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Theorems and mechanisms

Locate theorem and mechanism treatments without losing their hypotheses, applicability boundaries, failure cases, or proof status.

This index is a concise projection, not an independent explanation. Each record keeps the scope needed for use and links to its source page.

Coverage notes
  • A page category alone does not supply a theorem statement, hypotheses, proof, or mechanism record, so this view routes to the full treatment instead of inventing them.
  • Mechanisms appear only when a reviewed record identifies their defining data and scope.

Coverage note: records included through 2026-08-12

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  • Theorem Mechanism

    Holomorphic Functions and Cauchy Theory

    Why does holomorphy let local derivative data constrain contour integrals and global values?

    Principal question
    Why does holomorphy let local derivative data constrain contour integrals and global values?
    What the source page covers
    Holomorphic Functions and Cauchy Theory at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Holomorphic Functions and Cauchy Theory at reusable, hypothesis-aware mathematical-methods depth.
  • Theorem Mechanism

    Differential Forms, Integration, Orientation, and Stokes Theorem

    How do forms encode flux, integration, boundaries, and conservation laws coordinate-independently?

    Principal question
    How do forms encode flux, integration, boundaries, and conservation laws coordinate-independently?
    What the source page covers
    Differential Forms, Integration, Orientation, and Stokes Theorem at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Differential Forms, Integration, Orientation, and Stokes Theorem at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Smooth Manifolds, Tangent and Cotangent Bundles, and Tensor Fields
  • Theorem Mechanism

    Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems

    Which convergence notion and limit theorem justify an estimator or fluctuation approximation?

    Principal question
    Which convergence notion and limit theorem justify an estimator or fluctuation approximation?
    What the source page covers
    Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Probabilistic Convergence, Laws of Large Numbers, and Central Limit Theorems at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Probability Spaces, Random Variables, and Conditional Expectation
  • Theorem Mechanism

    Sturm–Liouville Problems and Eigenfunction Expansions

    When does a boundary-value problem yield orthogonal modes and a useful completeness relation?

    Principal question
    When does a boundary-value problem yield orthogonal modes and a useful completeness relation?
    What the source page covers
    Sturm–Liouville Problems and Eigenfunction Expansions at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Sturm–Liouville Problems and Eigenfunction Expansions at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Linear ODEs, Evolution Operators, and Wronskians
  • Theorem Mechanism

    de Rham Cohomology, Periods, Duality, and Intersection

    How do closed forms, periods, and dual cycles turn local differential data into global invariants?

    Principal question
    How do closed forms, periods, and dual cycles turn local differential data into global invariants?
    What the source page covers
    de Rham Cohomology, Periods, Duality, and Intersection at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    de Rham Cohomology, Periods, Duality, and Intersection at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Differential Forms, Integration, Orientation, and Stokes Theorem
  • Theorem Mechanism

    Lebesgue Integration and Convergence Theorems

    When do monotone or dominated convergence justify taking a limit through an integral?

    Principal question
    When do monotone or dominated convergence justify taking a limit through an integral?
    What the source page covers
    Lebesgue Integration and Convergence Theorems at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Lebesgue Integration and Convergence Theorems at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Measures and Measurable Functions
  • Theorem Mechanism

    Gaussian Vectors, Processes, Random Distributions, and Wick Structure

    Why are Gaussian correlations fixed by mean and covariance, and which infinite-dimensional claims need extra care?

    Principal question
    Why are Gaussian correlations fixed by mean and covariance, and which infinite-dimensional claims need extra care?
    What the source page covers
    Gaussian Vectors, Processes, Random Distributions, and Wick Structure at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Gaussian Vectors, Processes, Random Distributions, and Wick Structure at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Characteristic Functions, Moments, Cumulants, and Generating Functionals; Bilinear and Hermitian Forms, Adjoints, and Isometries
  • Theorem Mechanism

    Product Measures, Fubini–Tonelli, and Change of Variables

    When may iterated integrals be reordered, and how does a measure transform under a change of variables?

    Principal question
    When may iterated integrals be reordered, and how does a measure transform under a change of variables?
    What the source page covers
    Product Measures, Fubini–Tonelli, and Change of Variables at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Loop integration belongs to Scattering; anomalous functional Jacobians belong to Symmetry and Gauge Structure.
    Scope
    Product Measures, Fubini–Tonelli, and Change of Variables at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Lebesgue Integration and Convergence Theorems
  • Theorem Mechanism

    Normal Forms, Spectra, and Projectors

    Which finite-dimensional operators admit useful normal forms and spectral projectors, and what fails for non-normal maps?

    Principal question
    Which finite-dimensional operators admit useful normal forms and spectral projectors, and what fails for non-normal maps?
    What the source page covers
    Eigenvalues, algebraic versus geometric multiplicity, diagonalization, finite spectral theorem, Jordan form, SVD orientation, and projectors.
    Boundary
    Infinite-dimensional spectra and spectral measures belong to Functional and Spectral Analysis.
    Scope
    Eigenvalues, algebraic versus geometric multiplicity, diagonalization, finite spectral theorem, Jordan form, SVD orientation, and projectors.
    Assumptions
    Vector Spaces, Duals, Linear Maps, and Bases
  • Theorem Mechanism

    Fredholm and Dirac Index Theorems and Zero-Mode Counting

    How can analytic zero-mode data be related to topological characteristic data?

    Principal question
    How can analytic zero-mode data be related to topological characteristic data?
    What the source page covers
    Fredholm and Dirac Index Theorems and Zero-Mode Counting at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Anomaly calculations and physical consequences belong to Symmetry and Gauge Structure and Mathematical QFT.
    Scope
    Fredholm and Dirac Index Theorems and Zero-Mode Counting at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Spectra, Resolvents, Spectral Measures, and Functional Calculus; Characteristic Classes and Chern–Weil Theory; Spin Structures and Dirac Operators
  • Theorem Mechanism

    Self-Adjointness, Extensions, and Unitary Evolution

    When does a symmetric operator admit a self-adjoint realization and generate unitary time evolution?

    Principal question
    When does a symmetric operator admit a self-adjoint realization and generate unitary time evolution?
    What the source page covers
    Self-Adjointness, Extensions, and Unitary Evolution at reusable, hypothesis-aware mathematical-methods depth.
    Boundary
    Developed physical applications, theorem-first frameworks, and production software remain with their canonical destinations.
    Scope
    Self-Adjointness, Extensions, and Unitary Evolution at reusable, hypothesis-aware mathematical-methods depth.
    Assumptions
    Unbounded Operators, Domains, Closure, and Adjoints
  • Theorem Mechanism

    Wick’s Theorem and Free Gaussian Factorization

    How do operator ordering and Gaussian source differentiation produce bosonic and fermionic Wick factorization with correct signs?

    Principal question
    How do operator ordering and Gaussian source differentiation produce bosonic and fermionic Wick factorization with correct signs?
    What the source page covers
    The free operator and Gaussian factorization theorem, contractions, and convention-sensitive signs.
    Boundary
    Dyson expansion, interacting Wick expansion, Feynman rules, and diagrammatics belong to Scattering.
    Scope
    The free operator and Gaussian factorization theorem, contractions, and convention-sensitive signs.
    Assumptions
    The Generating Functional; Gaussian Fields and Sources
  • Theorem Mechanism

    The Källén–Lehmann Representation

    Which vacuum, translation and Poincaré covariance, completeness, spectrum, and Hilbert-space positivity assumptions yield the scalar Källén–Lehmann representation, and which parts do not require locality?

    Principal question
    Which vacuum, translation and Poincaré covariance, completeness, spectrum, and Hilbert-space positivity assumptions yield the scalar Källén–Lehmann representation, and which parts do not require locality?
    What the source page covers
    The physical derivation, normalization, and interpretation of the scalar Källén–Lehmann representation.
    Boundary
    Gauge-variant propagators need not have a positive density; rigorous spectral support belongs to Mathematical QFT.
    Scope
    The physical derivation, normalization, and interpretation of the scalar Källén–Lehmann representation.
    Assumptions
    Spectral Decomposition of Two-Point Functions
  • Theorem Mechanism

    Haag’s Theorem: Physical Meaning and Scope

    Under precisely which covariance, irreducibility, vacuum, equal-time, and representation assumptions does Haag’s theorem obstruct a global interaction picture?

    Principal question
    Under precisely which covariance, irreducibility, vacuum, equal-time, and representation assumptions does Haag’s theorem obstruct a global interaction picture?
    What the source page covers
    A careful physical theorem statement, hypothesis audit, common misstatements, and practical interpretation.
    Boundary
    Proof, theorem variants, representation-theoretic setting, and constructive implications belong to Mathematical QFT.
    Scope
    A careful physical theorem statement, hypothesis audit, common misstatements, and practical interpretation.
    Assumptions
    Fock Space, Vacuum, and Particle Number; Regulators, Cutoffs, and Continuum Limits; Hilbert Positivity and Unitary Evolution
  • Theorem Mechanism

    Reflection Positivity within Osterwalder–Schrader Reconstruction

    What positivity form does Euclidean time reflection define, what is it used to reconstruct, and which additional Osterwalder–Schrader hypotheses remain?

    Principal question
    What positivity form does Euclidean time reflection define, what is it used to reconstruct, and which additional Osterwalder–Schrader hypotheses remain?
    What the source page covers
    A physical introduction to the reflection operation, positivity form, reconstructed Hilbert-space intuition, and the fact that reflection positivity is only one OS axiom.
    Boundary
    Exact axiom packages, regularity variants, quotient construction, Hamiltonian reconstruction, analytic continuation, and proof belong to Mathematical QFT.
    Scope
    A physical introduction to the reflection operation, positivity form, reconstructed Hilbert-space intuition, and the fact that reflection positivity is only one OS axiom.
    Assumptions
    Euclidean Correlators and Schwinger Functions
  • Theorem Mechanism

    The Spin–Statistics Connection

    Which locality, covariance, spectrum, positivity, dimension, and field assumptions connect spin with exchange statistics?

    Principal question
    Which locality, covariance, spectrum, positivity, dimension, and field assumptions connect spin with exchange statistics?
    What the source page covers
    A physical theorem map, proof architecture, counterfactual tests, and explicit dimension or braid-statistics qualifications.
    Boundary
    Rigorous theorem variants, domain conditions, low-dimensional statistics, and superselection formulations belong to Mathematical QFT.
    Scope
    A physical theorem map, proof architecture, counterfactual tests, and explicit dimension or braid-statistics qualifications.
    Assumptions
    Microcausality and Relativistic Compatibility; Hilbert Positivity and Unitary Evolution; Canonical Quantization of the Free Dirac Field
  • Theorem Mechanism

    CPT: Hypotheses, Content, and Limits

    Under which locality, Lorentz covariance, spectral, positivity, and dimensional assumptions does CPT follow, and which common extensions require separate theorems?

