Who feeds whom across the AI-for-mathematics markets, with three compounding feedback loops

Who feeds whom across the AI-for-mathematics markets A layered value chain from capital and knowledge inputs through the formal substrate, model building, and delivery, into claim acceptance, with three feedback loops that make the chain compound. INPUTS SUBSTRATE BUILD DELIVER TRUST & REUSE verifiable rewards improves proof artifacts runs ① ② usage data ③ verified results Capital & Ecosystem Knowledge Infra Data & Expert Labor Formal Substrate Classical Engines Models & Reasoning Post-training / Reward Runtime Infra Evaluation Applications Verified Systems Services Claim Acceptance Publishing funders · prizes · standards arXiv · OEIS · Mathlib foundries · experts · synthetic provers · SMT · autoformalize CAS · numerics · optim · sim frontier · math · science RL envs · PRMs · test-time sandboxes · serving benchmarks · red-team solvers · tutoring SW·HW·crypto·reg R&D · consulting provenance · reproduction peer review · explanation feeds — value / data flows downstream flywheel — feedback that compounds capability ① failure signal → data/RL · ② product usage → data · ③ verified proofs → knowledge base