theorem statements, typically beginning with import Mathlib, import Aesop, and a theorem header ending in := by sorry; the released dataset keeps only pairs that pass Lean 4 syntax checking and a multi-LLMs-as-judge consistency pipeline (Guo et al., 8 Oct 2025). The syntax system groups statements by import set, runs Lean 4.15 in batches of 20 with a 300 s timeout, and reports an average per-statement runtime reduction from 6.215 s under individual execution to 0.808 s under grouped execution on 100k statements. For semantic consistency, an ensemble of QWQ-32B and Qwen3-32B is calibrated on a 4,000-statement benchmark and achieves Precision 0.8374, TPR 0.5967, TNR 0.9421, and FPR 0.0579, with both models required to agree on consistency for a statement to be retained.

Goedel-Prover uses Numina as the main informal source for large-scale theorem-proving data construction. Two formalizers are trained to map natural-language Numina problems into Lean 4 statements; after Compiling Correctness and Faithfulness and Completeness filtering, the paper reports that out of 928K informal statements, 760K have two valid formalized statements generated by Formalizer A and B, while 123K contain only one valid formalized statement (Lin et al., 11 Feb 2025). This yields 1.64M formal statements from Numina and private AoPS data, which form the core of Goedel-Pset-v1. Iterative proof discovery on this formalized pool produces a solved subset exceeding 800K statements in the abstract, while the detailed training table reports 928.2K Numina-origin solved statements by the final iteration. The same paper attributes Goedel-Prover-SFT’s 57.6% Pass@32 on miniF2F and 7 solved PutnamBench problems at Pass@512 to this large formalization-and-expert-iteration pipeline.

Numina-Math also appears as an RLVR substrate outside explicit theorem proving. "Let it Calm: Exploratory Annealed Decoding for Verifiable Reinforcement Learning" uses Numina-Math, described there as 860k math prompts, as the dataset for RLVR experiments on Qwen and Llama families (Yang et al., 6 Oct 2025). This suggests that Numina is functioning not only as a source for formalization but also as a standard reasoning corpus for verifiable-reward training.

4. Project Numina in formal reasoning infrastructure

Project Numina extends beyond datasets into formal systems and agents. "LeanGeo: Formalizing Competitional Geometry problems in Lean" presents a unified Lean 4 geometry system hosted under project-numina, with 52 new geometric definitions via abbrev, a point-centric SMT representation backed by LeanSMT and CVC5, and a theorem library described as containing 260 theorems (Song et al., 20 Aug 2025). LeanGeo-Bench comprises 122 problems drawn from UG, LB, SP, HSC, OP, and IMO sources, including 20 high-school competition problems from NuminaMath and 43 IMO geometry problems since 2000. In pass@4 evaluation, Gemini 2.5 Pro reaches 27.05%, o4-mini 22.13%, Grok 4 24.59%, Kimi K2 9.84%, Claude 4 10.66%, and Qwen3-235B-A22B 5.74%; no model solves any of the 62 olympiad-level problems in OP plus IMO. An RL pipeline using Kimina-Prover infrastructure improves LeanGeo-Bench pass@1 from 2.52% to 10.92%.

"Numina-Lean-Agent: An Open and General Agentic Reasoning System for Formal Mathematics" presents a different component of the same ecosystem: an MCP-based agent that uses Claude Code with Numina-Lean-MCP, including Lean-LSP-MCP, LeanDex, Informal Prover, and Discussion Partner (Liu et al., 20 Jan 2026). Using Claude Opus 4.5 as the base model, the system solves all problems in Putnam 2025, 12 / 12, under sequential tool use and without internet search. The paper emphasizes a paradigm shift away from training a dedicated formal prover: the central claim is that a general coding agent, augmented with Lean interaction and retrieval tools, can serve as the formal-mathematics reasoner.

The same work also demonstrates long-horizon collaboration on research-level mathematics. In a case study on the effective Brascamp–Lieb inequalities, the system participates in producing more than 8,000 lines of Lean code and autonomously introduces approximately 70 new definitions, lemmas, and theorems over less than two weeks of intermittent work (Liu et al., 20 Jan 2026). The paper describes this through recursive blueprint generation and refinement rather than a fixed theorem-proving loop. A notable technical point is that the agent sometimes corrects statements as part of formalization; the reported upperBound_empty_case lemma required an added hypothesis hβ_empty : β = 0, discovered during the proof process.

Taken together, LeanGeo and Numina-Lean-Agent define two complementary layers of Project Numina’s formal stack: one provides domain-specific formal infrastructure and benchmarks, and the other provides a tool-orchestrating agentic interface over Lean, retrieval, and informal reasoning.

