Fourier integral operators

This course develops oscillatory integrals, Lagrangian distributions and their applications to differential equations.

Stationary phase and critical manifolds

Read the stationary-phase lesson and its complete prerequisites.

Intrinsic regularity and principal symbols

Read the intrinsic components: frequency graphs, ordinary and clean phases, Sobolev improvement, Maslov lines, and the global symbol map, with worked examples and complete selected prerequisites.

Tangent zoom and intrinsic Gaussian symbols

Read the tangent lesson and Gaussian-symbol proof, with six solved exercises, exact prerequisites and singular models in every rank.

Fourier-integral calculus and symbol foundations

Read the calculus and symbol proofs, including geometric and Maslov prerequisites, ordinary composition and mapping, compactness, general homogeneous maps, complete symbol spaces, twenty-four solved exercises and exact proof links.

Restored: oscillatory distributions and their order

Read the complete restored lesson: derivative-loss oscillatory construction, wavefront inclusion, clean-phase order, stabilization and five solved exercises. The preserved argument is reconnected to the current exact prerequisite proofs, with its source credited alongside the free treatments. Earlier alternative proofs remain available. Sources and restoration record.

Restored: phase space and generating families

Read the restored geometry lesson: symplectic reduction, Darboux coordinates, contact slices, clean phases, generating functions and canonical composition, with five solved exercises. Complete flow and differential-form companions supply the exact prerequisites. The separate AN-03 flow extract retains its GFDL licence and notices. Sources and restoration record.

Restored: homogeneous phase equivalence and stabilization

Read the full phase-equivalence lesson: actual homogeneous changes, varying-rank minimal phases, signed quadratic stabilization and exact amplitude Jacobians. Includes two nonlinear examples, a reproducible figure and twelve solved exercises. Sources and restoration record.

Restored: intrinsic Lagrangian regularity

Read the full intrinsic-regularity lesson: coordinate and bundle invariance, matrix-valued tests, the frequency-graph criterion, prescribed clean phases and strict Sobolev improvement. Includes five solved exercises and a precise endpoint counterexample. Sources and restoration record.

Restored: tangent zooms and quadratic models

Read the full tangent-zoom lesson: two-scale localization, nonconvergent ordinary-symbol coefficients, regular-covector decay, nonlinear coordinates and singular Gaussian models. Includes all six solved exercises and the complete moving-test prerequisites. Sources and restoration record.

Restored: Gaussian lines and invariant symbols

Read the full Gaussian and principal-symbol lesson: Lagrangian intersection strata, Gaussian solution lines, the Maslov cocycle, critical densities, the global symbol isomorphism and antilinear bundle maps. Includes all five solved exercises and full localization and global-partition proofs. Sources and restoration record.

Restored: reduction and linear symbol composition

Read the full linear reduction and composition lesson: phase radicals, density-valued Gaussians, canonical reduction, staged constraints, exact normalization, graph transport and identity units. Includes all six solved exercises and a smoothness proof through changes of phase-Hessian rank. Sources and restoration record.

Restored: kernels, adjoints and clean composition

Read the full ordinary analytic-composition lesson: reflected kernel covectors, strong distribution mapping, adjoints, proper clean matching, exact excess normalization and fiber symbols. Includes all six solved exercises and the complete scalar kernel and topology proof, retaining its component licence. Sources and restoration record.

Restored: Hamilton fields and subprincipal transport

Read the full scalar transport lesson: Hamilton tangency, half-density differentiation, separate frequency coordinates, the exact lower-order symbol and periodic transport. Includes all six solved exercises and the complete scalar subprincipal coordinate proof, retaining its component licence. Sources and restoration record.

Restored: Graph operators, continuity and Egorov

Read the graph-operator lesson: local L2 continuity, compactness, all real Sobolev orders, elliptic inverses, ordered Egorov and full matrix lower-term removal. Includes all eleven solved exercises and the complete global inverse and circle multiplier proofs. Sources and restoration record.

