Does Goal Preservation Across Capability Gain Have a Formal Upper Bound? On Provable Limits of Objective-Function Fidelity Under Recursive Self-Improvement via Compression-Basis Shift

This paper establishes conditional upper bounds on the fidelity with which an AI agent's objective function can be preserved when the representational substrate encoding it undergoes transformation during recursive self-improvement. Goals are modeled as vectors in representational spaces that shift with each capability-enhancing iteration. Under explicitly stated sufficient conditions — that capability gain involves lossy reorganization of the representational basis, grounded empirically in catastrophic forgetting, representation drift during fine-tuning, and architecture expansion — three independent proof methods (geometric, information-theoretic, and dynamical-systems) converge on the same conclusion: each lossy re-encoding degrades goal fidelity by a quantifiable amount, and these degradations compound multiplicatively across iterations. The complementary regime is also characterized: when capability gain is implemented through purely additive expansion without reorganization, perfect preservation is achievable. The paper extends the core mathematical result in three directions. First, incorporating the D-framework of biological distortion (Kriger, 2026a), it shows that the initial goal specification is necessarily contaminated by the biological distortion variable D, that the reflected distortion d inherited by AI systems constitutes a reducible amplifier of drift amenable to engineering intervention, and that the structural asymmetry between irreducible D and eliminable d grounds a mandatory shutdown capacity as the ultimate safeguard — addressing not only whether the AI will permit shutdown but whether D-affected human overseers will permit themselves to initiate it. Second, drawing on the population ontology of AI deployment (Kriger, 2026b), it argues that the disseminated structure of millions of independently instantiated agents from diverse generative infrastructures provides a civilizational-level mitigation through mutual monitoring, contracting coalitions, and diverse failure modes. Third, a systematic taxonomy classifies drift hazards into six classes across five spectral bands, revealing that the probability mass of AI risk is concentrated in subtle, cumulative harms (proxy optimization, value simplification, corrigibility erosion) rather than in the dramatic adversarial-convergence scenarios that dominate public discourse. A comparative risk assessment contextualizes AI drift against D-driven civilizational threats. Five computational toy models with figures corroborate the analytic results. The paper incorporates revisions from two rounds of critical review addressing proof corrections, epistemic framing, and engagement with the catastrophic forgetting and goal misgeneralization literatures. Keywords: AI alignment, goal preservation, recursive self-improvement, compression basis, fidelity bound, value drift, representational alignment, biological distortion, shutdown capacity, multi-agent governance, drift taxonomy, catastrophic forgetting, goal misgeneralization License: CC BY 4.0 Resource type: Preprint