The Law of Persistent Distinction: A Universal Mathematical Theory of Information Storage Capacity

This paper develops a rigorous, substrate-independent mathematical theory of information storage capacity grounded in the primitive notion of dynamical distinguishability. Rather than assuming probabilistic channels, coding schemes, or specific physical substrates, the framework begins from a minimal structural quadruple consisting of a state space, a time evolution, a readout map, and an observation pseudometric. From this minimal structure, upper bounds on storage capacity are derived without presupposing encoding models or statistical assumptions. The work introduces several new structural constructs that formalize how distinguishability evolves under dynamics and observation. A canonical operator describes how message partitions collapse under observable drift, while complementary quantities measure readout separation and time-dependent distortion. Together, these tools allow the derivation of a sufficient stability condition for stored distinctions that follows directly from the triangle inequality. This condition isolates the precise tradeoff between initial separation and dynamical drift required for information persistence. A universal upper bound on storage capacity is established in terms of packing geometry of the observable image of the state space. Capacity is shown to be controlled by the metric structure induced by observation and by the rate at which distinguishability degrades under evolution. Classical Shannon coding bounds and Kolmogorov Sinai entropy appear as special cases under suitable instantiation of the general framework, including an explicit treatment of the binary symmetric channel. The theory further proves that distinguishability cannot be generated without structural cost. Any attempt to restore collapsed distinctions through active stabilization requires proportional complexity in the corrective mechanism. This leads to a structural dichotomy governing resource-bounded storage density and identifies invariant quotient richness as the fundamental quantity underlying persistent information storage. The paper proposes Dynamical Storage Geometry as a research program aimed at classifying dynamical systems by their capacity to support stable, observable distinctions per unit resource. The results provide a unified structural account of storage limits across mathematical, physical, and computational systems. Version: v3 (revised universal capacity bound, strengthened justification of structural conservation of distinguishability, and expanded Shannon example). Included in this record is the IQS Vault (Invariant-Quotient Storage Vault), a reference software implementation derived from the structural principles developed in this manuscript. IQS Vault is a local content addressed immutable object storage system implementing deterministic SHA-256 hashing, automatic deduplication, manifest tracking, and folder based ingestion automation. Each ingested file is canonically hashed and stored by its content fingerprint, ensuring that identical data is preserved as a single invariant object while subsequent copies reference the same stored identity. Manifest records capture ingestion metadata and reproducibility anchors, formalizing identity persistence independent of filename or location. The vault operationalizes invariant-distinction storage logic in computational form, translating the manuscript’s structural emphasis on distinguishability and invariant partitioning into a practical identity-preserving storage architecture.