KM Space Theory, Volume IV: Analytic Screening, Operator Defects, and Differential Equations

We develop the analytic layer of KM space theory by comparing screening before and after completion, closure, adjunction, spectral construction, elliptic realization, evolution, and solution formation. For bounded maps \[ κ_X:X→ KX, κ_Y:Y→ KY, \] and operators \[ D:DomD⊆ X→ Y, DK:DomDK⊆ KX→ KY, \] we separate the domain defect from the operator defect \[ ΔDom(D) :=DomDK/κ_X(DomD), Δ_D:=κ_YD-DKκ_X. \] No analytic operation is assumed to commute with screening. Exact criteria and counterexamples delimit completion, closure, adjoint, Fredholm, Sobolev, elliptic, semigroup, and nonlinear solution comparisons. A Duhamel formula computes evolution defects, linear reverse lifts form kernel torsors when their cokernel obstructions vanish, and a seven-term exact sequence controls composite comparisons. Weak boundary realizations are compared through trace, form, datum, compatibility, and conormal-factorization defects rather than through an untyped boundary symbol. Elliptic comparison separates symbol, parametrix, wavefront, global-end, and boundary-realization defects. The boundary layer includes a Stokes realization in which divergence inf–sup stability, pressure normalization, flux compatibility, and reverse lifting remain separate KM data. The evolution layer constructs the form-defined Stokes generator, Helmholtz projection defect, forced-velocity Duhamel formula, normalized pressure reconstruction, and trajectory-lifting obstruction. Boundedly damped waves are compared through separate elastic and damping defects, while Schrödinger groups are controlled by graph commutators, Duhamel integrals, spectral-projection transport, and reverse trajectory obstructions. Dissipation is not identified with exponential stability. Ordinary boundary triples identify Dirichlet, Neumann, and bounded Robin closed realizations through graph-continuous traces. Their defect spaces carry boundary lifting obstructions and torsors; Poisson and Weyl operators satisfy Green and Kreĭn resolvent identities and are natural only under simultaneous interior and boundary screening. Conditional Sobolev regularity upgrades already existing weak solutions, while explicit corner examples show that \(L^2\) data do not automatically yield global \(H^2\) regularity. Fractional Sobolev comparison is constructed from uniformly compatible local norms or interpolation of bounded endpoint morphisms. Positive self-adjoint operators generate Hilbert scales in which unitary resolvent intertwining transports all real orders, while compact embeddings, trace maps, graph domains, and reverse solution lifts remain separate hypotheses. Kato-type perturbations are treated as screened pairs: base and perturbation commutators combine into a total defect, relative bounds below one preserve graph topology, ambient defect extensions control resolvents and Riesz projections, and reverse resolvent solutions are governed by a cokernel obstruction and kernel torsor. Sectorial resolvent packages construct analytic semigroups and shifted fractional powers. Their defects admit Duhamel and Balakrishnan integral formulas, whereas maximal \(L^p\)-regularity remains a separately verified property. Elliptic regularity is separated from unique continuation. Local restriction kernels, Cauchy-invisible spaces, Carleman propagation packages, weighted screening defects, and seven-term comparison sequences make the propagation and reverse-lifting hypotheses explicit. Formal adjoint pairs are resolved into minimal and maximal closed realizations whose graph-domain quotients carry Green pairings. Closed extensions are classified by boundary annihilators, and screening of a chosen realization is reduced to an explicit map of boundary quotients. On compact boundaries, order-zero Fredholm trace screening is characterized by invertibility of the principal boundary symbol. On noncompact boundaries, local ellipticity must be supplemented by a uniform coercive end estimate or a parametrix modulo compact operators. Parameter families yield \(K^0\)-valued boundary indices without assuming separate kernel and cokernel bundles across rank jumps. At the topological-vector-space level, completion is separated into image, kernel-creation, and Hausdorff-cokernel defects. Strict short exact sequences survive completion. Completed projective tensors preserve quotient screenings, whereas completed injective tensors preserve isometric subspace screenings. Bochner \(L^p\)-realizations preserve strict Banach quotients and measurable lifting torsors. Fixed screenings commute with weak time differentiation, while a time-dependent screening contributes the explicit defect \(K'(t)u(t)\). Closed invariant redundancy spaces characterize quotient semigroup descent. Ordered Lie–Trotter factors carry separate finite-step, commutator, quotient, and lifting defects. Nonlinear flows are separated into vector-field, flow, maximal-time, linearized, and reconstruction defects. Thus our subject is the comparison between \[ K∘ F F∘ K, \] rather than functional analysis without KM data. Maximal monotone relations, pseudomonotone Galerkin methods, topological degree, semilinear evolution, nonlinear symmetry groupoids, Lyapunov–Schmidt reduction, and pitchfork bifurcation are incorporated only through explicitly typed comparison packages. Core solvability does not imply a source lift, nonempty nonlinear fibers are not automatically torsors, and spectral stability does not imply nonlinear stability. For split cores, an operator decomposes into core, contamination, return, and redundant blocks. A Schur complement computes the corrected core, and an explicit cokernel class gives the obstruction to lifting a core solution. For closed redundancies with strict graph-domain quotient exactness, restriction, full, and quotient operators form a short exact sequence of two-term Banach complexes. Its canonical six-term homology sequence gives Fredholm two-out-of-three, index additivity, resolvent inclusion, and exact Riesz-space descent without choosing complements. On noncompact elliptic cores, interior estimates are combined with compact transition control, coercive or invertible end models, and adjoint estimates. Essential-spectrum and index comparison require explicitly transparent redundant restrictions. Typed analytic comparison records compose through a defect cocycle, and their domain maps carry a seven-term exact sequence with a genuine two-out-of-three law. The volume retains eight chapters in analytic dependency order. Throughout, a priori estimates are not promoted to existence theorems, weak solutions are not promoted to strong solutions, formal symmetry is not promoted to self-adjointness, index stability is not promoted to separate kernel and cokernel stability, and vanishing lifting obstructions do not select canonical lifts. ## Keywords **KM space theory; analytic screening; operator defects; unbounded operators; Fredholm theory; elliptic operators; boundary-value problems; Sobolev spaces; semigroup theory; differential equations; nonlinear solution spaces; reverse lifting; obstruction classes; kernel torsors; spectral comparison.**