Structural Collapse as Information Loss: The Exponential Decay Mechanism under Accumulating Constraints

Large language models exhibit abrupt reasoning collapse when structural contradictions accumulate in their context---a phenomenon distinct from gradual degradation under factual noise. A systematic δ = 0 control experiment (6 models, 4 vendors, 500--128,000 tokens) shows that under zero structural contradiction, models with sufficient baseline capability maintain perfect accuracy across all tested context lengths, establishing that the phenomenon commonly attributed to "context rot" is driven primarily by contradiction accumulation (δ > 0), not by context length itself. The same exponential structure appears in Boolean satisfiability (SAT), where e^(−δ) is a mathematical identity with the first moment of the solution count---not an approximation. We propose a general framework grounded in the first moment method from combinatorics: defining cumulative information loss δ = Σ I(constraint_i) in nats, we show that the multiplicative survival potential S = N_eff · (μ/μ_c) · e^(−δ) captures collapse under accumulating constraints across structurally distinct domains.

In SAT, where δ is exactly computable, the framework yields parameter-free predictions. Three constraint types (random clauses, XOR pairs, implication chains) produce exponential decay in solution counts at rates determined entirely by their information content: the predicted decay rate ratio α_XOR/α_random = ln 2 / |ln(7/8)| = 5.19× is confirmed experimentally at 5.04 ± 0.25 (CV = 5%) across six system sizes. A single XOR pair (0.693 nats) eliminates as many solutions as 5.2 random clauses---constraint quality, not quantity, governs collapse.

In LLM reasoning (11 models, 5 vendors, 4B--70B+ parameters), the same structure appears empirically: (i) context margin (μ) and structural contradiction (δ) interact multiplicatively---two individually non-lethal stressors become lethal in combination, ruling out additive degradation models; (ii) a single structural contradiction causes more damage than 10 factual contradictions in models with sufficient baseline accuracy, paralleling the SAT quality-over-quantity finding; (iii) an RLHF-derived alignment layer acts as a cushion, absorbing contradictions until a system-specific critical threshold δ_c is exceeded.

Extension to the second moment method (Paley--Zygmund inequality) with pair correlation function g(β) = 3/4 + (1/8)(1−β)³ explains 74% of the gap between the first-moment upper bound (α ≈ 5.19) and the true 3-SAT threshold (α ≈ 4.27); the remaining 26% is attributable to replica symmetry breaking. All results are machine-verified in Lean 4 (16 modules, 160 propositions, no sorry, no axiom).