Multi-agent LLM systems decompose complex tasks across specialist agents, yet joint prompt optimization remains difficult because agents are coupled through structured information handoffs. We identify interface mismatch as a structural failure mode that existing optimizers cannot diagnose: their scalar attribution signals conflate reasoning quality with format incompatibility, and the inter-agent contract is left entirely implicit. We propose BID-SPO (Bidding-based Shapley Prompt Optimization), which makes the inter-agent contract an explicit optimization variable. Each agent declares an Output Bid and an Input Demand, and a contract-aware cooperative-game characteristic function yields an attribution signal whose Shapley decomposition exposes three interpretable dimensions (global task contribution, downstream interface alignment, and upstream adaptation cost), turning attribution into a targeted diagnosis for prompt repair. A bid-conformance constraint keeps declared contracts feasible with respect to actual agent behavior throughout optimization. Across seven benchmarks and different model configurations, BID-SPO outperforms MASPO and AgentDropout, with accuracy gains of +2.93% under the Qwen3-8B + Gemini-2.5-pro configuration and +4.38% under the DeepSeek-v4 flash / pro configuration. Our findings emphasize that reliable multi-agent LLM optimization requires optimizing inter-agent interfaces as explicit objects, not just improving agents in isolation.

Continuous-space reasoning methods replace discrete intermediate tokens with probability-weighted soft embeddings, aiming to preserve semantic information that greedy Chain-of-Thought decoding discards. We show that this promise is limited by local representation geometry and the latent semantic manifold. When the next-token distribution is low-entropy and concentrated, a soft token becomes nearly collinear with the Top-1 embedding and collapses toward greedy decoding. When the distribution is high-entropy, diffuse, or stochastically perturbed, the continuous state may leave the greedy path but drift away from the local scale of naturally occurring candidate representations. We diagnose these two regimes with step-level probes over representation divergence, local norm calibration, and behavior-cloud membership. To address them, we propose Manifold Thinking, a training-free inference framework that combines structured tangent movement away from the Top-1 direction with local norm-calibrated projection back to the candidate-token neighborhood. Across mathematical and programming benchmarks, Manifold Thinking improves over discrete CoT baselines and baselines covering the latest stochastic Soft Thinking strategies, suggesting that useful continuous reasoning requires geometric correction rather than unconstrained linear token mixing alone.