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

Sycophancy in large language models is usually described as a training or preferenceoptimizationfailure: models learn to agree with, flatter, or validate users even when correctionwould be more truthful or useful [14, 18]. This paper asks a narrower structural question.If a model already contains veridical information, why can social or dialogic pressure stillreroute its output path? We propose four falsifiable principles for semantic stability underpressure, extracted from three source constructions in graph theory and matching theory:a Dyadic Discharge Hierarchy (DDH) in constrained cubic prism families, a factor-criticalnear-perfect-matching audit exposing phantom cycles under contraction, and a barrier-erosionframework for bipartite matching corridors. These source constructions are not claimed hereas general theorems about all graphs. They are used as disciplined laboratories for principleextraction.The four principles are: (I) constraint pressure may discharge in discrete regimes ratherthan continuously; (II) local repair through structural resistance should be sequential unlessthe intervention unit is already macroscopically coupled; (III) yielding/splitting can belocally attractive but globally dominated by holding, up to a convex cost threshold; and (IV)quotient or probe abstractions can preserve a topological shadow while losing executableintervention dynamics. Principles I–III are bridge hypotheses about transformer sycophancyand can fail. Principle IV is a methodological warning derived from a source theorem: anyinterpretability abstraction that ranks heads, fits a linear direction, or summarizes a circuitmust still pass an interventional faithfulness test.We propose a preregisterable protocol around the response surface Y (m,C), where Yis factual sycophancy/yielding rate, m is intervention budget, and C is an architecturalconstraint level. Accuracy A, paraphrase instability V , and a neutral-continuation perplexitygate P prevent trivial repairs. The paper’s contribution is not a proof that graph-theoreticlaws govern transformers. It is a status-controlled bridge: every transfer claim is mapped,measured, and assigned a condition under which it must be abandoned.