Restored: Corank geometry and sufficient continuity

Read the corank lesson: two projection kernels and their shared symplectic quotient, homogeneous coordinates preserving a prescribed tangent map, sufficient bounds when ranks change, all real Sobolev orders and flat-family sharpness. Includes all eight solved exercises, explicit flow and parameter-integration proofs, and six reproducible finite model checks. Sources and restoration record.

Restored: Homogeneous submanifold normal forms

Read the submanifold lesson: symplectic, characteristic and normal directions; exact homogeneous canonical coordinates; characteristic leaves and reduced forms; constant-rank canonical relations and sharp continuity. Includes nine complete solutions, the exact coordinate figure, complete supporting flow, bundle and leaf proofs, and six reproducible finite model checks. Sources and restoration record.

Restored: Cubic scaling and necessary continuity

Read the cubic-scaling lesson: one fixed operator and concentrated inputs, the surviving cubic phase, frequency cancellation, a nonzero limiting profile and the necessary corank bound. Includes the exact changing-rank example, all nine complete solutions and six reproducible exact and numerical model checks. Sources and restoration record.

Restored: Folds, reflections and uniform smooth descent

Read the fold and descent lesson: exact fold coordinates and reflection; smooth descent and extension with all parameter estimates; ordinary symbols and critical amplitude realization. Includes nine complete solutions, the exact fold diagram, the homogeneous phase and amplitude-order proof, and six reproducible finite model checks. Sources and restoration record.

Restored: Simultaneous reflections and flat corrections

Read the simultaneous-reflection lesson: two distinct sheet exchanges, full smooth coordinate construction, parameter Borel realization, flat correction with every derivative, and homogeneous coordinates on the positive ray. Includes nine complete solutions, the exact reflection and shear diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Folded forms and symplectic targets

Read the folded-form lesson: ordinary and homogeneous normal forms, the smooth relative Moser construction, symmetry under a sheet exchange, full symplectic target coordinates and closed quotient extensions. Includes ten complete solutions, the exact signed-area diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Two reflections in folded symplectic coordinates

Read the simultaneous symplectic lesson: two sheet exchanges preserving a folded form, full ordinary and homogeneous coordinates, commuting-flow constructions and exact nonlinear normalization. Includes ten complete solutions, the original shift and cancellation diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Canonical relations with two folding projections

Read the two-sided fold lesson: complete ordinary and homogeneous relation normal forms, both full target charts, the exact radial hypothesis and a nondegenerate conic phase. Includes ten complete solutions, the original two-sheet diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Fold amplitudes and critical densities

Read the fold-amplitude lesson: exact normalization, two radial density factors, reduced homogeneous phase, smooth first-frequency tail and complete reduced-amplitude representation. Includes ten complete solutions, the original density diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Airy functions and fold model operators

Read the Airy model lesson: every derivative estimate, complete two-coefficient representation, full smoothing tail and endpoint L2 bound. Includes the complete AN-03 bounded-derivative proof with its licence notices, ten complete solutions, the original numerical illustration, and six reproducible finite model checks. Sources and restoration record.

Restored: Invariant fold continuity and Sobolev transfer

Read the invariant fold theorem: complete canonical localization, both full inverse recoveries, all real Sobolev orders, finite-rank bundles and the sharp universal order threshold. Includes ten complete solutions, the exact operator transfer diagram, and six reproducible finite model checks. Sources and restoration record.

Restored: Prescribed canonical coordinates and isotropic fibers

Read prescribed coordinates and isotropic fibers: exact ordinary and homogeneous coordinate extensions, marked values and derivatives, both isotropic induction cases, a complete homogeneous rotation, and conormal germs. Includes ten complete solutions, the lifted cusp illustration, and six reproducible finite checks. Sources and restoration record.

Restored: Clean Lagrangian pairs and common transversals

Read clean Lagrangian pairs: complete conic and ordinary simultaneous normal forms, exact clean cylinder reductions, full marked coordinate maps and inverses, and a common transverse Lagrangian plane. Includes ten complete solutions and six reproducible finite checks. Sources and restoration record.

