Topological Obstruction to Forgetting in Finite Distinction Systems: Quotient Invariants, Non-Hermitian Skin Effects, and Topological Side-Ledgers

Official website: distinctiontheory.orgPublic portal for the start guide, papers, claim status, failure registry, prior-art boundary, and citation resources. Canonical GitHub repository:https://github.com/yiningwu-research/Distinction-Theory FDS-P7 develops the protected complement of FDS-P4. P4 showed that capacity recovery is not distinction recovery: task-relevant preimage distinctions erased by non-injective local truncation cannot be reconstructed without inverse information. P7 studies the complementary case in which task identity is not stored as an ordinary local record but is carried by an accessible invariant quotient. The paper formalizes invariant side-ledgers through a quotient map q : X → Q and the induced random variable Qinv = q(X). If a task sector V factors through this quotient, then local coarse-graining may lose task distinctions while invariant-aware representation preserves task identity. A noisy invariant recovery bound gives a Fano-style limit on residual task-sector uncertainty under imperfect readout. The physical bridge is the Non-Hermitian Skin Effect (NHSE), treated as one model-class realization rather than a universal mechanism. Point-gap winding, generalized Brillouin-zone structure, boundary-localized modes, and boundary-sensitive spectra provide an auditable setting for invariant-supported persistence. P7 does not claim free memory, perpetual motion, or a violation of the Second Law. Protected invariants relocate entropy and resource accounting rather than deleting it. The operational signature is a dual-channel feature: suppressed invariant-aware forgetting together with a resource or entropy-ledger reorganization near a protection-breaking transition. The paper also extracts a design principle for Physical AI: resource-constrained agents should not merely preserve more local detail; they should preserve the quotient that preserves task identity.