5. NUMINA as a 3D multimodal benchmark

In "NUMINA: A Natural Understanding Benchmark for Multi-dimensional Intelligence and Numerical Reasoning Abilities" (Zeng et al., 20 Sep 2025), NUMINA denotes a ScanNet-based 3D vision-language benchmark focused on indoor scenes and fine-grained numerical reasoning. The dataset contains 74,526 QA pairs, of which 46,194 are numerical and 28,332 non-numerical. It defines three task types—Fact Validation, Prompt Matching, and Numerical Inference—and three core numerical dimensions—Quantity, Distance, and Volume. The annotations are multi-scale: object-level metadata include category labels, instance IDs, centroid coordinates, axis-aligned bounding-box dimensions, min/max coordinates, and box volume; pairwise annotations include convex hull distance

$S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$5

scene-level aggregation supports counting and relational questions.

The benchmark is built through NUMINA-Flow, an automated pipeline that combines rule-based numerical ground truth extraction with GPT-4o template generation, Qwen2.5-72B rewriting for non-numerical ScanQA-derived questions, rule-based verification, and human inspection. Templates are generated at approximately 10 per task category, FV labels are balanced, PM correct answers are uniformly distributed across A–E, and faulty items are regenerated up to 5 times. Human inspection covers 20,000 randomly sampled QA pairs and reports 99.5% correctness (Zeng et al., 20 Sep 2025).

Evaluation uses the Chat-Scene framework with Mask3D for 3D segmentation, DINO-V2 for 2D features, and object-level language projection into an LLM decoder. Accuracy is used for FV and PM. For numerical inference, the paper defines Threshold Accuracy,

$S(\mathbf{SA}^h) = S_1^h + S_2^h + \gamma S_3^h,$6

with TA@5, TA@10, and TA@20 reported; the main table emphasizes TA@5.

The empirical conclusion is sharply negative for current open models. Non-numerical PM reaches approximately 82–86% accuracy, with Qwen2.5-7B at 85.70%. In FV, quantity questions are around 65–72%, distance 51–55%, and volume 80–83%. In NI, however, performance is extremely low: quantity TA@5 is about 8–11%, distance TA@5 about 1–2.7%, and volume TA@5 about 1.8–3.2% (Zeng et al., 20 Sep 2025). Contrapositive FV tests show only small differences, on average about 1–2%, indicating that logical consistency is substantially better than metric grounding. The paper therefore frames NUMINA as evidence that contemporary MLLMs can describe 3D scenes but do not reliably measure them.

6. Cross-cutting themes, limitations, and significance

A common misconception is to treat NUMINA as a single model family. The literature instead uses the name for multiple independent artifacts whose only strict commonality is a concern with precise alignment under structured constraints. In text-to-video generation, the constraint is numerical object count; in formal mathematics, it is syntactic validity, semantic consistency, and proof verification; in the 3D benchmark, it is grounded numerical inference over scene geometry (Sun et al., 9 Apr 2026, Guo et al., 8 Oct 2025, Zeng et al., 20 Sep 2025).

Several shared methodological patterns nonetheless recur. One is the use of latent or formal structure as an intermediate control layer. Video NUMINA extracts a countable latent layout from attention and then regenerates under guided cross-attention. Numina-ATF inserts Lean compilation and semantic judging into autoformalization. Numina-Lean-Agent elevates tool-mediated interaction with Lean, retrieval, and informal reasoning into the central mechanism of proof development. NUMINA-Flow turns geometric metadata into benchmark questions through templating, rewriting, and verification (Sun et al., 9 Apr 2026, Liu et al., 20 Jan 2026, Zeng et al., 20 Sep 2025).

Another shared pattern is explicit verification. In the video setting, the paper evaluates count correctness with GroundingDINO and tracks temporal consistency; in Numina-ATF, syntax is checked by Lean 4.15 and consistency by an ensemble judge; in theorem proving, success is Lean acceptance of proofs; in the 3D benchmark, answers are checked against numerical ground truth derived from ScanNet geometry (Sun et al., 9 Apr 2026, Guo et al., 8 Oct 2025, Liu et al., 20 Jan 2026, Zeng et al., 20 Sep 2025). This suggests a family of research programs in which evaluation is not merely preference-based but tied to structural or symbolic correctness.

The limitations are equally domain-specific. Video NUMINA does not tackle very large counts or complex relational constraints and can fail when attention fragments a single object into parts. LeanGeo shows that even strong frontier models remain at zero on olympiad-level geometry sections OP and IMO. Numina-Lean-Agent still suffers from long-context degradation, type-theory friction, and proofs that are functionally correct but not “Mathlib-like.” The 3D NUMINA benchmark focuses on geometric and numerical cognition rather than social or interactive reasoning, and current models achieve TA@5 below 3% on distance and volume inference (Sun et al., 9 Apr 2026, Song et al., 20 Aug 2025, Liu et al., 20 Jan 2026, Zeng et al., 20 Sep 2025).

Taken together, these works suggest that the name NUMINA has become attached to research efforts emphasizing exactness where contemporary generative or multimodal models are weakest: counting in video, formal fidelity in mathematics, verified proof search, and metric reasoning in 3D scenes. A plausible implication is that the label functions less as a single technical brand than as a marker of precision-oriented AI research across multiple modalities and verification regimes.