    Principal question
    Under which locality, Lorentz covariance, spectral, positivity, and dimensional assumptions does CPT follow, and which common extensions require separate theorems?
    What the source page covers
    A physical theorem statement, hypothesis ledger, discrete-operation convention map, and failure modes.
    Boundary
    Rigorous PCT theorem variants and proof belong to Mathematical QFT; discrete-symmetry phenomenology belongs to Gauge Theories.
    Scope
    A physical theorem statement, hypothesis ledger, discrete-operation convention map, and failure modes.
    Assumptions
    Microcausality and Relativistic Compatibility; Poincaré Covariance and the Spectrum Condition; The Dirac Field
  • Theorem Mechanism

    Goldstone's theorem

    The canonical owner states the hypotheses and pole argument; this record routes readers there and warns against applying the conclusion without those hypotheses.

    Hypotheses
    Use the canonical page for the complete statement and pole derivation.
    Failure warning
    Gauge symmetries, finite volume, low-dimensional obstructions, spacetime-symmetry breaking, and nonrelativistic systems require separate statements.
    Canonical exposition
    Open the canonical exposition
    Aliases
    Goldstone theorem, Nambu–Goldstone theorem
    Scope
    Relativistic continuous global-symmetry breaking under the precise hypotheses stated by the canonical owner.
    Assumptions
    The symmetry is global rather than a gauge redundancy.; The charge, vacuum, locality, spectral, and infinite-volume hypotheses are checked by the owning treatment.
  • Theorem Mechanism

    Goldstone's Theorem: Hypotheses and Pole Argument

    Under which locality, current, state, infinite-volume, and spectral assumptions does a broken continuous global symmetry require gapless spectral weight?

    Principal question
    Under which locality, current, state, infinite-volume, and spectral assumptions does a broken continuous global symmetry require gapless spectral weight?
    What the source page covers
    The relativistic current-order-parameter proof, Goldstone pole, charge-existence subtlety, and an explicit hypothesis ledger.
    Boundary
    Nonrelativistic counting, spacetime breaking, low-dimensional obstructions, and the Higgs exception are treated separately; model dynamics belongs downstream.
    Scope
    The relativistic current-order-parameter proof, Goldstone pole, charge-existence subtlety, and an explicit hypothesis ledger.
    Assumptions
    Finite Volume, Thermodynamic Limits, and Pure Phases; Quantum Currents, Improvements, and Conservation
  • Theorem Mechanism

    Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions

    When does the naive one-mode-per-broken-generator rule fail or require modification?

    Principal question
    When does the naive one-mode-per-broken-generator rule fail or require modification?
    What the source page covers
    Relativistic counting assumptions, type-A/type-B orientation, finite-density pairing, spacetime symmetry and inverse-Higgs qualifications, low-dimensional no-breaking results, and long-range-interaction caveats.
    Boundary
    Many-Body owns developed nonrelativistic and finite-density counting; CFT owns scale or conformal breaking; EFT construction belongs to Renormalization and EFT.
    Scope
    Relativistic counting assumptions, type-A/type-B orientation, finite-density pairing, spacetime symmetry and inverse-Higgs qualifications, low-dimensional no-breaking results, and long-range-interaction caveats.
    Assumptions
    Goldstone's Theorem: Hypotheses and Pole Argument
  • Theorem Mechanism

    Cutkosky Cutting Rules

    How are discontinuities of Feynman integrals computed by putting internal lines on shell?

    Principal question
    How are discontinuities of Feynman integrals computed by putting internal lines on shell?
    What the source page covers
    Propagator discontinuity, cut measure, largest-time orientation, one-loop examples, signs, and relation to the optical theorem
    Boundary
    Renormalized cuts; Generalized unitarity reconstruction; Finite-temperature cutting rules
    Canonical treatment
    Cutkosky Cutting Rules
    Scope
    Propagator discontinuity, cut measure, largest-time orientation, one-loop examples, signs, and relation to the optical theorem
    Assumptions
    The Optical Theorem and Cut Interpretation; Landau Equations and Physical Singularities
  • Theorem Mechanism

    The Optical Theorem and Cut Interpretation

    How does forward-scattering unitarity equate an absorptive part with an inclusive total rate?

    Principal question
    How does forward-scattering unitarity equate an absorptive part with an inclusive total rate?
    What the source page covers
    Forward limit, total cross section, intermediate-state phase space, diagrammatic cut intuition, and normalization checks
    Boundary
    Cutkosky proof; Experimental luminosity; Hadronic optical theorem applications
    Scope
    Forward limit, total cross section, intermediate-state phase space, diagrammatic cut intuition, and normalization checks
    Assumptions
    S-Matrix Unitarity; Cross Sections and Decay Rates
  • Theorem Mechanism

    Soft Theorems

    How does emission of a low-energy massless quantum factorize from a hard scattering process?

    Principal question
    How does emission of a low-energy massless quantum factorize from a hard scattering process?
    What the source page covers
    Leading soft factors, Abelian and non-Abelian orientation, gravity comparison, external-line origin, and subleading caveats
    Boundary
    Asymptotic-symmetry proof; Memory effects; Loop-corrected soft theorems as a catalog
    Canonical treatment
    Soft Theorems
    Scope
    Leading soft factors, Abelian and non-Abelian orientation, gravity comparison, external-line origin, and subleading caveats
    Assumptions
    Soft and Collinear Singularities
  • Theorem Mechanism

    Bloch–Nordsieck and KLN Cancellation

    Why can suitably inclusive probabilities remain finite even when individual real and virtual contributions diverge?

    Principal question
    Why can suitably inclusive probabilities remain finite even when individual real and virtual contributions diverge?
    What the source page covers
    Degenerate states, real–virtual cancellation, inclusiveness, mass singularities, KLN hypotheses, and finite-resolution interpretation
    Boundary
    Local subtraction algorithms; Non-global observables; Confinement
    Scope
    Degenerate states, real–virtual cancellation, inclusiveness, mass singularities, KLN hypotheses, and finite-resolution interpretation
    Assumptions
    Soft and Collinear Singularities; The Optical Theorem and Cut Interpretation
  • Theorem Mechanism

    Forward-Limit Positivity Bounds

    Why must certain low-energy forward-amplitude derivatives be positive in a causal, analytic, unitary UV completion?

    Principal question
    Why must certain low-energy forward-amplitude derivatives be positive in a causal, analytic, unitary UV completion?
    What the source page covers
    Pole subtraction, crossing, positive absorptive part, derivative moments, assumptions, and simple scalar EFT example
    Boundary
    General EFT basis construction; Current best bounds; Gravity with an unresolved forward pole
    Scope
    Pole subtraction, crossing, positive absorptive part, derivative moments, assumptions, and simple scalar EFT example
    Assumptions
    Forward Scattering Sum Rules; Causality, Growth, and Analytic Domains
  • Theorem Mechanism

    Local Field Redefinitions and the Equivalence Theorem

    Under what conditions do local perturbative field redefinitions leave S-matrix elements and physical observables unchanged?

    Principal question
    Under what conditions do local perturbative field redefinitions leave S-matrix elements and physical observables unchanged?
    What the source page covers
    Invertible local redefinitions, action and source changes, Jacobians, LSZ effects, order-by-order coefficient shifts, redundant operators, observable invariance, and limitations with boundaries or nonlocal maps
    Boundary
    A theorem that all off-shell quantities are invariant; Anomaly-producing transformations owned by Volume 3
    Scope
    Invertible local redefinitions, action and source changes, Jacobians, LSZ effects, order-by-order coefficient shifts, redundant operators, observable invariance, and limitations with boundaries or nonlocal maps
    Assumptions
    Integration by Parts and Equation-of-Motion Redundancy; LSZ Reduction: Poles, Residues, and Stable External States
  • Theorem Mechanism

    Decoupling Theorems and Threshold Corrections

    Under which assumptions do heavy particles decouple, and how do threshold corrections survive in low-energy parameters?

    Principal question
    Under which assumptions do heavy particles decouple, and how do threshold corrections survive in low-energy parameters?
    What the source page covers
    Appelquist–Carazzone logic, mass-dependent versus mass-independent schemes, matching of fields and couplings, power-suppressed effects, logarithmic threshold terms, assumptions, and known exceptions
    Boundary
    A claim that every heavy state decouples; Model-specific threshold spectra
    Scope
    Appelquist–Carazzone logic, mass-dependent versus mass-independent schemes, matching of fields and couplings, power-suppressed effects, logarithmic threshold terms, assumptions, and known exceptions
    Assumptions
    Matching Conditions Beyond Tree Level; Scheme Transformations and RG Invariants
  • Theorem Mechanism

    Consistency of Perturbative Yang–Mills Theory

    How do Slavnov–Taylor identities and BRST cohomology secure renormalizability and physical unitarity in Yang–Mills theory?

    Principal question
    How do Slavnov–Taylor identities and BRST cohomology secure renormalizability and physical unitarity in Yang–Mills theory?
    What the source page covers
    Allowed counterterms, Slavnov identity, gauge-parameter independence, cancellation of unphysical polarizations, BRST physical space, anomaly caveat, and proof architecture
    Boundary
    Full algebraic-renormalization proof; General S-matrix unitarity derivation; Anomalous chiral gauge theories
    Scope
    Allowed counterterms, Slavnov identity, gauge-parameter independence, cancellation of unphysical polarizations, BRST physical space, anomaly caveat, and proof architecture
    Assumptions
    Gauge-Fixed Yang–Mills Action and Ghost Sector; Symmetry Constraints and the Space of Counterterms
  • Theorem Mechanism

    Longitudinal Vector Bosons and the Equivalence Theorem

    How does the Higgs sector preserve high-energy consistency in longitudinal weak-boson scattering?

    Principal question
    How does the Higgs sector preserve high-energy consistency in longitudinal weak-boson scattering?
    What the source page covers
    Longitudinal polarization growth, Goldstone equivalence theorem, cancellation of energy growth, partial-wave unitarity, Higgs-role diagnosis, and theorem assumptions
    Boundary
    Generic partial-wave formalism; Full loop electroweak corrections; Strong electroweak models
    Scope
    Longitudinal polarization growth, Goldstone equivalence theorem, cancellation of energy growth, partial-wave unitarity, Higgs-role diagnosis, and theorem assumptions
    Assumptions
    The Higgs Doublet and Electroweak Symmetry Breaking; Partial-Wave Unitarity
  • Theorem Mechanism

    Elasticity, Factorization, and Their Hypotheses

    Why do higher conserved charges forbid particle production and reduce many-body scattering to two-body factors?