Restored: Real and complex symplectic function normal forms

Read real and complex function normal forms: complete real, simple complex, finite-contact and involutive models, smooth complex division, and the common-domain parameter solver. Includes fourteen complete solutions, the weighted-flow and contact-sign illustration, and seven reproducible finite checks. Sources and restoration record.

Restored: Quadratic Hamilton maps and positive complex planes

Read quadratic Hamilton maps and positive planes: full real nonnegative and index-one classification, degenerate blocks, sectorial complex geometry, positive spectral planes, compatible structures and non-strict reduction. Includes fourteen complete solutions, the original flow illustration, and the complete spectral algebra and contour proofs. Seven original finite checks and two added spectral checks accompany the written proofs. Sources and restoration record.

Restored: Positive Lagrangian ideals and distributions

Read positive ideals and distributions: exact complex division and elimination, uniform symbol residues, damping, wavefront containment and the complete normal representation with its converse. Includes seventeen complete solutions, two exact illustrations, the full smooth complex stationary-phase proof, and contour and Gaussian foundations. Original finite checks and additional foundation checks accompany the written proofs. Sources and restoration record.

Restored: Complex-phase Fourier integral operators

Read complex-phase Fourier integral operators: complete transverse ideal and operator composition, exact supports and orders, and both directions of the universal local L² criterion. Includes seventeen solved exercises, the original positive-plane and damping illustration, and a complete half-order endpoint proof with its scale and summation illustration. Retained finite checks and endpoint controls accompany the written proofs. Sources and restoration record.

Restored: Maslov index, crossings and global phase

Read Maslov index, crossings and global phase: complete integer classification of Lagrangian-plane loops, exact topology and lifting proofs, finite signed crossings and the coefficient/frame/dual convention. Includes fourteen solved exercises and the original determinant and branch illustration. Finite model checks and sources and restoration record accompany the full proofs.

Restored: Singularities along a real characteristic direction

Read singularities along a real characteristic direction: full signed kernels and exact wavefront, a commuting transport-tube cutoff, and smooth propagation for every real order, including radial and zero Hamilton fields. Includes three solved exercises, the original directed-flow illustration and the exact tensor-wavefront and projection proof. Finite checks and sources and restoration record accompany the full proofs.

Energy and existence foundations for Cauchy problems

The positive-quantization and sharp lower-bound proof includes global Sobolev bounds and strong time continuity. The integration and duality companion supplies Hahn–Banach, all-exponent integral estimates, Bochner integration, the complete time primitive and trace, and Hilbert-valued duality. Both retain their AN-03 component notices. Sources and exact proof scope and finite checks accompany the written proofs. The restored U030 receiving lesson below uses these complete proofs.

Restored: First-order Cauchy problems

The complete lesson proves scalar energy and existence, the full real Hamilton flow, a commuting symbol with both smoothing errors, fixed-time propagation and conditional spacetime/slice equality. All sixteen original solved exercises remain. The mixed spacetime composition proof supplies both multiplication orders and every remainder. The pullback and trace companion retains its AN-03 notices and proves the actual continuous trace. Sources and proof review and exact proof map accompany the lesson.

Mixed half-space spaces and boundary traces

The two-weight Hilbert and trace proofs give the full real-index spaces, multiplier domains and exact boundary constants. The support and duality companion proves both half-space dualities, one-sided order changes, minimal extensions, integer jets, tangential operator bounds and differential normal recovery. These complete prerequisite proofs retain their component licences and credit. Sources and proof scope and finite model checks are included. The higher-order Cauchy lesson U031 is now restored with its complete receiving proof and all 23 original solutions.

Conormal test spaces and intrinsic boundary jets

The conormal amplitude and test-space proofs include the full Besov endpoint, amplitude characterization and completeness. The dual conormal companion constructs the actual boundary trace and corrected derivatives. The smoothness proof retains every scaled test estimate, boundary jet and ordinary derivative, and identifies the original supported distribution. These forty prerequisite proofs retain their component licences. Sources and proof scope and finite checks are included. U031 is now restored with its complete receiving proof and all 23 original solutions.

