Post-training methods use training-side signals to decide which supervision to emphasize, but those signals need not identify behavior that is capability-bearing in the target environment. We probe this with DIAA (Dual-Implicit Adversarial Amplification), a per-sample σ = sigmoid(-βΔ) SFT/RL hybrid: after every literal Python code fence is deleted from the anchor data, the model still emits unexecuted Tool-Integrated Reasoning (TIR) form on 29% of MATH-500 prompts, three times vanilla GRPO, and suppressing the form recovers accuracy. A paired within-prompt intervention decomposes the removable drag along GRPO → SFT → DIAA as +2.2 → +8.5 → +10.6 pp, identifying SFT imitation and σ-amplification as comparable sources; a teacher RFT with 0% TIR likewise trains a student emitting TIR at 45.8%. We propose Intervention-Certified Amplification (ICA), which gates the per-sample weight by a certificate derived from paired rollouts, σ_eff(x) = σ(x)·c(x; θ_c). Frozen and online instantiations cut TIR to 13.8% and 11.2%, more than halve residual drag while improving accuracy, and carry the joint improvement to held-out GSM8K without retuning the certificate.

Pre-training on large, high-quality datasets is essential for improving the reasoning abilities of Large Language Models (LLMs), particularly in specialized fields like mathematics. However, the field of Multimodal LLMs (MLLMs) lacks a comprehensive, open-source dataset for mathematical reasoning. To fill this gap, we present InfiMM-WebMath-40B, a high-quality dataset of interleaved image-text documents. It consists of 24 million web pages, 85 million image URLs, and 40 billion text tokens, all carefully extracted and filtered from CommonCrawl. We outline our data collection and processing pipeline in detail. Models trained on InfiMM-WebMath-40B demonstrate strong performance in both text-only and multimodal settings, setting a new state-of-the-art on multimodal math benchmarks such as MathVerse and We-Math.