    Principal question
    Why do higher conserved charges forbid particle production and reduce many-body scattering to two-body factors?
    What the source page covers
    Rapidity variables, elastic scattering, absence of production, factorization, ordering in 1+1 dimensions, and assumptions, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Massless scattering subtleties in full; Nonintegrable corrections; Rigorous asymptotic completeness. Those subjects remain with their canonical volume or Research owner.
    Scope
    Rapidity variables, elastic scattering, absence of production, factorization, ordering in 1+1 dimensions, and assumptions, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Classical and Quantum Conserved Charges; LSZ Reduction: Poles, Residues, and Stable External States
  • Theorem Mechanism

    Volume Reduction, Large-N Equivalence, and Their Hypotheses

    Under which symmetry and limit assumptions can a large-N gauge theory become independent of spacetime volume?

    Principal question
    Under which symmetry and limit assumptions can a large-N gauge theory become independent of spacetime volume?
    What the source page covers
    Factorization, translation and center symmetry, reduction statement, symmetry-breaking obstruction, twisted/deformed orientation, and finite-N corrections, with assumptions, conventions, observables, and failure modes made explicit.
    Boundary
    Lattice implementation; Universal validity; Current numerical status without dated evidence. Those subjects remain with their canonical volume or Research owner.
    Scope
    Factorization, translation and center symmetry, reduction statement, symmetry-breaking obstruction, twisted/deformed orientation, and finite-N corrections, with assumptions, conventions, observables, and failure modes made explicit.
    Assumptions
    Large-N Limits, Normalizations, and Orders of Limits; Large-N Factorization and Master-Field Claims
  • Theorem Mechanism

    The Nielsen–Ninomiya Obstruction

    Which locality, translation, Hermiticity, chirality, and topology assumptions force lattice fermion doubling, and which formulation relaxes which assumption?

    Principal question
    Which locality, translation, Hermiticity, chirality, and topology assumptions force lattice fermion doubling, and which formulation relaxes which assumption?
    What the source page covers
    A dimension-qualified theorem statement, assumptions, momentum-space topological intuition, zero counting, proof roadmap, loophole taxonomy, and non-implications for interacting or gauge constructions.
    Boundary
    Index-theorem and bundle mathematics belongs to Volumes 1 and 16, continuum anomaly structure to Volume 3, and particular lattice responses to the following pages.
    Scope
    A dimension-qualified theorem statement, assumptions, momentum-space topological intuition, zero counting, proof roadmap, loophole taxonomy, and non-implications for interacting or gauge constructions.
    Assumptions
    Naive Fermions and Species Doubling
  • Theorem Mechanism

    Transfer Matrices between Euclidean and Hamiltonian QFT

    Under which positivity, locality, boundary, and anisotropic-limit assumptions does a Euclidean lattice action define a transfer matrix and a Hamiltonian?

    Principal question
    Under which positivity, locality, boundary, and anisotropic-limit assumptions does a Euclidean lattice action define a transfer matrix and a Hamiltonian?
    What the source page covers
    Transfer-kernel construction, temporal gauge where appropriate, reflection positivity, positive transfer operators, logarithmic Hamiltonians, anisotropic temporal limits, normalization and ordering terms, Euclidean spectral extraction, and failure modes when positivity is absent.
    Boundary
    General canonical-functional equivalence belongs to Volume 2, reflection positivity foundations to Chapter 1, Euclidean simulation algorithms to Chapter 5, and theorem-level reconstruction to Volume 16.
    Scope
    Transfer-kernel construction, temporal gauge where appropriate, reflection positivity, positive transfer operators, logarithmic Hamiltonians, anisotropic temporal limits, normalization and ordering terms, Euclidean spectral extraction, and failure modes when positivity is absent.
    Assumptions
    Regulated Hamiltonian Field Theory; Reflection Positivity and Transfer-Matrix Criteria
  • Theorem Mechanism

    Averaged Null Energy and Positivity

    Why is averaged null energy positive in a unitary relativistic CFT, and what are the precise operator, state, and smearing assumptions?

    Principal question
    Why is averaged null energy positive in a unitary relativistic CFT, and what are the precise operator, state, and smearing assumptions?
    What the source page covers
    ANEC operator definition, positivity statements, representative proof routes, relation to reflection positivity and causality, saturation and null states, domain issues, and extensions or limitations.
    Boundary
    Jet definitions and scattering event shapes remain in Volume 4; this chapter owns intrinsic CFT detector operators, light-ray expansions, positivity, and collider bounds; experimental or mutable frontier claims remain downstream.
    Scope
    ANEC operator definition, positivity statements, representative proof routes, relation to reflection positivity and causality, saturation and null states, domain issues, and extensions or limitations.
    Assumptions
    Detector Operators and Energy Flow at Null Infinity; Hilbert Positivity and Unitary Evolution
  • Theorem Mechanism

    The State–Operator Correspondence

    When and in what sense does each local operator define a state, and which states are recovered by local insertions?

    Principal question
    When and in what sense does each local operator define a state, and which states are recovered by local insertions?
    What the source page covers
    The state–operator map, vacuum and infinity insertions, primary and descendant states, normalization, completeness assumptions, and limitations from sectors or extended operators.
    Boundary
    General Euclidean axioms, reflection positivity, and operator definitions remain in Volume 2; thermal cylinder physics remains in Volume 11; executable spectral checks require separately supplied code and data.
    Scope
    The state–operator map, vacuum and infinity insertions, primary and descendant states, normalization, completeness assumptions, and limitations from sectors or extended operators.
    Assumptions
    Radial Time and Quantization on Spheres; Local and Composite Operator Insertions
  • Theorem Mechanism

    Highest-Weight Modules, Null States, and the Kac Determinant

    How are Virasoro highest-weight modules built, and where do their Gram determinants signal null vectors and reducibility?

    Principal question
    How are Virasoro highest-weight modules built, and where do their Gram determinants signal null vectors and reducibility?
    What the source page covers
    Verma modules, descendant partitions, Shapovalov forms, Kac determinant, degenerate weights, explicit null vectors, irreducible quotients, and unitary-series qualifications.
    Boundary
    Nonunitary, logarithmic, and noncompact theories move to Chapter 6; general symmetry and anomaly definitions stay in Volume 3; modular and thermal constraints are synthesized in Chapter 13.
    Scope
    Verma modules, descendant partitions, Shapovalov forms, Kac determinant, degenerate weights, explicit null vectors, irreducible quotients, and unitary-series qualifications.
    Assumptions
    The Virasoro Algebra and the Stress Tensor; Descendant States and Gram Matrices
  • Theorem Mechanism

    OPE Convergence, Associativity, and Domain Control

    Where does the conformal OPE converge, what controls its remainder, and how does associativity become crossing rather than a merely formal rearrangement?

    Principal question
    Where does the conformal OPE converge, what controls its remainder, and how does associativity become crossing rather than a merely formal rearrangement?
    What the source page covers
    Radial convergence domains, nested-sphere geometry, absolute convergence under stated hypotheses, truncation estimates, analytic continuation between channels, and associativity as equality on overlapping domains.
    Boundary
    The local Wilson OPE is defined in Volume 2; general distribution and tensor analysis remain in Volume 1; amplitude observables remain in Volume 4; block-generation code and benchmark data require separately supplied code and data and Reference.
    Scope
    Radial convergence domains, nested-sphere geometry, absolute convergence under stated hypotheses, truncation estimates, analytic continuation between channels, and associativity as equality on overlapping domains.
    Assumptions
    From the Local OPE to Conformal Data; Completeness and the Operator Basis
  • Theorem Mechanism

    The Lorentzian Inversion Formula

    Under which analyticity and Regge assumptions does a Lorentzian inversion integral reconstruct spin-analytic CFT data?

    Principal question
    Under which analyticity and Regge assumptions does a Lorentzian inversion integral reconstruct spin-analytic CFT data?
    What the source page covers
    Double-discontinuity inversion, integration domain and measure, partial-wave coefficient function, analyticity in spin, pole residues, low-spin arcs or subtractions, positivity consequences, and theorem hypotheses.
    Boundary
    General real-time prescriptions remain in Volume 2 and scattering dispersion or Regge theory in Volume 4; this chapter owns their CFT-correlator consequences, while mutable frontier claims live in Research.
    Scope
    Double-discontinuity inversion, integration domain and measure, partial-wave coefficient function, analyticity in spin, pole residues, low-spin arcs or subtractions, positivity consequences, and theorem hypotheses.
    Assumptions
    Double-Twist Families and Anomalous Dimensions; Conformal Partial Waves and the Shadow Formalism
  • Theorem Mechanism

    Unitarity Bounds and Null States

    Under an assumed positive-energy unitary conformal module, how do algebraic positivity and shortening constrain scaling dimensions?

    Principal question
    Under an assumed positive-energy unitary conformal module, how do algebraic positivity and shortening constrain scaling dimensions?
    What the source page covers
    The representation-theoretic statement of spin-dependent unitarity bounds, shortening at saturation, null descendants, free-field or conservation equations, and the exact positive-form assumptions; Chapter 2 owns the CFT construction of the radial adjoint, reflection-positive inner product, and level-by-level Gram matrices.
    Boundary
    General Lie theory and Lorentz representations remain in Volume 1; general global-symmetry, Ward-identity, and anomaly machinery remains in Volume 3; RG criteria for fixed points remain in Volume 5; theorem-level edge cases remain in Volume 16.
    Scope
    The representation-theoretic statement of spin-dependent unitarity bounds, shortening at saturation, null descendants, free-field or conservation equations, and the exact positive-form assumptions; Chapter 2 owns the CFT construction of the radial adjoint, reflection-positive inner product, and level-by-level Gram matrices.
    Assumptions
    Primaries, Descendants, and Conformal Multiplets; Hilbert Positivity and Unitary Evolution
  • Theorem Mechanism

    Local RG and Weyl Consistency Conditions

    How do spacetime-dependent couplings and commutativity of Weyl transformations constrain beta functions, anomalies, and candidate monotonic quantities?

