Personalized neural audio watermarking systems embed user-specific payloads into audio to trace the user associated with a leaked or misused copy. Existing evaluations mainly assume access to a single personalized copy, but real-world distribution may expose multiple copies of the same recording. We find that simple sample-wise averaging breaks traceability: averaging only two copies produces tracing-failure rates of 87.7-99.7% across the five systems while largely preserving audio quality. Decoder outputs show distinct responses to coalition payload composition across systems, with high majority agreement in TimbreWM but substantially lower agreement in WMCodec. In a more sophisticated payload-aware setting, mixture weighting can further steer the decoder toward a selected nonmember, achieving a target success rate of 92.3% in TimbreWM. Preliminary confidence screening rejects many collusion-induced assignments while retaining most valid single-copy outputs. Code and results are available at https://github.com/Silwan5z/neural-audio-watermark-collusion-attack. This work highlights multi-copy traceability as an additional evaluation dimension for personalized neural audio watermarking alongside conventional single-copy robustness.
Ranking functions are a fundamental technique for proving loop termination, but constructing them manually is challenging and requires substantial expertise. We propose RAINFALL, a neuro-symbolic framework which follows the generate-verify-refine paradigm and tightly integrates LLMs with symbolic verification. RAINFALL formulates ranking function synthesis as a code-completion task by inserting placeholders into the input program and constraining the LLM to fill four predefined slots for each loop. The completed program is then checked by a symbolic verifier. If verification succeeds, a valid ranking function is obtained; otherwise, the verifier's diagnostics are translated into natural-language feedback that guides iterative refinement of the LLM's output. We evaluate RAINFALL on two public benchmark suites comprising 358 terminating C programs. Using MiniMax V2.7, RAINFALL synthesizes verifiable ranking functions for 323 programs (90.22%), outperforming all promising baselines. In particular, it solves 12 more programs than PROTON 2.1, the best-performing baseline and winner of the SV-COMP 2026 Termination category. Furthermore, combining RAINFALL with PROTON 2.1 increases the success rate to 96.65% (346/358 programs). We also conduct ablation study confirming individual contributions of placeholder contracts, few-shot examples, symbolic verification, and diagnostic information feedback. Experiments across LLMs further confirm the effectiveness of RAINFALL. Our results suggest a generic, sound framework for ranking-function synthesis and, beyond termination, LLM-assisted synthesis of formally checkable liveness certificates, both of which represent new directions of software engineering research.