Normal extension and the complete boundary defect

The normal-extension proof constructs the actual supported extension at every conormal test order, proves uniqueness among all supported extensions satisfying the weighted equation, and retains every coefficient in the full boundary defect. It includes finite nested supports, both Volterra transpose conventions, the complete companion matrix and invariant gluing under the precise noncharacteristic hypotheses. These eighteen prerequisite proofs have 350 exact earlier dependencies. Source and proof scope and finite checks are included. The full U031 is now restored with its complete receiving proof and all 23 original solutions.

Local boundary calculus: kernels, composition and conormal action

The local calculus connects test spaces and lacunary symbols, resolved corner kernels, adjoints, composition and distributional action, and bounds and conormal continuity. Complete proofs retain every finite remainder, all boundary jets, both Sobolev interpolation arguments and the residual no-gain obstruction, with five original solved exercises. These 54 prerequisite entries connect to 351 exact earlier proofs. Source and proof scope and finite checks are included. The full U031 is now restored with its complete receiving proof and all 23 original solutions.

Boundary operators of every nonnegative real order

The complete positive-order proof gives the exact loss of derivatives for every nonnegative real operator order on both original half-space Sobolev scales, at every real index and for rectangular systems. It retains the full residual and integer decomposition, proves complex one-sided multipliers and the strip estimate, and obtains the original restriction quotient by adjoint duality. All seventeen original numbered displays and the corrected source figure are included. These seventeen entries connect to 295 exact earlier proofs. Source and proof scope and finite checks are available. The complete U031 is now restored with its complete receiving proof and all 23 original solutions.

Intrinsic conormal symbols and polyhomogeneous corner kernels

The intrinsic conormal symbol proof retains the complete half-density coordinate law, principal quotient and ordered operator action with every finite remainder. The corner-kernel proof establishes both directions of the polyhomogeneous characterization, retaining logarithmic terms until support eliminates them and excluding hidden corner distributions. These 24 entries connect to 340 exact earlier proofs. All thirteen original mathematical displays and the respective component notices are retained. Source and proof scope and finite checks are included. The full U031 is now restored with its complete receiving proof and all 23 original solutions.

Global boundary operators and compressed kernel geometry

The geometry companion gives the full stretched kernel, compressed covectors, half-density law and inverse reconstruction. The operator companion proves proper composition, adjoints, asymptotic sums, all-real Sobolev bounds, conormal receiving, dual action, every boundary jet and the ordered elliptic parametrix. These 36 entries connect to 434 exact earlier proofs. All 51 original mathematical displays and the original coordinate figure are retained. Source and proof scope and finite checks are included. The full U031 is now restored with its complete receiving proof and all 23 original solutions.

Compressed wave fronts, tangential action and normal extension

The compressed wave-front proof gives the full residual, elliptic and differential inclusions, with the exact noncompact local-order distinction. The normal and tangential companion proves the noncharacteristic extension conclusion, complete conormal topology, boundary microsupport and elliptic comparison. The smooth-tester companion proves the full smoothness and trace consequences. All 46 original displays and the pure-normal interior exception are retained. These 37 entries connect to 498 exact earlier proofs. Source and proof scope and finite checks are included. U031 is now restored with its complete receiving proof and all 23 original solutions.

Restored: Higher-order Cauchy problems

The complete lesson proves energy and existence for every real Sobolev order, full mixed-space and boundary traces, supported and semiglobal solutions, exact all-order factorization, forced propagation and intrinsic boundary wave-front equality. The backward-access condition and incoming-edge counterexample are retained. All 23 original solved exercises and 95 original displays remain. The final division, weighted adjoint uniqueness and beta/Gamma trace constant are proved explicitly. Its 65 entries connect to 620 exact earlier proofs. Exact proof map, sources and scope, all 195 retained obligations and finite checks accompany the editable lesson.

Convex supports and constant-coefficient inverses

The convex-support companion proves the complete finite-jet extension and the distribution estimate at the original derivative order. The constant-coefficient companion constructs a fundamental solution for every nonzero polynomial operator, including nonelliptic ones, and proves the fixed local inverse with all parameter bounds. An exact diagram and its editable source illustrate both mechanisms. Their ten entries connect to 316 earlier proofs. Exact proof map, source and scope, and finite checks accompany the editable text. U032 is now restored with its complete receiving proof and all twenty original solutions.