    Principal question
    How do spacetime-dependent couplings and commutativity of Weyl transformations constrain beta functions, anomalies, and candidate monotonic quantities?
    What the source page covers
    Local coupling sources, local RG operator, Weyl consistency conditions, anomaly-coefficient gradients, flavor rotations, scheme covariance, fixed-point limits, and the boundary between identities and monotonicity theorems.
    Boundary
    General anomaly descent remains in Volume 3, local RG and EFT flow mechanics in Volume 5, and curved-spacetime QFT in Volume 14; this chapter owns fixed-point Weyl data and CFT deformation constraints.
    Scope
    Local coupling sources, local RG operator, Weyl consistency conditions, anomaly-coefficient gradients, flavor rotations, scheme covariance, fixed-point limits, and the boundary between identities and monotonicity theorems.
    Assumptions
    The Trace Ward Identity and Weyl Anomaly; Local Couplings, Trace Identities, and the Local Renormalization Group
  • Theorem Mechanism

    Boundary Entropy and Defect Monotonicity

    Which boundary or defect entropy quantities are universal, and under what hypotheses do they obey monotonicity along defect RG flows?

    Principal question
    Which boundary or defect entropy quantities are universal, and under what hypotheses do they obey monotonicity along defect RG flows?
    What the source page covers
    Boundary entropy and defect free-energy candidates, universal terms, subtraction schemes, defect RG flows, known monotonicity statements by dimension and codimension, counterexamples, and theorem hypotheses.
    Boundary
    General extended-operator, boundary, interface, fusion, and generalized-symmetry definitions remain in Volume 3; this chapter owns their conformal data and bootstrap; executable searches require separately supplied code and data.
    Scope
    Boundary entropy and defect free-energy candidates, universal terms, subtraction schemes, defect RG flows, known monotonicity statements by dimension and codimension, counterexamples, and theorem hypotheses.
    Assumptions
    Conformal Boundaries and Defects; Ultraviolet and Infrared Fixed Points: Criteria and Evidence
  • Theorem Mechanism

    Monotonicity Theorems and Flow Constraints

    Which anomaly, entropy, or sphere quantities provably order conformal endpoints of RG flows in each dimension?

    Principal question
    Which anomaly, entropy, or sphere quantities provably order conformal endpoints of RG flows in each dimension?
    What the source page covers
    c-, F-, and a-type statements, sum rules, dilaton arguments, relative entropy links as context, stationarity, weak versus strong forms, boundary and defect analogues as handoffs, and counterexample boundaries.
    Boundary
    General anomaly descent remains in Volume 3, local RG and EFT flow mechanics in Volume 5, and curved-spacetime QFT in Volume 14; this chapter owns fixed-point Weyl data and CFT deformation constraints.
    Scope
    c-, F-, and a-type statements, sum rules, dilaton arguments, relative entropy links as context, stationarity, weak versus strong forms, boundary and defect analogues as handoffs, and counterexample boundaries.
    Assumptions
    Local RG and Weyl Consistency Conditions; Sphere Partition Functions and Universal CFT Data
  • Theorem Mechanism

    Tauberian Theorems and Asymptotic Spectral Data

    Which asymptotic statements about CFT spectra follow from singular correlator limits via Tauberian theorems, and with what averaging?

    Principal question
    Which asymptotic statements about CFT spectra follow from singular correlator limits via Tauberian theorems, and with what averaging?
    What the source page covers
    Positive spectral measures, Laplace-like transforms, integrated density asymptotics, remainder bounds, smearing and averaging scales, heavy-state OPE averages, and the distinction between pointwise and averaged conclusions.
    Boundary
    General real-time prescriptions remain in Volume 2 and scattering dispersion or Regge theory in Volume 4; this chapter owns their CFT-correlator consequences, while mutable frontier claims live in Research.
    Scope
    Positive spectral measures, Laplace-like transforms, integrated density asymptotics, remainder bounds, smearing and averaging scales, heavy-state OPE averages, and the distinction between pointwise and averaged conclusions.
    Assumptions
    The Lightcone OPE and Large-Spin Expansion; Asymptotic Scales, Remainders, Uniformity, and Optimal Truncation
  • Theorem Mechanism

    Graded Spacetime Symmetry and the Supersymmetry Theorems

    Under which locality, spectrum, analyticity, and scattering assumptions can an interacting relativistic theory admit a nontrivial graded extension of spacetime symmetry?

    Principal question
    Under which locality, spectrum, analyticity, and scattering assumptions can an interacting relativistic theory admit a nontrivial graded extension of spacetime symmetry?
    What the source page covers
    The supersymmetric extension theorems, their hypotheses and loopholes, graded Lie brackets, Jacobi constraints, internal versus spacetime generators, and the precise boundary between algebraic classification and existence of a QFT.
    Boundary
    Volume 1 retains spinor and representation mathematics, Volume 3 retains ordinary symmetry and charge machinery, Volume 9 owns conformal crossing, and Volume 4 owns amplitude methods.
    Scope
    The supersymmetric extension theorems, their hypotheses and loopholes, graded Lie brackets, Jacobi constraints, internal versus spacetime generators, and the precise boundary between algebraic classification and existence of a QFT.
    Assumptions
    Lie Groups, Lie Algebras, and Exponential and Adjoint Maps; Lorentz Field Representations and Poincaré Particle Representations
  • Theorem Mechanism

    F- and D-Term Breaking, Vacuum Energy, and the Goldstino

    How do positive vacuum energy, nonzero F or D expectation values, and supercurrent matrix elements diagnose spontaneous supersymmetry breaking and identify the goldstino?

    Principal question
    How do positive vacuum energy, nonzero F or D expectation values, and supercurrent matrix elements diagnose spontaneous supersymmetry breaking and identify the goldstino?
    What the source page covers
    Vacuum-energy algebra, incompatible auxiliary equations, order parameters, goldstino theorem, fermion mass zero mode, supercurrent pole, caveats from gauge symmetry and boundaries, and local-versus-global vacuum status.
    Boundary
    Volume 5 retains EFT validity, Volume 7 retains vacuum decay, Volume 6 owns phenomenological models and empirical constraints, and no soft path is promoted into a theorem about nonsupersymmetric QCD.
    Scope
    Vacuum-energy algebra, incompatible auxiliary equations, order parameters, goldstino theorem, fermion mass zero mode, supercurrent pole, caveats from gauge symmetry and boundaries, and local-versus-global vacuum status.
    Assumptions
    Gauge–Matter Systems, F- and D-Term Potentials, and FI Data; The Four-Dimensional N=1 Super-Poincaré Algebra
  • Theorem Mechanism

    Nonrenormalization Theorems: Wilsonian, 1PI, and Infrared Scope

    What exactly is nonrenormalized in a supersymmetric theory, and why can infrared effects make the 1PI statement differ from the Wilsonian theorem?

    Principal question
    What exactly is nonrenormalized in a supersymmetric theory, and why can infrared effects make the 1PI statement differ from the Wilsonian theorem?
    What the source page covers
    Perturbative superpotential nonrenormalization, supergraph and spurion proofs, Wilsonian locality, wavefunction and Kähler corrections, 1PI infrared nonlocality, anomalies, and nonperturbative exceptions.
    Boundary
    Volume 3 retains anomaly foundations, Volume 5 retains scheme and RG machinery, Volume 7 retains generic instanton zero modes, and Chapters 8 and 10 own theory-specific exact dynamics.
    Scope
    Perturbative superpotential nonrenormalization, supergraph and spurion proofs, Wilsonian locality, wavefunction and Kähler corrections, 1PI infrared nonlocality, anomalies, and nonperturbative exceptions.
    Assumptions
    Holomorphic Couplings and Background Superfields; Supergraphs, D-Algebra, and Quantum Effective Actions
  • Theorem Mechanism

    Spectral Pairing, Ground States, and Supersymmetry Breaking

    Why are positive-energy bosonic and fermionic states paired, and which unpaired zero-energy states diagnose unbroken supersymmetry?

    Principal question
    Why are positive-energy bosonic and fermionic states paired, and which unpaired zero-energy states diagnose unbroken supersymmetry?
    What the source page covers
    Supercharge isomorphisms between positive-energy eigenspaces, kernels, zero-mode normalizability, ground-state wavefunctions, spontaneous breaking in quantum mechanics, and pairing failures caused by domains.
    Boundary
    Volume 1 retains Hilbert-complex and index mathematics, Volume 7 retains generic instanton and tunneling methods, Volume 11 owns thermal traces, and executable calculations must be accompanied by code, inputs, and validation checks.
    Scope
    Supercharge isomorphisms between positive-energy eigenspaces, kernels, zero-mode normalizability, ground-state wavefunctions, spontaneous breaking in quantum mechanics, and pairing failures caused by domains.
    Assumptions
    Supercharges, Partner Hamiltonians, and Positive Energy
  • Theorem Mechanism

    R-Symmetry and Nelson–Seiberg-Type Criteria

    Under which genericity, field-content, and vacuum assumptions do R-symmetry criteria constrain F-term supersymmetry breaking?

    Principal question
    Under which genericity, field-content, and vacuum assumptions do R-symmetry criteria constrain F-term supersymmetry breaking?
    What the source page covers
    Continuous R symmetries, generic superpotentials, necessary and sufficient statements under hypotheses, field-counting arguments, spontaneous R breaking, exceptional models, runaways, and limits of theorem application.
    Boundary
    Volume 5 retains EFT validity, Volume 7 retains vacuum decay, Volume 6 owns phenomenological models and empirical constraints, and no soft path is promoted into a theorem about nonsupersymmetric QCD.
    Scope
    Continuous R symmetries, generic superpotentials, necessary and sufficient statements under hypotheses, field-counting arguments, spontaneous R breaking, exceptional models, runaways, and limits of theorem application.
    Assumptions
    O’Raifeartaigh and Fayet–Iliopoulos Model Laboratories; Holomorphic Couplings and Background Superfields
  • Theorem Mechanism

    Q-Cohomology, Hodge Decomposition, and Zero-Energy States

    Under which analytic hypotheses do Q-cohomology, harmonic representatives, and zero-energy states describe the same physical sector?

    Principal question
    Under which analytic hypotheses do Q-cohomology, harmonic representatives, and zero-energy states describe the same physical sector?
    What the source page covers
    Supercharges as differentials, cochain grading, kernels and images, Hodge decomposition, harmonic representatives, de Rham realization, closed-range and boundary qualifications, and cohomological protection.
    Boundary
    Volume 1 retains Hilbert-complex and index mathematics, Volume 7 retains generic instanton and tunneling methods, Volume 11 owns thermal traces, and executable calculations must be accompanied by code, inputs, and validation checks.
    Scope
    Supercharges as differentials, cochain grading, kernels and images, Hodge decomposition, harmonic representatives, de Rham realization, closed-range and boundary qualifications, and cohomological protection.
    Assumptions
    Supercharges, Partner Hamiltonians, and Positive Energy; Chains, Homology, Cohomology, and Exact Sequences; de Rham Cohomology, Periods, Duality, and Intersection
  • Theorem Mechanism

    Finite-Dimensional and Equivariant Localization

    What does finite-dimensional equivariant localization actually prove, and which ingredients survive only formally in a QFT path integral?

