Where Does the Noise Come From? A Variance-Components Decomposition of Non-Determinism in LLM Brand Answers

Background: Researchers increasingly use repeated identical prompts to audit stochastic variation in large language model (LLM) brand recommendations, yet no standardized protocol exists for setting iteration counts, selecting stability metrics, or establishing reliability thresholds. Objective: We formalize the Dice Roll Method as a reusable protocol for repeated-query auditing of LLM brand recommendations, grounded in a generative model of temperature-scaled nucleus sampling. Methods: Total response variance is decomposed into sampling, prompt-phrasing, run-to-run, and model-version components. The stack: a negative-binomial mixed model with iterations as repeated measures; Cliff's delta as the distribution-free effect size; dependence-preserving bootstrap; simulation-based power; a generalizability-theory decomposition; drift diagnostics on pinned snapshots. We reanalyse five brand-recommendation auditing studies: approximately 190,000 observations, 270+ brands, 6 languages, iteration counts 5 to 40. Results: Three tiers of iteration guidance emerge from the D-study: exploratory (n = 5, G = 0.58), confirmatory (n = 10, G = 0.74), and rigorous (n = 15, G = 0.81), tied to effect-size and generalizability targets. The four metric families (count, set, embedding, fairness-adjusted PASOR) are complementary, motivating a compact metric battery over single indicators. A pre-registered external validation on three independent corpora (Motoki et al., 100-round; Rozado, 24 models; llm-stability) reproduces the D-study reliability prediction in 37 of 39 cells with no failures and the n = 5 power value to two decimals; the fixed tiers do not transfer, supporting a pilot-then-solve reading. Conclusion: The protocol gives repeated-query auditing of LLM brand recommendations a statistically principled footing under the conditional, non-Gaussian structure of real autoregressive generation.

Large language models (LLMs) increasingly mediate how people form impressions of organisations, yet most monitoring is done in English, assuming an English query returns a representative picture. We measure how far that holds. We queried three grounded LLMs (GPT-5.4, Gemini 3.1 Pro, Perplexity Sonar Pro) about 66 brands from eleven Northern, Baltic, and Central European markets, in twelve languages across four families (Germanic, Uralic, Baltic, Slavic), generating 35,640 responses. Multilingual embeddings (BGE-M3) allow cross-language comparison without translation. Three results emerge. First, AI-constructed reputation is language-bound: mean cross-language cosine similarity is 0.825, same-family responses are more similar than cross-family (0.844 vs 0.820; d = 0.31), and sentiment varies by language (F = 268.5, eta^2 = 0.077), with Uralic and Baltic languages most positive and Germanic, including English, most critical; clustering recovers the Slavic and Baltic families (cophenetic 0.915). Second, query language shifts which brands are recommended far more than how they are described: moving from an English query to a brand's home language raises recommendation share by 0.80 for local champions but only 0.15 for global multinationals (t = -8.84, p < 0.001), with no comparable reversal in sentiment. An English-only audit therefore understates a local champion's AI visibility. Third, response stability varies more with model choice than with language (eta^2_model = 0.32 vs eta^2_language = 0.01, on a five-iteration replication over a 20-brand subset). These results indicate that English-only AI reputation monitoring leaves a measurable language blind spot, concentrated in the visibility of locally headquartered brands.

Large language models (LLMs) are powerful black-box systems, making it difficult to discern whether their answers reflect stable internal beliefs or superficial pattern matching. We identify cross-contextual consistency as an underutilized behavioral property of LLMs: a credible answer should remain stable when the same task is placed under topic-aligned, content-neutral contextual variation. Building on this intuition, we operationalize Cross-Contextual Consistency (C3) by comparing model generations under original and perturbed prompts. Across 26 models and six benchmarks spanning reasoning, factuality, and code generation, we find that answers with smaller cross-contextual shifts are more likely to be correct or factual. We demonstrate that C3 provides a complementary axis of evaluation and can serve as a benchmark usefulness diagnostic, identifying which portions of a benchmark remain informative even when aggregated scores are widely considered "saturate".