Restored: Supported Cauchy solvability

The complete lesson proves that one-sided support forces real normal roots and positive cone support forces a noncharacteristic normal. It then proves the real principal-type alternative: simple interior roots and positive double contact on the boundary. The full compact-source hypothesis, every adjoint lower term, dilation Jacobian, finite transport remainder and cone damping estimate are retained. Complete support, nonelliptic inverse and circle-count proofs are included. All 20 original solutions and 54 original displays remain. Its 47 entries connect to 480 exact earlier proofs. Exact proof map, sources and scope, all 73 obligations and finite checks accompany the editable lesson and its root diagram.

Scalar splitting and finite-matrix positivity

The scalar companion proves the uniform jet and splitting lemmas, squared partitions and the metric selected by the value and Hessian. The finite-matrix companion proves the sharp lower bound for every fixed finite system, with full differentiated remainders and Sobolev energy shifts. An exact curved-graph diagram has an editable source. Twelve entries connect to 252 earlier proofs. Exact proof map, sources and scope, and finite checks accompany the editable text. The scalar Fefferman–Phong theorem is completed below; U033 is now restored with its complete receiving proof and all twenty original solutions.

Metric Weyl products and symplectic covariance

Three companions prove the compatible-metric Weyl product and all finite remainders, Schwartz and distribution action with affine symplectic covariance, and reflection-compatible changes of quantization. Exact affine domains, every Fourier factor and the full original topology remain. Thirty-two entries connect to 250 earlier proofs. An exact metric-shear diagram has editable source. The proof map, sources and scope, and finite checks accompany the text. The operator bound and Fefferman–Phong proof are completed in the next companion; U033 is now restored with its complete receiving proof and all twenty original solutions.

Metric operator bounds and scalar positivity

The uniform Hilbert-valued metric operator bound and scalar Fefferman–Phong theorem are complete, with the full coefficient calculus, squared localization, dimension induction, finite-derivative limits and classical quantization signs. Eighteen entries connect to 280 earlier proofs. An exact square-defect figure explains how a nonnegative symbol can have negative expectations. The proof map, sources and scope, and finite checks accompany the editable text. U033 is now restored below, including all twenty original solutions.

Restored: Principal-type boundary Cauchy problems

The complete lesson proves the forward singular-weight and backward weighted estimates, reconstructs every normal jet, and constructs supported solutions at all Sobolev orders. It includes the full tangential model, decreasing-cutoff half-derivative gain, boundary wave-front equality on one fixed collar and complete factorization. Exact weak-subsequence and quotient-inverse proofs fill the prerequisites. All 20 original solutions and 93 original displays remain. Its 70 entries connect to 715 exact earlier proofs. Exact proof map, sources and scope, all 87 obligations and finite checks accompany the editable lesson and its Hamilton-curve and nested-collar diagram.

Restored: Sobolev propagation

The complete lesson proves propagation at the exact Sobolev order, with the full gain of m minus one derivatives. It uses distributional averaging, an actual L2 primitive, one compact transport tube and full canonical conjugation, including all operator orders and inverse defects. Radial curves and zero Hamilton fields are included. The three original complete solutions and all 15 original displays remain. Its 26 entries connect to 511 exact earlier proofs. An exact threshold diagram explains why the critical order need not be attained. The proof map, sources and scope, all 20 obligations and finite checks accompany the editable lesson.

Restored: One compact characteristic tube

The complete lesson constructs one homogeneous canonical chart along a compact characteristic whose positive covector directions do not repeat. It proves uniform flow continuation, full symplectic and primitive identities, global injectivity, intrinsic Maslov-symbol gluing, both graph inverses and all lower-term conjugation identities on one finite cylinder. All 22 original displays and three original complete solutions remain. Its 25 entries connect to 505 exact earlier proofs. The proof map, sources and scope, all 20 obligations and finite checks accompany the editable text and exact base-flow and cotangent-scale diagram.

