DSFB Structural Semiotics Calculus Endoduction as a Fourth Mode of Inference: Formal Syntax, Composition Rules, and Provable Properties of Endoductive Inference over Residual Trajectories

We introduce the DSFB Structural Semiotics Calculus (DSSC), a typed algebraic frameworkthat unifies residual sign semantics, admissibility envelopes, grammar-state machines, motifrewriting, and endoductive inference into a single coherent formal system. Building directlyon the algebraic primitives of Algebraic Deterministic Dynamics (ADD) and the Drift–SlewFusion Bootstrap (DSFB) estimation layer, the calculus supplies deterministic compositionrules, operational semantics in the style of structured operational semantics (SOS), and prov-able meta-properties without stochastic assumptions. Endoduction is formalized as a fourthinference mode, distinct from deduction, induction, and abduction, with explicit soundnessand bounded completeness guarantees under discrete-time, finite-dimensional, bounded-noiseassumptions. The framework yields replay-deterministic, non-interfering observers whoseoutputs are typed episodes over abstract residual trajectories. We prove: (i) finite-timeenvelope exit under ADD dynamics (generalized from prior work); (ii) monotonicity of theheuristics bank under augmentation; (iii) bounded fragmentation under bounded noise; (iv)categorical non-interference via a functor O : Traj → Ep; and (v) commutativity of ADDrewriting with grammar transitions. Every prior DSFB and ADD paper is an instance ofthis single calculus. This paper serves as the canonical foundational reference for all DSFBinstantiations.Keywords: deterministic inference, structural semiotics, residual calculus, endoduction,typed operational semantics, non-interfering observer, admissibility envelope, algebraic de-terministic dynamics, grammar automata, motif rewriting.