    Principal question
    What does finite-dimensional equivariant localization actually prove, and which ingredients survive only formally in a QFT path integral?
    What the source page covers
    Equivariant differentials, invariant integration, fixed loci, normal bundles, Euler classes, compactness, orientations, isolated and nonisolated formulas, zero weights, and the exact analogy boundary to field theory.
    Boundary
    Volume 1 retains equivariant and index mathematics, Volume 3 retains BRST/BV, Volume 14 owns dynamical curved spacetime and supergravity, and Chapter 16 owns computed exact observables.
    Scope
    Equivariant differentials, invariant integration, fixed loci, normal bundles, Euler classes, compactness, orientations, isolated and nonisolated formulas, zero weights, and the exact analogy boundary to field theory.
    Assumptions
    Differential Forms, Integration, Orientation, and Stokes Theorem; Fredholm and Dirac Index Theorems and Zero-Mode Counting; Lie Groups, Lie Algebras, and Exponential and Adjoint Maps
  • Theorem Mechanism

    BPS Bounds, Shortening, and Multiplet Recombination

    How do positivity and central charges imply BPS bounds, shortening, null states, and recombination of unitary supermultiplets?

    Principal question
    How do positivity and central charges imply BPS bounds, shortening, null states, and recombination of unitary supermultiplets?
    What the source page covers
    Positive supercharge anticommutator matrices, mass and scaling bounds, preserved-supercharge fractions, shortened state counts, threshold multiplets, character identities, and recombination under parameter variation.
    Boundary
    Volume 1 retains spinor and representation mathematics, Volume 3 retains ordinary symmetry and charge machinery, Volume 9 owns conformal crossing, and Volume 4 owns amplitude methods.
    Scope
    Positive supercharge anticommutator matrices, mass and scaling bounds, preserved-supercharge fractions, shortened state counts, threshold multiplets, character identities, and recombination under parameter variation.
    Assumptions
    Extended Supersymmetry, R-Symmetry, and Central Charges; Massive and Massless Unitary Supermultiplets; Bilinear and Hermitian Forms, Adjoints, and Isometries
  • Theorem Mechanism

    BPS Particles and Central Charges

    How do conserved charges and the supersymmetry algebra bound particle masses, and which additional conditions establish existence, stability, and protection?

    Principal question
    How do conserved charges and the supersymmetry algebra bound particle masses, and which additional conditions establish existence, stability, and protection?
    What the source page covers
    Particle central charges, mass bounds, preserved supercharges, shortened particle multiplets, charge lattices, phase conventions, marginal decay, protected indices, and the separation of algebraic bound from dynamical state existence.
    Boundary
    Volume 1 retains quotient geometry, Volume 3 retains generic spontaneous symmetry breaking, Volume 7 retains generic solitons and zero modes, and dimension-specific dynamics stay in later chapters.
    Scope
    Particle central charges, mass bounds, preserved supercharges, shortened particle multiplets, charge lattices, phase conventions, marginal decay, protected indices, and the separation of algebraic bound from dynamical state existence.
    Assumptions
    BPS Bounds, Shortening, and Multiplet Recombination
  • Theorem Mechanism

    Infinite-Volume KMS States, Passivity, and Phase Multiplicity

    How are equilibrium, passivity, extremality, and phase multiplicity formulated when the thermodynamic limit removes the global Gibbs trace?

    Principal question
    How are equilibrium, passivity, extremality, and phase multiplicity formulated when the thermodynamic limit removes the global Gibbs trace?
    What the source page covers
    Local-observable KMS states, complete passivity, extremal and mixed thermal phases, clustering, disjoint representations, superselection by boundary conditions, and a physics-facing map to operator-algebraic results.
    Boundary
    This chapter owns thermal-state, KMS, Matsubara, and spectral dictionaries; Volume 8 retains regulator-specific Euclidean reconstruction machinery, Volume 9 owns thermal-CFT specializations, and Volume 16 owns theorem-first operator-algebraic status.
    Scope
    Local-observable KMS states, complete passivity, extremal and mixed thermal phases, clustering, disjoint representations, superselection by boundary conditions, and a physics-facing map to operator-algebraic results.
    Assumptions
    Thermal Density Operators and the KMS Condition; Thermodynamic Limits, Phases, and Ensemble Equivalence
  • Theorem Mechanism

    Onsager Reciprocity and Entropy Production

    How do microscopic time-reversal properties and local entropy production constrain dissipative transport matrices?

    Principal question
    How do microscopic time-reversal properties and local entropy production constrain dissipative transport matrices?
    What the source page covers
    Thermodynamic forces and fluxes, Onsager–Casimir reciprocity, entropy-current divergence, positive semidefinite transport matrices, magnetic and rotational exceptions, and limitations of entropy-current arguments.
    Boundary
    This chapter owns relativistic dissipative formulations and their hypothesis-explicit stability, causality, and PDE status; Chapter 13 owns fluctuating and generalized sectors, Chapter 18 owns collision implementations, and Volume 15 owns holographic realizations.
    Scope
    Thermodynamic forces and fluxes, Onsager–Casimir reciprocity, entropy-current divergence, positive semidefinite transport matrices, magnetic and rotational exceptions, and limitations of entropy-current arguments.
    Assumptions
    Relativistic Dissipative Hydrodynamics; KMS Relations and Fluctuation–Dissipation; Internal, Spacetime, Discrete, and Antiunitary Symmetries
  • Theorem Mechanism

    Dynamical KMS and Topological Symmetries

    Which dynamical KMS and topological or BRST-like structures encode local equilibrium and SK normalization, and what extra assumptions are required beyond microscopic unitarity?

    Principal question
    Which dynamical KMS and topological or BRST-like structures encode local equilibrium and SK normalization, and what extra assumptions are required beyond microscopic unitarity?
    What the source page covers
    Discrete dynamical KMS transformations, thermal translations, topological a=0 sector, ghost completions, classical supersymmetry limits, entropy-current consequences, anomalies, and local-versus-global equilibrium qualifications.
    Boundary
    Volume 3 retains anomaly and higher-form-current foundations; this chapter owns hydrodynamic EFT and universal constitutive consequences, Volume 12 owns integrable-model and material realizations, and Volume 15 owns holographic constructions.
    Scope
    Discrete dynamical KMS transformations, thermal translations, topological a=0 sector, ghost completions, classical supersymmetry limits, entropy-current consequences, anomalies, and local-versus-global equilibrium qualifications.
    Assumptions
    Schwinger–Keldysh Effective Actions for Fluids; Unitarity, Normalization, and Largest-Time Identities
  • Theorem Mechanism

    Unitarity, Normalization, and Largest-Time Identities

    How do microscopic unitarity, Hermiticity, and trace normalization constrain a Schwinger–Keldysh generating functional and its effective action?

    Principal question
    How do microscopic unitarity, Hermiticity, and trace normalization constrain a Schwinger–Keldysh generating functional and its effective action?
    What the source page covers
    Z[J,J]=1, reality conjugation, causal zeros, largest-time identities, cutting relations, topological a=0 sector, positivity qualifications, and the distinction between closed-unitary and reduced-open dynamics.
    Boundary
    Volume 2 retains in-out versus in-in foundations; this chapter owns normalized closed-time-path practice and response, Chapter 8 owns self-consistent two-time evolution, and Chapter 16 owns reduced open-system dynamics.
    Scope
    Z[J,J]=1, reality conjugation, causal zeros, largest-time identities, cutting relations, topological a=0 sector, positivity qualifications, and the distinction between closed-unitary and reduced-open dynamics.
    Assumptions
    Closed-Time-Path Generating Functionals in Practice; Hilbert Positivity and Unitary Evolution
  • Theorem Mechanism

    The H-Theorem and Kinetic Entropy Production

    Under which collision, positivity, reversibility, and molecular-chaos assumptions does a kinetic entropy increase, and what equilibrium family saturates the result?

    Principal question
    Under which collision, positivity, reversibility, and molecular-chaos assumptions does a kinetic entropy increase, and what equilibrium family saturates the result?
    What the source page covers
    Classical and quantum kinetic entropy functionals, H-theorem inequalities, collision invariants, local equilibrium, chemical constraints, entropy production, and limitations for off-shell, coherent, or memory evolution.
    Boundary
    This chapter owns generic kinetic reduction, closures, and validity; Chapter 17 owns gauge-plasma effective kinetics, Chapter 18 owns collision staging, and model-specific quantum-matter transport belongs to Volume 12.
    Scope
    Classical and quantum kinetic entropy functionals, H-theorem inequalities, collision invariants, local equilibrium, chemical constraints, entropy production, and limitations for off-shell, coherent, or memory evolution.
    Assumptions
    Collision Kernels, Conservation, and Detailed Balance
  • Theorem Mechanism

    Detailed Balance and Fluctuation–Dissipation

    Which time-reversal and probability-current conditions characterize detailed balance, and how do they imply equilibrium fluctuation–dissipation identities?

    Principal question
    Which time-reversal and probability-current conditions characterize detailed balance, and how do they imply equilibrium fluctuation–dissipation identities?
    What the source page covers
    Detailed balance, reversible and irreversible drift, stationary probability current, stochastic time reversal, entropy production, MSRJD equilibrium symmetry, and fluctuation–dissipation consequences.
    Boundary
    Volume 1 retains stochastic-process mathematics; Volume 5 retains static RG fixed points; this chapter owns field-theoretic stochastic dynamics and dynamic universality, while model realizations belong to Volume 12.
    Scope
    Detailed balance, reversible and irreversible drift, stationary probability current, stochastic time reversal, entropy production, MSRJD equilibrium symmetry, and fluctuation–dissipation consequences.
    Assumptions
    Fokker–Planck Evolution and Stationary Measures; The MSRJD Response Functional
  • Theorem Mechanism

    Transport Sum Rules and Ultraviolet Constraints

    Which exact commutators, Ward identities, dispersion relations, and ultraviolet asymptotics constrain integrated transport spectral weight?