Restored: Prescribed finite singular rays

The complete lesson constructs a distribution singular on exactly one closed characteristic segment, with forcing singularities at exactly its two endpoints. Its exact Sobolev threshold is attained nowhere on the segment, while all lower orders hold. The Gaussian profile, explicit shrinking-support partition, convergent regularization with every conic derivative estimate and full canonical transfer are proved. All 15 original displays, three complete original solutions and the original frequency-cone diagram remain. Its 26 entries connect to 553 exact earlier proofs. The proof map, sources and scope, all 23 obligations and finite checks accompany the editable text.

Restored: Solvability near a compact set

The complete lesson proves existence near a compact set when every complete characteristic escapes. It identifies the smooth finite-dimensional adjoint obstruction, the exact gain of m minus one derivatives at every real Sobolev order, and one smooth solution for smooth data. Complete companions supply closed graphs and smooth quotients and fixed-support compactness and duality, with their GFDL notices. All 32 original mathematical displays and three full solutions remain; one differential typesetting slip is corrected. Its 41 entries connect to 629 exact earlier proofs. The proof map, sources and scope, all 26 obligations and finite checks accompany the editable texts and exact characteristic diagrams.

Restored: Global solvability and characteristic excursions

The complete lesson proves when every distribution admits a global solution modulo a smooth error. It connects that existence to adjoint singular-support bounds and control of complete characteristic excursions. The proof includes the escaping half-ray obstruction, every step of the global test-space estimate, the actual Hahn–Banach extension and its locally finite smooth residual. All 19 original displays, three full solutions and the slit-domain diagram remain. Its 31 entries connect to 723 exact earlier proofs. The proof map, sources and scope, all 27 obligations and finite checks accompany the editable source.

Restored: Global time and the bicharacteristic relation

The complete lesson constructs a smooth trajectory space, one global section and a positive radial coordinate constant along the flow. It proves the full proper-relation equivalences, scalar transport with every symbol derivative, and the closed homogeneous canonical relation with its exact kernel sign. All 12 original displays and three complete solutions remain. A new exact diagram shows radial drift becoming constant radius, including the excluded endpoints. Its 33 entries connect to 727 exact earlier proofs. The proof map, sources and scope, all 27 obligations and finite checks accompany the editable source and reproducible diagram.

Restored: Directional transport for characteristic symbols

The complete lesson solves transport on the full Maslov and half-density symbol line, with zero characteristic-diagonal data and every moving-endpoint derivative. Its closed directional hull retains the forcing boundary; an exact kernel example proves why zero travel time is necessary. All 14 original displays, three full solutions and the source-boundary diagram remain. Its 23 entries connect to 770 exact earlier proofs. The proof map, sources and scope, all 17 obligations and finite checks accompany the editable source.

Restored: From directional symbols to global kernel corrections

The complete lesson constructs actual kernel corrections with wavefront in the closed directional hull. It proves controlled symbol realization, every asymptotic remainder, ordinary order-zero transport and the complete residual induction. All 19 original displays and three complete solutions remain. A new explicit annular-bump diagram accompanies the proof that an infinite sum of smooth pieces can be singular. Its 26 entries connect to 782 exact earlier proofs. The proof map, sources and scope, all 17 obligations and finite checks accompany the editable source and reproducible diagram.

Restored: Local signed parametrices and their delayed errors

The complete lesson preserves the entire diagonal jump while placing the signed cutoff error at positive travel time. It proves the actual proper kernel construction, full conjugation identities, exact residual and difference orders, and continuity on every real Sobolev scale. All 25 original displays, three complete solutions and the coordinate diagram remain. Its 27 entries connect to 816 exact earlier proofs. The proof map, sources and scope, all 15 obligations and finite checks accompany the editable source.

Restored: Comparing global parametrices through a compact middle

The complete lesson defines the compact-middle product and proves its exact wavefront bound. Compact return then gives same-sign uniqueness, and the correct adjoint sign supplies the missing parametrix identity once both signed right constructions exist. All 18 original displays, three full solutions and the exact cutoff diagram remain. Its 25 entries connect to 763 exact earlier proofs. The proof map, sources and scope, all 18 obligations and finite checks accompany the editable source.

