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

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UPDATE (Jan 10, 2026): Verification Complete. This version includes a fully verified Lean 4 kernel with 1,647 formal proofs. We have successfully removed the standard library dependencies (propext, Quot.sound), rendering the theory constructively computable. The kernel now natively computes the Standard Model mass spectrum (including the Top Quark) to within 1σ of experimental values using integer arithmetic, and resolves the Muon anomaly via the derived fractal correction. 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.

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