    Principal question
    Which exact commutators, Ward identities, dispersion relations, and ultraviolet asymptotics constrain integrated transport spectral weight?
    What the source page covers
    Transport sum rules, subtractions, equal-time commutators, OPE tails, thermodynamic contacts, channel differences, finite-density modifications, and use of sum rules as diagnostics rather than reconstructions.
    Boundary
    This chapter owns response definitions, Kubo limits, transport observables, projection methods, and inference contracts; Chapters 17–18 own hot-QCD calculations and evidence, while executable extraction must be accompanied by code, inputs, and validation checks.
    Scope
    Transport sum rules, subtractions, equal-time commutators, OPE tails, thermodynamic contacts, channel differences, finite-density modifications, and use of sum rules as diagnostics rather than reconstructions.
    Assumptions
    Spectral Functions and Transport Peaks; Thermal Spectral Positivity and Sum Rules
  • Theorem Mechanism

    Thermodynamic Limits, Phases, and Ensemble Equivalence

    Which limits and convexity conditions justify a thermodynamic phase, spontaneous symmetry breaking, and equivalence or inequivalence of ensembles?

    Principal question
    Which limits and convexity conditions justify a thermodynamic phase, spontaneous symmetry breaking, and equivalence or inequivalence of ensembles?
    What the source page covers
    Volume and source order of limits, pure and mixed phases, clustering, boundary-condition selection, nonanalytic thermodynamic potentials, ensemble equivalence criteria, long-range exceptions, and finite-size scaling cautions.
    Boundary
    Volume 1 retains probability, measure, asymptotic, and convex-analysis foundations; Volume 2 retains vacuum functional-integral definitions; rigorous infinite-system existence belongs to Volume 16.
    Scope
    Volume and source order of limits, pure and mixed phases, clustering, boundary-condition selection, nonanalytic thermodynamic potentials, ensemble equivalence criteria, long-range exceptions, and finite-size scaling cautions.
    Assumptions
    Partition Functions and Thermodynamic Response
  • Theorem Mechanism

    Thermal Spectral Positivity and Sum Rules

    Which positivity, antisymmetry, moment, and ultraviolet constraints does a thermal spectral function obey, and which of them fail for gauge-variant, non-Hermitian, or finite-density operators?

    Principal question
    Which positivity, antisymmetry, moment, and ultraviolet constraints does a thermal spectral function obey, and which of them fail for gauge-variant, non-Hermitian, or finite-density operators?
    What the source page covers
    Positive-metric Lehmann weights, sign properties, equal-time commutator sum rules, subtracted moments, OPE or ultraviolet asymptotics, susceptibility relations, and explicit exceptions in indefinite-metric or charged channels.
    Boundary
    This chapter owns thermal-state, KMS, Matsubara, and spectral dictionaries; Volume 8 retains regulator-specific Euclidean reconstruction machinery, Volume 9 owns thermal-CFT specializations, and Volume 16 owns theorem-first operator-algebraic status.
    Scope
    Positive-metric Lehmann weights, sign properties, equal-time commutator sum rules, subtracted moments, OPE or ultraviolet asymptotics, susceptibility relations, and explicit exceptions in indefinite-metric or charged channels.
    Assumptions
    Thermal Propagators and Spectral Representations; Microcausality and Relativistic Compatibility
  • Theorem Mechanism

    Strong Hyperbolicity, Stability, and Causal Propagation

    How are linear mode stability, front velocity, characteristic causality, strong hyperbolicity, local well-posedness, nonlinear stability, and shock admissibility distinguished?

    Principal question
    How are linear mode stability, front velocity, characteristic causality, strong hyperbolicity, local well-posedness, nonlinear stability, and shock admissibility distinguished?
    What the source page covers
    Dispersion roots, rest and boosted stability, group versus front velocity, characteristic matrices, strong and symmetric hyperbolicity, local existence hypotheses, nonlinear energy estimates, weak solutions, and evidence ceilings.
    Boundary
    This chapter owns relativistic dissipative formulations and their hypothesis-explicit stability, causality, and PDE status; Chapter 13 owns fluctuating and generalized sectors, Chapter 18 owns collision implementations, and Volume 15 owns holographic realizations.
    Scope
    Dispersion roots, rest and boosted stability, group versus front velocity, characteristic matrices, strong and symmetric hyperbolicity, local existence hypotheses, nonlinear energy estimates, weak solutions, and evidence ceilings.
    Assumptions
    Israel–Stewart, BRSSS, and DNMR Transient Hydrodynamics; BDNK First-Order Causal Hydrodynamics
  • Theorem Mechanism

    Lyapunov Growth and Chaos Bounds

    Under exactly which analyticity, boundedness, factorization, thermal-scale, and time-window assumptions may an exponential OTOC growth rate be defined and bounded?

    Principal question
    Under exactly which analyticity, boundedness, factorization, thermal-scale, and time-window assumptions may an exponential OTOC growth rate be defined and bounded?
    What the source page covers
    Many-body Lyapunov regimes, MSS-type analyticity argument, factorization hierarchy, dissipation and scrambling times, finite-coupling corrections, saturation, and explicit non-applicability to arbitrary operators or finite systems.
    Boundary
    This chapter owns universal distinctions among equilibration, ETH, scaling, chaos, and scrambling; Volume 12 owns model and platform realizations, Volume 13 owns information-theoretic treatment, and Volume 15 owns holographic realizations.
    Scope
    Many-body Lyapunov regimes, MSS-type analyticity argument, factorization hierarchy, dissipation and scrambling times, finite-coupling corrections, saturation, and explicit non-applicability to arbitrary operators or finite systems.
    Assumptions
    Out-of-Time-Order Correlators and Contour Regularization; Holomorphic Functions and Cauchy Theory
  • Theorem Mechanism

    Luttinger's Theorem, Fermi Volume, and Failure Modes

    Under which symmetry, analyticity, adiabaticity, and topological hypotheses does particle density fix Fermi volume?

    Principal question
    Under which symmetry, analyticity, adiabaticity, and topological hypotheses does particle density fix Fermi volume?
    What the source page covers
    Luttinger counting from Green functions or flux insertion, spin and Brillouin-zone conventions, zeros, broken symmetry, fractionalization, topological order, and precise failure modes.
    Boundary
    Volume 5 owns RG construction and Volume 11 owns generic transport. This chapter owns Fermi-surface kinematics, Landau theory, Luttinger hypotheses, and matter-specific instabilities.
    Scope
    Luttinger counting from Green functions or flux insertion, spin and Brillouin-zone conventions, zeros, broken symmetry, fractionalization, topological order, and precise failure modes.
    Assumptions
    The Fermi Gas and Fermi-Surface Kinematics; Dyson Equations and Self-Energies
  • Theorem Mechanism

    Universal Relations and Tan Contact

    How does the short-distance contact control momentum tails, energy derivatives, rf response, and pressure relations across quantum gases?

    Principal question
    How does the short-distance contact control momentum tails, energy derivatives, rf response, and pressure relations across quantum gases?
    What the source page covers
    Tan contact definitions, adiabatic and virial relations, high-momentum/frequency tails, operator-product interfaces, normalization across statistics, and range corrections.
    Boundary
    Volumes 4 and 5 own scattering and few-body EFT machinery. This chapter owns the use of matched two- and three-body data in quantum gases; current resonance and loss records belong to Research.
    Scope
    Tan contact definitions, adiabatic and virial relations, high-momentum/frequency tails, operator-product interfaces, normalization across statistics, and range corrections.
    Assumptions
    Short-Range Scattering Data as Many-Body Inputs; Spectral Moments and Many-Body Sum Rules
  • Theorem Mechanism

    Finite-Density Goldstone Counting

    How do charge-density commutators and broken spacetime or internal symmetries change Goldstone counting at finite density?

    Principal question
    How do charge-density commutators and broken spacetime or internal symmetries change Goldstone counting at finite density?
    What the source page covers
    The Watanabe–Brauner commutator matrix, type-A/type-B modes, quadratic dispersion, inverse-Higgs qualifications, finite-density examples, and orders of thermodynamic and zero-momentum limits.
    Boundary
    Volume 2 owns Fock-space and path-integral foundations, Volume 3 owns symmetry and Ward identities, and Volume 5 owns generic EFT construction; this chapter owns their nonrelativistic many-body realization and matching.
    Scope
    The Watanabe–Brauner commutator matrix, type-A/type-B modes, quadratic dispersion, inverse-Higgs qualifications, finite-density examples, and orders of thermodynamic and zero-momentum limits.
    Assumptions
    Densities, Currents, and Nonrelativistic Ward Identities; Goldstone Counting, Low-Dimensional Obstructions, and Spacetime Exceptions
  • Theorem Mechanism

    Migdal's Theorem and Vertex-Correction Validity

    Which energy, momentum, coupling, and band-structure hierarchy suppresses electron–phonon vertex corrections in Migdal theory?

    Principal question
    Which energy, momentum, coupling, and band-structure hierarchy suppresses electron–phonon vertex corrections in Migdal theory?
    What the source page covers
    Migdal parameter, vertex-correction scaling, adiabatic ratio, forward scattering, low-density and narrow-band failures, and material validity tests.
    Boundary
    Volume 11 owns generic response and Volume 15 owns holographic realizations. This chapter owns microscopic paired-matter theory, gauge-invariant observables, topology, and experimental inference.
    Scope
    Migdal parameter, vertex-correction scaling, adiabatic ratio, forward scattering, low-density and narrow-band failures, and material validity tests.
    Assumptions
    Electron–Phonon Fields and Retarded Interactions; The Fermi Gas and Fermi-Surface Kinematics
  • Theorem Mechanism

    Tomita–Takesaki Modular Operators and Flow

    How do the Tomita operator, polar decomposition, modular operator, and modular automorphism group arise from a cyclic separating state?

    Principal question
    How do the Tomita operator, polar decomposition, modular operator, and modular automorphism group arise from a cyclic separating state?
    What the source page covers
    This page owns S, J, Δ, their domains and polar decomposition, modular automorphisms, KMS structure, and state-algebra dependence.
    Boundary
    It does not own the full theorem proof or replace unbounded-operator closures with formal finite-dimensional matrix identities.
    Scope
    This page owns S, J, Δ, their domains and polar decomposition, modular automorphisms, KMS structure, and state-algebra dependence.
    Assumptions
    Operator Algebras and Positive Functionals: a Bridge; Modular Hamiltonians: Definitions and Domains
  • Theorem Mechanism

    Entropic Monotones in Two Dimensions

    How do interval entropy, Lorentz symmetry, strong subadditivity, and fixed-point scaling yield a two-dimensional entropic c-function?