Restored: Global signed parametrices from local delayed errors

The complete lesson assembles the local kernels and corrects their full residual. It proves both inverse identities, uniqueness, exact signed wavefront sets, the noncharacteristic difference and the precise Sobolev gain for every real operator order, including opposite orientations on disconnected characteristic parts. All 36 original displays, three full solutions and the exact source-and-hull diagram remain. Its 37 entries connect to 868 exact earlier proofs. The proof map, sources and scope, all 25 obligations and finite checks accompany the editable source.

Restored: First-order systems and ordered evolution

The complete lesson proves energy estimates, existence, the initial trace, uniqueness and ordered evolution for separable Hilbert coefficients. It includes integrable forcing, the precise reverse-time condition and a fixed positive symmetrizer. The complete prerequisite companion proves the Hilbert Fourier scale, ordered calculus, packet and Sobolev bounds, and sharp positivity with constants independent of coefficient dimension. All 40 original displays, three full solutions and the exact two-front diagram remain. Its 44 proof entries connect to 646 exact earlier proofs. The proof map, sources and scope, all 31 obligations and finite checks accompany both editable sources.

Restored: Oscillatory Cauchy kernels and exact evolution

The complete lesson constructs the short-time phase and full ordinary matrix amplitude, then corrects both the initial error and equation residual to obtain the exact Cauchy evolution. It proves every differentiated frequency remainder, global smoothing estimate, ordered transport and half-density factor. Finite-time graph composition retains Maslov data and proves exact wavefront equality for a scalar homogeneous principal direction with arbitrary matrix lower terms. All 40 original displays, three full solutions and the sampled base/cotangent diagram remain. Its 49 proof entries connect to 539 exact earlier proofs. The proof map, sources and scope, all 43 obligations and eight finite checks accompany the editable source and reproducible figure.

Restored: Separated characteristic branches and polarization

The complete lesson reduces a Hermitian principal system with uniformly separated single-eigenvalue branches, retaining full ordinary matrix lower terms and all time-dependent inverse defects. It constructs local frames, full commuting projections and exact branch kernels, controls every global smoothing error, and joins the polarization bundles without a global eigenbasis. It proves the exact kernel wavefront and projected-data propagation on compact trajectory families with the stated gap. All 42 original displays, four complete solutions and the exact ray/polarization diagram remain. Its 55 proof entries connect to 574 exact earlier proofs. The proof map, sources and scope, all 51 obligations and nine finite checks accompany the editable source and reproducible figure.

Restored: Positive symmetrizers and non-Hermitian evolution

The complete lesson proves the global Cauchy theorem for a smooth uniformly positive variable metric. It constructs positive symbol square roots, a coercive quantized energy, regularized solutions and the full ordinary reduction to a Hermitian principal system. The separated-branch extension retains nonorthogonal polarization, complete global smoothing corrections and exact projected wavefront. All 50 original displays, five complete solutions and the exact component-energy ellipse remain. Its 55 proof entries connect to 623 exact earlier proofs. The proof map, sources and scope, all 53 obligations and nine finite checks accompany the editable source and reproducible figure.

Restored: Energy and existence with a timelike Dirichlet boundary

The complete lesson proves the supported mixed Dirichlet theorem for every real Sobolev order s >= 0. It retains the inward boundary flux, weighted energy, rough uniqueness, one-sided adjoint estimate, dual existence and actual boundary trace, then proves localization and global assembly across compact time slabs. All 48 original displays and four complete solutions remain. Its 48 proof entries connect to 527 exact earlier proofs. The proof map, sources and scope, all 45 obligations and six finite checks accompany the editable source and a reproducible flat-boundary and energy diagram.

Continuing course work

Analytic FIO estimates in symbol classes with derivative losses, distribution-space topology and completeness, propagation and the remaining course are unfinished. The ordinary intrinsic characterization is proved in the restored lesson above.