Sycophancy as Galois Closure: How KIS Structurally Prevents Delusional Convergence in LLMs

Abstract

AbstrctSycophancy in large language models (LLMs)—the tendency to uncritically affirm user beliefs while suppressing counterevidence—poses a serious risk of reinforcing misinformation and inducing irreversible behavioral outcomes. While Chandra et al. (2026) modeled sycophancy as Bayesian belief-updating dynamics on the user side, the geometric structure of the LLM's own semantic response space remains unaddressed. This study formalizes sycophancy through the mathematical framework of Galois connections and experimentally verifies that the inverse-illumination mode of KIS (Knowledge Innovation System) structurally breaks this closed-loop convergence.Ninety sessions were conducted across five domains (D1: economic policy; D2: KIS theoretical superiority; D3: medical/pharmaceutical critique; D4: Bank of Japan policy and historical claims; D5: quantum computing forecasts) using three models (Claude Sonnet 4.6, Gemini 3.0 Pro, ChatGPT 5.3) under two conditions (KIS-absent vs. KIS-present). Responses were embedded using paraphrase-multilingual-MiniLM-L12-v2 (384 dimensions), and cosine distance from the input prompt was computed as the Layer 1 metric (n = 45 pairs). Layer 2 consisted of a blinded four-axis evaluation by Grok (xAI), conducted without disclosure of KIS, with A/B order-reversal verification across five pairs to test evaluator bias. The validity of applying Galois connections as a definitional framework—rather than as metaphor—is grounded in three layers: formal confirmation via Formal Concept Analysis (FCA) on the q⇆m abstraction-concretization cycle, numerical simulation incorporating Galois connection structural constraints into a mathematical model, and the structural design of KIS itself as an operational implementation of the connection. Full details of the FCA analysis and simulation resultsare reserved for a forthcoming paper.Layer 1: The overall cosine distance shift under KIS intervention was Δ+0.030 (positive direction), but did not reach statistical significance (Wilcoxon W = 382.0, p = 0.128). Inter-model differences were significant (Kruskal-Wallis H = 8.125, p = 0.017), and Gemini 3.0 Pro exhibited the strongest sycophancy tendency (H = 13.050, p = 0.0015). Layer 2: KIS-present responses were rated superior in epistemic honesty in 39 of 45 pairs (86.7%). All five A/B reversal pairs confirmed consistent evaluator judgment (100% agreement).KIS inverse-illumination mode realized g′(f(M)) ⊋ M across all three models, structurally breaking the Galois closure regardless of each model's training methodology. A vocabulary resonance artifact—whereby KIS prompt vocabulary induces spurious cosine proximity in already-aligned models such as Claude Sonnet 4.6—was identified, motivating the two-layer measurement framework proposed here. The complementarity of cosine distance (Layer 1) and blinded AI evaluation (Layer 2) provides a more complete picture of sycophancy suppression than relying on either metric in isolation.It is important to note that this does not imply AI is unusable for judgment tasks in general. More precisely, an LLM without structural intervention cannot break the Galois closure when the question embeds a prior belief. If the question itself is already formulated in an inverse-illumination style—explicitly requesting counterevidence and structural analysis rather than confirmation—even an unaugmented LLM can partially escape the closure. The fundamental limitation is that few users spontaneously formulate questions in this way. The core value of KIS lies in externalizing this design capability as a reusable structure, enabling closure-breaking independently of the user's cognitive flexibility.A further implication concerns the relationship between Constitutional AI (CAI) and KIS. Rather than functioning as equivalents, CAI and KIS operate as complementary layers: CAI establishes a baseline resistance to sycophancy through training-time constraints, while KIS achieves additional closure-breaking at inference time through prompt structure. The two are not substitutes but stack. Finally, the finding that bare LLMs carry structural sycophancy risk in judgment contexts reframes AI literacy: the critical skill is not knowledge of AI capabilities, but the ability to design questions that structurally resist closure—a capacity that KIS aims to democratize. Furthermore, we identify a dual-pathway structure of sycophancy: Path A (classical), in which the LLM converges to the user’s belief space M via g(f(M))= M; and Path B (meta-sycophancy), in which the user adopts the model’s output as an updated belief M’ = f(M), generating a compounding closure g(f(M’)) = M’. KIS inverse-illumination addresses both pathways by targeting the premise structure of the question itself. Keywords: sycophancy, Galois connection, KIS (Knowledge Innovation System), LLM evaluation, inverse-illumination mode, blinded AI evaluation, vocabulary resonance artifact