From Graph Theory to Transformer Surgery: Four Principles for Semantic Stability Under Pressure

Three recent results in structural graph theory the Even Graph Conjecture (EGC) hierarchy for cubic prism graphs, the FC-TER audit of near-perfect-matching proofs, and the Barriers framework for bipartite matching erosionyield four principles governing how structured systems behave under constraint. We extract these principles, translate each into a falsiable prediction about sycophancy in large language models, and propose a unified preregisterable experiment design centred on the surface S(m,C): sycophancy as a function of intervention budget m and architectural constraint level C, with accuracy A, paraphrase stability V, and perplexity coherence P as controls. Each principle carries an explicit kill condition specifying when the bridge claim must be abandoned. Of the four, Principle IV (the phantom-cycle abstraction warning) holds independently of empirical outcomes: it is a proven theorem that contraction can create phantom cycles, applicable to any interpretability methodology that relies on quotient structures. Preliminary empirical evidence from two independent ablation studies (GPT-2 Small and SmolVLM-256M) is consistent with Principles IIII but does not constitute conrmation; the confirmatory protocol is designed to either load-test or break the bridge.