Symbolic Topology: Field-Relative Invariants for Coherent Transformation in Symbolic State Spaces

This paper introduces Symbolic Topology, a field-relative mathematical framework for describing how coherent symbolic configurations evolve, preserve selected invariants under transformation pressure, and encounter obstructions to local-to-global consistency. It develops the gauge-theoretic chart anticipated by Symbolic Engineering, in which continuity is the conserved quantity of symbolic transformation, and proposes three candidate invariant families into which that continuity decomposes: identity integrity, capability coherence and recognition continuity. The decomposition has the form described by decompositional dual-aspect monism: an undivided whole yields separable components whose correlations record their common ground. Symbolic fields are defined over a declared contextual base, so every claim of global coherence is relative to that base and its cover, without presupposing a single manifold that contains every symbolic domain. The invariants are modelled through a symbolic action functional, a connection relating local symbolic frames, and a provisional generator algebra whose order-sensitivity yields testable predictions: the same transformations applied in different sequences should produce measurably different trajectories. A model-relative admissibility condition classifies transitions by whether the candidate invariants remain within tolerances fixed in advance. Noether’s first and second theorems are given distinct roles: global symmetries may yield conserved quantities, while local gauge symmetries yield constraint identities. The paper therefore formulates a research programme and the conditions required for full derivations; it does not claim that the invariants have already been derived as Noether currents. The connection is proposed as the transport component of the recursion operator of the Topology of Meaning, and relations to Recursive Enclosure are stated as testable correspondences. The Poetic Intensity Measurement Framework provides an initial instantiation in one domain. Context Symbolic Topology belongs to the RIZOM Working Papers in Symbolic Topology, a programme studying how coherence persists, reorganises and fails across transformation in symbolic, developmental and organisational fields. It is the second chart of the atlas begun in Symbolic Engineering, which states continuity as a conservation principle in a geometric register; the present paper supplies the gauge-theoretic chart and the programme for decomposing that continuity. Its correspondences connect it to two neighbouring frameworks: the Topology of Meaning, whose recursion operator the connection partly specifies, and Recursive Enclosure, whose conditions of recursive invariance and perturbation stability it compares with its own admissibility condition. Towards an Anti-Foundational Physics (forthcoming) draws on its candidate invariants for the conservation of coherence in interior fields.