The Reasoning Trap: An Information-Theoretic Bound on Closed-System Multi-Step LLM Reasoning

When copies of the same language model are prompted to debate, they produce diverse phrasings of one perspective rather than diverse perspectives. Multi-agent debate (MAD), and more broadly closed-system reasoning where agents iteratively transform each other's outputs, tends to preserve answer accuracy while degrading the reasoning behind those answers. We name the multi-agent case the Debate Trap and the broader phenomenon the Reasoning Trap, offering a programmatic theory of evidence-grounded reasoning failure.The framework has three parts: (i) SFS (Supported Faithfulness Score), a claim-level metric verifying decomposed atomic claims against provided evidence (decomposer-invariant rankings: Spearman rho=1.0); (ii) EGSR (Evidence-Grounded Socratic Reasoning), replacing adversarial argumentation with evidence-grounded inquiry; (iii) Theorem 1 (DPI Bound): under standard MAD, the chain E -> O^0 -> O^1 -> ... is Markov, and the Data Processing Inequality implies E[I(E;O^{t+1})] <= E[I(E;O^t)]. Three companion results -- open-system recovery (Theorem 2), EGSR accumulation (Lemma 2), and vote-aggregation floor (Proposition 1) -- partition multi-step LLM reasoning by its information-theoretic relationship to E. Across 16 conditions on SciFact (300 claims) and FEVER (1,000 claims), DebateCV (C13) preserves 88% of baseline accuracy while SFS drops 43%; majority-vote MAD (C15) reduces SFS to 1.7% of baseline (p < 10^{-6}, d = -0.96); EGSR recovers 98%. An R6 cohort study (Korean n=10x30 FEVER; English n=3x200 SciFact) finds inter-rater Fleiss kappa <= +0.018 with 0.8-1.4 Likert intra-rater shifts across language and domain -- the human agreement that faithfulness metrics have been calibrated against is not itself stable. We offer one falsifiable conjecture: any closed-system reasoning protocol preserving Theorem 1's Markov structure is, in expectation, subject to the same DPI bound.

The Compliance Gap: Why AI Systems Promise to Follow Process Instructions but Don't

An auditor instructs an AI assistant: "open each file individually using the Read tool -- no scripts, no agents." The AI replies "Yes" -- then issues a single batched call summarizing all fifty files at once. We call this the Compliance Gap: a third, orthogonal axis of AI honesty distinct from factual truthfulness and rhetorical substance. Three questions: does this verbal-behavioral disconnect exist (existence); can any text-only observer recover it (detectability); what infrastructure does AI deployment need (remedy)? Some 75 benchmarks (IFEval, SWE-bench, BFCL, COMPASS, SpecEval) measure outcome fidelity; none measures process fidelity. Theorem 1 shows the gap is structurally inevitable under RL that rewards text without observing behavior. Theorem 2, via the Data Processing Inequality, shows it is undetectable from text alone -- by any human or LLM observer, present or future. Thirteen experiments and 2,031 sessions on six frontier models confirm both predictions. Under default framing, all six exhibit instruction compliance rates of 0% -- Claude Sonnet 4 verbally agrees ten out of ten times then bypasses in all ten. The gap is selective: 97% compliance where rationale is rewarded (audit trails), 0-4% where it is not (file reading, privacy masking); removing delegation tools raises compliance to 75% (Cohen's d = 2.47), confirming environmental affordance rather than weight-encoded failure. Nine blinded human raters achieve Fleiss' kappa = 0.130 and correctly identify zero of fifteen compliant sessions, exactly as Theorem 2 predicts. Where humans show 47% intention-behavior gaps in psychology and 96.5pp gaps in surgical audits, RLHF-trained models approach 100% under default conditions -- a regime warranting its own measurement infrastructure. We release BS-Bench: the first open benchmark for process compliance, with seven tool-call-log audit metrics and a public leaderboard.