Synchronization Resistance: Measuring Multi-LLM Agreement Difficulty

Multi-LLM systems are increasingly deployed in settings where the cost of unnoticed disagreement is high: clinical decision support, regulatory submissions, financial compliance. Existing methods for measuring agreement across models, including categorical voting, pairwise embedding similarity, and human-rated evaluation, collapse the question into a single number that hides where models actually disagree and how hard the disagreement is to overcome. This paper validates Synchronization Resistance (SR), a measurement instrument grounded in the Kuramoto order parameter R(K), that reports per-prompt agreement difficulty as a continuous scalar derived from a coupling sweep. The broader Convergence Detection framework was formally pre-registered on the Open Science Framework as registration `xqy5w` on 2026-03-17. The five Battery 3C-specific hypotheses — H1 human-intuition correlation, H2 R(K) curve classifiability, H3 two-point shortcut accuracy, H4 replication stability, H5 category discrimination — were committed to the public OSF project `xv9j6` on 2026-04-07T01:00:00Z, before any data collection, as a public-timestamped extension of the pre-registered framework. The instrument was Bounce v3.1, exercising six local language models via Ollama at temperature zero across 22 prompts and an 11-point coupling sweep. Three hypotheses passed cleanly (H2: 18/20 prompts at R²>0.90; H3: MAE=0.07; H4: 0/4 status flips). H1 returned ρ=0.30 (p=0.26), inconclusive. H5 failed by both pre-registered criteria (Jonckheere-Terpstra p=0.06 against the Bonferroni-corrected α=0.01 threshold; the F<C<X<S ordering was violated, with Creative>Controversial). The inconclusive and failure outcomes are themselves informative: SR detects model-level disagreement that does not align with human categorical labels, which is the operational value of the instrument rather than a defect. Pre-registration, results, and the T2 reproducibility receipt are available on OSF. Pre-registration: The broader Convergence Detection framework was formally pre-registered as OSF registration xqy5w on 2026-03-17 (DOI 10.17605/OSF.IO/XQY5W). Battery 3C-specific deposit: the five SR hypotheses were committed to the public OSF project xv9j6 (DOI 10.17605/OSF.IO/XV9J6) on 2026-04-07T01:00:00Z, before any data collection — a public-timestamped extension within the formally pre-registered framework. Results landed 2026-04-08T01:30:00Z against the same project. Reproducibility: T2 receipt manifest hash fe68dc99cb81f26221ccfdd6adabd12f22927e3dc64dddde2a4f6ff70636c562. Patent and trademark rights notice. This work is licensed under CC BY-NC-ND 4.0 for copyright purposes only. Per CC BY-NC-ND 4.0 §2(b)(2), patent and trademark rights are not licensed under this Public License. Methods, instruments, or systems described in this work may be subject to pending or issued patents held by BAION Systems LLC or Steven A. Mullins Jr.; patent licensing terms, where applicable, are governed separately from this deposit.