    Principal question
    How do interval entropy, Lorentz symmetry, strong subadditivity, and fixed-point scaling yield a two-dimensional entropic c-function?
    What the source page covers
    This page owns the c-function definition, derivative normalization, monotonicity hypotheses, UV and IR limits, and relation to central charge.
    Boundary
    It does not transfer the proof to nonrelativistic, nonunitary, finite-temperature, or higher-dimensional flows without replacement assumptions.
    Scope
    This page owns the c-function definition, derivative normalization, monotonicity hypotheses, UV and IR limits, and relation to central charge.
    Assumptions
    Information Measures Along RG Flows
  • Theorem Mechanism

    Positivity, Monotonicity, and Data Processing

    Which positivity and inclusion or channel hypotheses imply monotonicity of relative entropy and data processing in QFT?

    Principal question
    Which positivity and inclusion or channel hypotheses imply monotonicity of relative entropy and data processing in QFT?
    What the source page covers
    This page owns equality direction, completely positive state-preserving maps, algebra restriction, support conditions, and theorem-versus-model status.
    Boundary
    It does not infer data processing for nonlinear coarse graining, postselection, or regulator changes that define no common channel.
    Scope
    This page owns equality direction, completely positive state-preserving maps, algebra restriction, support conditions, and theorem-versus-model status.
    Assumptions
    Operator Algebras and Positive Functionals: a Bridge; Relative Entropy for QFT States
  • Theorem Mechanism

    Rindler Wedges and the Bisognano–Wichmann Theorem

    Under which covariance, spectrum, locality, and vacuum hypotheses does wedge modular flow coincide with Lorentz boosts?

    Principal question
    Under which covariance, spectrum, locality, and vacuum hypotheses does wedge modular flow coincide with Lorentz boosts?
    What the source page covers
    This page owns the Bisognano–Wichmann statement, boost normalization, wedge geometry, Unruh temperature relation, and theorem hypotheses.
    Boundary
    It does not own the proof or extend geometric modular flow to arbitrary regions, excited states, or Lorentz-violating regulators.
    Scope
    This page owns the Bisognano–Wichmann statement, boost normalization, wedge geometry, Unruh temperature relation, and theorem hypotheses.
    Assumptions
    Microcausality and Relativistic Compatibility; Spacetime Currents, Stress Tensors, and Charge Algebras; Tomita–Takesaki Modular Operators and Flow
  • Theorem Mechanism

    Fidelity, Chernoff Bounds, and State Overlap

    Which fidelity, overlap, and Chernoff quantities are well-defined for continuum states, and how do they bound discrimination under constraints?

    Principal question
    Which fidelity, overlap, and Chernoff quantities are well-defined for continuum states, and how do they bound discrimination under constraints?
    What the source page covers
    This page owns algebraic and regulated fidelity conventions, transition probabilities, Chernoff exponents, support dependence, and metric relations.
    Boundary
    It does not interchange squared and unsquared fidelity or use a global vector overlap as local-state fidelity without specifying the algebra.
    Scope
    This page owns algebraic and regulated fidelity conventions, transition probabilities, Chernoff exponents, support dependence, and metric relations.
    Assumptions
    Relative Entropy for QFT States
  • Theorem Mechanism

    The Split Property and Approximate Tensor Products

    Under what separation and phase-space conditions can two local algebras be interpolated by a type-I factor and treated as approximately independent?

    Principal question
    Under what separation and phase-space conditions can two local algebras be interpolated by a type-I factor and treated as approximately independent?
    What the source page covers
    This page owns split inclusions, collar regions, type-I interpolants, product-state preparation, approximate tensor products, and dependence on separation scale.
    Boundary
    It does not equate the split property with exact geometric factorization or claim a universal error bound without nuclearity and state-dependent input.
    Scope
    This page owns split inclusions, collar regions, type-I interpolants, product-state preparation, approximate tensor products, and dependence on separation scale.
    Assumptions
    Clustering, Vacuum Assumptions, and Long-Range Correlations; Von Neumann Factors and Type-III Local Algebras
  • Theorem Mechanism

    Nuclearity, Phase-Space Bounds, and Split Distance

    How do nuclearity maps and phase-space bounds control the existence and quantitative quality of split inclusions at finite separation?

    Principal question
    How do nuclearity maps and phase-space bounds control the existence and quantitative quality of split inclusions at finite separation?
    What the source page covers
    This page owns energy-damped local-state maps, nuclearity indices, split distance, temperature and mass dependence, and their role as phase-space diagnostics.
    Boundary
    It does not present nuclearity as one model-independent inequality or infer a numerical split error from qualitative compactness alone.
    Scope
    This page owns energy-damped local-state maps, nuclearity indices, split distance, temperature and mass dependence, and their role as phase-space diagnostics.
    Assumptions
    The Split Property and Approximate Tensor Products
  • Theorem Mechanism

    Additivity, Haag Duality, and Information Completeness

    When do additivity and Haag duality make local and complementary observable assignments information-complete, and where do boundary sectors obstruct them?

    Principal question
    When do additivity and Haag duality make local and complementary observable assignments information-complete, and where do boundary sectors obstruct them?
    What the source page covers
    This page owns additivity variants, A(O′) versus A(O)′, dual nets, disconnected regions, and missing boundary or topological operators.
    Boundary
    It does not assume Haag duality in gauge or topological theories or absorb the classification of generalized-symmetry defects owned by Volume III.
    Scope
    This page owns additivity variants, A(O′) versus A(O)′, dual nets, disconnected regions, and missing boundary or topological operators.
    Assumptions
    Regions, Causal Complements, and Nets of Observables
  • Theorem Mechanism

    Strong Subadditivity and Entropic Inequalities

    How do strong subadditivity and related inequalities apply when QFT regions overlap, touch, or require regulated subsystem choices?

    Principal question
    How do strong subadditivity and related inequalities apply when QFT regions overlap, touch, or require regulated subsystem choices?
    What the source page covers
    This page owns region assignments for SSA, proof hypotheses, conditional entropy combinations, ultraviolet cancellation, and equality cases.
    Boundary
    It does not apply tensor-product inequalities to incompatible algebras or combine unmatched regulator terms.
    Scope
    This page owns region assignments for SSA, proof hypotheses, conditional entropy combinations, ultraviolet cancellation, and equality cases.
    Assumptions
    Entropy of a Regulated Subregion
  • Theorem Mechanism

    Modular Analyticity and Chaos Bounds

    Which analyticity, boundedness, and modular-KMS hypotheses yield growth bounds for modular correlators, and what do those bounds diagnose?

    Principal question
    Which analyticity, boundedness, and modular-KMS hypotheses yield growth bounds for modular correlators, and what do those bounds diagnose?
    What the source page covers
    This page owns modular-time analytic strips, normalized correlators, growth exponents, state and algebra dependence, and comparison with physical chaos bounds.
    Boundary
    It does not equate modular growth with real-time many-body chaos or infer a Lyapunov velocity from an exponent.
    Scope
    This page owns modular-time analytic strips, normalized correlators, growth exponents, state and algebra dependence, and comparison with physical chaos bounds.
    Assumptions
    Modular KMS Relations and Modular Correlators
  • Theorem Mechanism

    Reeh–Schlieder Property and Limits of Localization

    How does Reeh–Schlieder cyclicity coexist with locality, and why does local state approximation not imply cheap creation or superluminal signaling?

    Principal question
    How does Reeh–Schlieder cyclicity coexist with locality, and why does local state approximation not imply cheap creation or superluminal signaling?
    What the source page covers
    This page owns the operational reading of vacuum cyclicity, local approximation of global vectors, unavoidable tails, success probability, and energy qualifications.
    Boundary
    It does not own the theorem proof and rejects the inference that dense local excitations constitute deterministic finite-energy remote control.
    Scope
    This page owns the operational reading of vacuum cyclicity, local approximation of global vectors, unavoidable tails, success probability, and energy qualifications.
    Assumptions
    Vacua, States, and Representations; Microcausality and Relativistic Compatibility; Regions, Causal Complements, and Nets of Observables; Von Neumann Factors and Type-III Local Algebras
  • Theorem Mechanism

    Recovery Maps and Approximate Markovianity

    What recovery guarantee follows from small conditional mutual information, and which algebra, channel, fidelity, and energy assumptions control it?

    Principal question
    What recovery guarantee follows from small conditional mutual information, and which algebra, channel, fidelity, and energy assumptions control it?
    What the source page covers
    This page owns existence and performance of recovery maps, information–disturbance interpretation, fidelity bounds, local domains, and approximation errors.
    Boundary
    It does not claim a unique, local, energy-feasible recovery operation from a dimension-free existence bound.
    Scope
    This page owns existence and performance of recovery maps, information–disturbance interpretation, fidelity bounds, local domains, and approximation errors.
    Assumptions
    Conditional Mutual Information and Quantum Markov Structure
  • Theorem Mechanism

    Relative Entropy and Modular Horizon Laws

    Under which algebraic and modular assumptions do relative entropy and modular energy yield horizon first-law, monotonicity, or generalized-entropy statements?

    Principal question
    Under which algebraic and modular assumptions do relative entropy and modular energy yield horizon first-law, monotonicity, or generalized-entropy statements?
    What the source page covers
    The page owns wedge and horizon algebras, modular Hamiltonians where known, relative-entropy finiteness, first variations, monotonicity, and causal-horizon scope.
    Boundary
    Generic modular theory remains in Volume XIII, nonlinear gravitational extremization follows later, and unknown modular Hamiltonians are not replaced by local ansätze.
    Scope
    The page owns wedge and horizon algebras, modular Hamiltonians where known, relative-entropy finiteness, first variations, monotonicity, and causal-horizon scope.
    Assumptions
    Fine-Grained, Coarse-Grained, and Algebraic Entropy; Relative Entropy for QFT States; Rindler Wedges and the Bisognano–Wichmann Theorem
  • Theorem Mechanism

    Propagation of the Hadamard Property

    Under which hyperbolic evolution and initial-data hypotheses does Hadamard singular structure propagate from a neighborhood of one Cauchy surface throughout spacetime?

