The Surprising Universality of LLM Outputs: A Real-Time Verification Primitive
Authors: Alex Bogdan, Adrian de Valois-Franklin
Organizations: Evolutionairy AI Toronto, Canada
Abstract
We report a striking statistical regularity in frontier LLM outputs that enables a CPU-only scoring primitive running at 2.6 microseconds per token, with estimated latency up to 100,000× (five orders of magnitude) below existing sampling-based detectors. Across six contemporary models from five independent vendors, two generation sizes, and five held-out domains, token rank-frequency distributions converge to the same two-parameter Mandelbrot ranking distribution, with 34 of 36 model-by-domain fits exceeding R2=0.94 and 35 of 36 favoring Mandelbrot over Zipf by AIC. The shared family does not collapse the models into statistical duplicates. Fitted Mandelbrot parameters remain cleanly separable between models: the cross-model spread in q (1.63 to 3.69) exceeds its per-model bootstrap standard deviation (0.03 to 0.10) by more than an order of magnitude, yielding tens of standard deviations of separation per few thousand output tokens. Two capabilities follow. First, statistical model fingerprinting: text from a vendor-delivered LLM can be tested against its claimed model family without cryptographic watermarks or access to model internals, supporting provenance verification and silent-substitution audits. Second, a model-agnostic reference distribution for black-box output assessment, from which we derive a single-pass scoring primitive that composes with model log probabilities when available and degrades to a rank-only mode usable on closed APIs. Pilot results on FRANK, TruthfulQA, and HaluEval map where the primitive helps (lexical anomalies, unsupported entities) and where it structurally cannot (reasoning errors in domain-appropriate vocabulary). We position the primitive as a first-pass triage layer in compound evaluation stacks, not as a replacement for sampling-based or source-conditioned verifiers.
Large language models (LLMs) are increasingly consumed through opaque serving chains - API aggregators, resellers, and inference providers - in which the client has no technical means to confirm that the model answering is the model advertised, and recent audits show that a substantial fraction of commercial endpoints deviate from the vendor's reference weights. Existing identification techniques require long generated texts, token-level log-probabilities, adversarially crafted prompts, or the model owner's cooperation. We show that far weaker evidence suffices. We define a behavioral fingerprint of an LLM as the empirical distribution of its answers to trivial one-word prompts - "name a random number between 1 and 100" - collected across four languages at a cost of one output token per query. Measuring 165 models served via a large commercial aggregator (OpenRouter), we find that (i) these distributions are highly non-uniform (median cell entropy 1.0 bit) and model-specific: split halves of the same model's samples lie an order of magnitude closer than samples of different models; (ii) Jensen-Shannon divergence between fingerprints recovers model lineage, assigning a model to its documented family with 59.5% leave-one-out accuracy against an 18.4% chance rate; and (iii) a biometric-style verification protocol achieves a 7.3% equal error rate with the full 40-cell battery, and below 11% with eight probe cells - roughly a hundred single-token queries per audit. We further report ecosystem anomalies, including a proprietary-branded flagship endpoint distributionally indistinguishable from an open-weight Qwen model. The protocol, prompts, raw data, and analysis code are released for reproduction and operational use.
Large language models (LLMs) are high-value assets that can be derived through redeployment, fine-tuning, quantization, or further alignment. Because deployed LLMs are commonly exposed only through query APIs, ownership verification must often rely on black-box text responses. This setting is difficult: generations are open-ended and can vary across repeated queries, while existing black-box fingerprints rely on signals that are fragile under a final-response interface, including full-text matching, soft behavioral features, or model-specific prompts designed not to transfer. We propose TCF (Targeted Counterfactual Fingerprinting), a black-box LLM fingerprinting framework that converts open-ended generation comparison into constrained-answer targeted counterfactual transfer. TCF restricts each verification query to a finite answer space, reducing the surface-form ambiguity that enters the verification score, and optimizes a prompt perturbation toward a counterfactual target different from the protected model's clean answer on the original prompt. Verification reduces to checking whether the suspect model's parsed final answer matches the recorded target. We introduce the source-model counterfactual margin (SCM), a protected-model-only quantity that certifies the target is unlikely before the perturbation and likely after it; SCM controls target selection, perturbation stopping, and fingerprint filtering. Under explicit derived-preservation and independent-transfer budgets motivated by local behavioral closeness, we derive a target-accuracy gap between derived and independent models. Across four LLM families, TCF achieves an average AUC of 0.9861, improving over TRAP, ProFLingo, and ZeroPrint by 0.07 to 0.19.
LLM provenance testing asks whether a suspect LLM belongs to the same lineage as a source. Existing black-box methods largely infer this relationship from response-level characteristics, but these characteristics may shift under adaptation or deployment even when the underlying meaning remains unchanged, weakening the reliability of provenance evidence. To address this limitation, we introduce induced decision regions by mapping open-ended outputs into a finite decision space, thereby abstracting away surface-form variation and reframing provenance testing as measuring the inheritance of decision regions. Empirical analysis shows that the source's induced regions are preserved more strongly in related models than in unrelated models. Building on this signal, we propose Stemma, a practical black-box LLM fingerprinting method that operationalises stability, robustness, and specificity as complementary probe-selection principles for reliably estimating induced decision region inheritance. Across 770 source-suspect pairs drawn from 56 public checkpoints and spanning diverse model-weight transformations, Stemma achieves 0.967 AUC and 87.8% TPR at 1% FPR, substantially outperforming four representative baselines. It further achieves 0.995 AUC and 93.5% TPR at 1% FPR on 1,260 pairs covering 91 deployment instances, demonstrating robustness to diverse inference-time deployment settings.