EIP-1559 has become the dominant transaction fee mechanism in permissionless blockchains, regulating scarce block space through an adaptive base fee. The design balances a trade-off between responsiveness to demand fluctuations and stability of long-term resource usage. On a single chain, this trade-off is well understood and empirically stable. In multi-chain systems, however, cross-chain flows introduce delayed economic interactions across otherwise independent fee markets. These interactions alter the control dynamics and create stability risks that existing models do not capture. Deployed systems have already polarized. Polkadot enforces conservative update rates, effectively disabling fee adaptivity and reverting the market to inefficient first-price auctions. Cosmos adopts aggressive update rates without formal guarantees. To fill this gap, we propose a control-theoretic framework that formalizes multi-chain fee markets as a nonlinear dynamical system with delayed feedback. We identify delay-amplified over-correction as the structural root of instability. Experiments show that cross-chain coupling erodes stability relative to monolithic blockchains, reducing the safe parameter space for base fee update rates by up to 84.8%. Moreover, instability propagates across chains, contaminating the otherwise stable peer chains. At Ethereum's update rate, which leaves a monolithic chain stable, the multi-chain system exhibits extreme load oscillation between full and near-empty blocks, with over half of all blocks at full capacity. Mainstream Cosmos chains already exceed their safe bounds under their default configurations, and the resulting load oscillation raises their block failure rate by an order of magnitude. Guided by this mechanism, we design a lightweight protocol-level mitigation that removes the over-correction and stabilizes dynamics in multi-chain environments.

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