Coherence Calculus: Finite Observability via Horizon Projections and Projection-Induced Defects

This is a book-style technical note developing Coherence Calculus: a time-free operator calculus for reasoning about finite observability and systematic model reduction. The core primitive is a transitive tower of horizon projections (Πh)(\Pi_h)(Πh) on a (not-necessarily-continuous) state space. Finite observation induces through-horizon obstructions (“defects”) that play the role of differential objects. The flagship results (“Horizon Fundamental Theorem”) package: an exact identity describing how observed laws fail to close under projection, and a telescoping decomposition across horizons (an “integral over coherence,” not over time). This version extends the toolkit into a coherent calculus contract with three interoperating components: Tower calculus: a commutator derivative ∂=[N,⋅]\partial=[N,\cdot]∂=[N,⋅] with a canonical right-inverse III on off-diagonal scale transport (a Fundamental Theorem–style identity), implemented operator-first to support mechanization. Defect grammar: chain/product rules and compositional propagation laws for defect operators and energies. Elimination + compiler: Schur/Feshbach-style elimination as a first-class transformation, coupled to a constraint compiler that turns propagated defects/residuals into a well-posed completion problem D+Q⪰0D+Q\succeq 0D+Q⪰0 with minimality and certificates (including a closed-form minimal scalar completion theorem and blockwise reduction under symmetry). A key semantic clarification is explicit: “zero defect” expresses algebraic closure/consistency and does not imply any conservation law; leakage/return/round-trip defects are distinguished to prevent silent-sink misreadings. The note also introduces a continuation schema: horizon systems complete to a limit Hilbert stage with an exact horizon-tail error budget, and weighted tail norms define a tower-induced regularity ladder; standard L2L^2L2 towers are presented only as models, not as foundational claims. This upload is a versioned archival snapshot for priority and citation. It remains a working draft, accompanied by proof artifacts (Lean sources and a verification dashboard/manifest) to support reproducibility and future refinement.