    Principal question
    Under which hyperbolic evolution and initial-data hypotheses does Hadamard singular structure propagate from a neighborhood of one Cauchy surface throughout spacetime?
    What the source page covers
    The physical propagation-of-singularities argument linking local Cauchy data to global Hadamard admissibility, with operator, gauge, and global-hyperbolicity assumptions explicit.
    Boundary
    Volume I owns the general propagation-of-singularities theorem and Volume XVI owns proof-first Hadamard propagation for scalar, spinor, and gauge systems.
    Scope
    The physical propagation-of-singularities argument linking local Cauchy data to global Hadamard admissibility, with operator, gauge, and global-hyperbolicity assumptions explicit.
    Assumptions
    Hadamard Admissibility and the Two-Point Wavefront Criterion; Green Operators, Causal Propagators, and State-Dependent Two-Point Functions
  • Theorem Mechanism

    Quantum Trapped Surfaces and Semiclassical Singularity Theorems

    Which semiclassical singularity results use quantum trapped surfaces or generalized-entropy conditions, and what causality, completeness, regularity, and focusing hypotheses do they require?

    Principal question
    Which semiclassical singularity results use quantum trapped surfaces or generalized-entropy conditions, and what causality, completeness, regularity, and focusing hypotheses do they require?
    What the source page covers
    The page owns definitions of quantum trapped cuts, theorem assumption ledgers, null completeness conclusions, semiclassical validity conditions, and comparison with classical theorems.
    Boundary
    The page does not prove every cited geometric theorem, infer singularity resolution, or extend a theorem past the regime where generalized entropy is controlled.
    Scope
    The page owns definitions of quantum trapped cuts, theorem assumption ledgers, null completeness conclusions, semiclassical validity conditions, and comparison with classical theorems.
    Assumptions
    Quantum Expansion and Covariant Entropy Bounds; Raychaudhuri Evolution, Null Focusing, and Renormalized Stress; Averaged Null Energy in Curved Spacetime
  • Theorem Mechanism

    Semiclassical First Laws and Physical-Process Variations

    Which stationary and physical-process first laws follow from geometric variation, stress flux, and relative entropy, and how do their domains and perturbative orders differ?

    Principal question
    Which stationary and physical-process first laws follow from geometric variation, stress flux, and relative entropy, and how do their domains and perturbative orders differ?
    What the source page covers
    The page owns equilibrium-state versus physical-process variations, horizon flux balance, Noether charge, modular first order, boundary terms, and quasi-stationary hypotheses.
    Boundary
    A first law does not establish a second law, rapid nonlinear horizons are outside the derivation, and complete geometric proofs remain specialist handoffs.
    Scope
    The page owns equilibrium-state versus physical-process variations, horizon flux balance, Noether charge, modular first order, boundary terms, and quasi-stationary hypotheses.
    Assumptions
    Relative Entropy and Modular Horizon Laws; Noether-Charge Entropy and Higher-Curvature Terms; Conservation, Local Covariance, and the Backreaction Source
  • Theorem Mechanism

    The Generalized Second Law: Hypotheses and Proven Scope

    For which causal horizons, states, algebras, energy assumptions, and semiclassical approximations has generalized-entropy nondecrease been established rather than conjectured?

    Principal question
    For which causal horizons, states, algebras, energy assumptions, and semiclassical approximations has generalized-entropy nondecrease been established rather than conjectured?
    What the source page covers
    The page owns a theorem-domain ledger for causal horizons and proven axiomatic settings, generalized-entropy change, relative-entropy arguments, and explicit unresolved extensions.
    Boundary
    The page does not promote established causal-horizon results to a universal theorem for arbitrary trapping or dynamical horizons, higher-curvature theories, or quantum gravity.
    Scope
    The page owns a theorem-domain ledger for causal horizons and proven axiomatic settings, generalized-entropy change, relative-entropy arguments, and explicit unresolved extensions.
    Assumptions
    Generalized Entropy and UV Renormalization; Relative Entropy for QFT States; The Semiclassical Einstein Equation
  • Theorem Mechanism

    Exact Toy-Code Reconstruction Theorems

    What do exact tensor and finite-dimensional code theorems establish, and which gravitational assumptions are absent?

    Principal question
    What do exact tensor and finite-dimensional code theorems establish, and which gravitational assumptions are absent?
    What the source page covers
    The exact encoding, erasure pattern, logical algebra, complementary recovery, network geometry, and theorem hypotheses of canonical toy codes.
    Boundary
    Volume VIII owns tensor-network algorithms and Volume XIII owns QEC theorems; this page owns toy holographic interpretation only.
    Scope
    The exact encoding, erasure pattern, logical algebra, complementary recovery, network geometry, and theorem hypotheses of canonical toy codes.
    Assumptions
    Error Models, Codes, and Recovery Conditions; Entanglement Structure and Tensor-Network Ansätze
  • Theorem Mechanism

    Maximin Constructions and Extremal-Surface Existence

    When does a maximin construction establish the existence and causal properties of an HRT surface?

    Principal question
    When does a maximin construction establish the existence and causal properties of an HRT surface?
    What the source page covers
    The achronal-slice maximization and within-slice minimization, stability, homology, focusing assumptions, barriers, and theorem domain.
    Boundary
    Volume I owns Lorentzian geometry and Volume XIV owns focusing hypotheses; this page owns their holographic maximin application, not unrestricted existence.
    Scope
    The achronal-slice maximization and within-slice minimization, stability, homology, focusing assumptions, barriers, and theorem domain.
    Assumptions
    Covariant Extremal Surfaces and HRT; Causal Wedges and Subregion Reconstruction
  • Theorem Mechanism

    Complementary Recovery, Area Terms, and Center Data

    How do complementary recovery, shared center data, and generalized-entropy area terms fit together in a holographic code?

    Principal question
    How do complementary recovery, shared center data, and generalized-entropy area terms fit together in a holographic code?
    What the source page covers
    The operator-algebra recovery condition, complementary channel, center or superselection data, area-term role, and state-dependence qualification.
    Boundary
    Volume XIII owns complementary recovery; Chapter 14 owns proposed crossed-product structures; this page does not posit an unqualified universal area operator.
    Scope
    The operator-algebra recovery condition, complementary channel, center or superselection data, area-term role, and state-dependence qualification.
    Assumptions
    Complementary Recovery and Cleaning Relations; Leading Semiclassical JLMS and Code-Subspace Claims
  • Theorem Mechanism

    Entanglement Wedges, Nesting, and Information Inequalities

    How do extremal-surface geometry and boundary inclusion produce entanglement-wedge nesting and holographic entropy inequalities?

    Principal question
    How do extremal-surface geometry and boundary inclusion produce entanglement-wedge nesting and holographic entropy inequalities?
    What the source page covers
    The wedge definition, domain of dependence, nesting under boundary inclusion, strong subadditivity geometry, focusing assumptions, and quantum corrections.
    Boundary
    Volume XIII owns information inequalities; Chapter 15 owns reconstruction from wedges; this page owns geometric wedge and inequality statements only.
    Scope
    The wedge definition, domain of dependence, nesting under boundary inclusion, strong subadditivity geometry, focusing assumptions, and quantum corrections.
    Assumptions
    Covariant Extremal Surfaces and HRT; Maximin Constructions and Extremal-Surface Existence
  • Theorem Mechanism

    Holographic Entropy Inequalities and Entropy Cones

    Which entropy vectors are allowed by classical holographic geometry, and how do cone inequalities differ from universal quantum inequalities?

    Principal question
    Which entropy vectors are allowed by classical holographic geometry, and how do cone inequalities differ from universal quantum inequalities?
    What the source page covers
    The holographic entropy cone, graph and cut models, multipartite inequalities, extreme rays, dimension or party dependence, and classical large-N scope.
    Boundary
    Volume XIII owns multipartite information and general entropy cones; this page owns the extra geometric constraints and their holographic status.
    Scope
    The holographic entropy cone, graph and cut models, multipartite inequalities, extreme rays, dimension or party dependence, and classical large-N scope.
    Assumptions
    Entanglement Wedges, Nesting, and Information Inequalities; Multipartite Information and Entropy Cones
  • Theorem Mechanism

    Haag–Ruelle Scattering-State Construction

    How do almost-local operators with separated velocity supports create incoming and outgoing multiparticle states in a massive local QFT?

    Principal question
    How do almost-local operators with separated velocity supports create incoming and outgoing multiparticle states in a massive local QFT?
    What the source page covers
    Haag–Ruelle approximants, mass-gap and stability hypotheses, velocity-support separation, commutator decay, strong limits, Fock inner products, and independence from interpolating operators.
    Boundary
    Volume IV retains LSZ calculations and cross sections; this page proves existence of scattering states and does not assume asymptotic completeness.
    Scope
    Haag–Ruelle approximants, mass-gap and stability hypotheses, velocity-support separation, commutator decay, strong limits, Fock inner products, and independence from interpolating operators.
    Assumptions
    Jost Points, Edge-of-the-Wedge, and Locality; Clustering, Vacuum Uniqueness, and Mass-Gap Implications; Particles, Mass-Shell Spectrum, and One-Particle Subspaces
  • Theorem Mechanism

    The Wightman Reconstruction Theorem

    Under which distributional axioms does a hierarchy of vacuum n-point functions reconstruct a Hilbert space, vacuum, Poincaré representation, and operator-valued fields?

    Principal question
    Under which distributional axioms does a hierarchy of vacuum n-point functions reconstruct a Hilbert space, vacuum, Poincaré representation, and operator-valued fields?
    What the source page covers
    The reconstruction theorem from the Borchers algebra quotient: positive sesquilinear form, null ideal, completion, dense domain, field action, covariance, locality, cyclicity, and uniqueness up to unitary equivalence.
    Boundary
    The theorem reconstructs a Wightman realization from a complete hierarchy; it does not produce that hierarchy, prove cutoff removal, or imply equivalence with a Euclidean or net formulation.
    Scope
    The reconstruction theorem from the Borchers algebra quotient: positive sesquilinear form, null ideal, completion, dense domain, field action, covariance, locality, cyclicity, and uniqueness up to unitary equivalence.
    Assumptions
    Theorem-First Claim Records: Objects, Hypotheses, Conclusions, and Status; Wightman Fields, Domains, and Axioms; Wightman Functions and Spectral Support
  • Theorem Mechanism

    Osterwalder–Schrader Reconstruction

    Under which complete OS hypotheses can Euclidean Schwinger functions be continued to a local, positive-energy Wightman theory, and in what sense is the result unique?

    Principal question
    Under which complete OS hypotheses can Euclidean Schwinger functions be continued to a local, positive-energy Wightman theory, and in what sense is the result unique?
    What the source page covers
    The OS reconstruction chain from reflected Hilbert space through analytic continuation, Poincaré representation, operator-valued fields, locality, vacuum, and equality of the reconstructed Schwinger boundary values.
    Boundary
    Constructive QFT must first supply Schwinger functions satisfying the hypotheses; the theorem neither removes a cutoff nor states the converse for every Wightman theory without additional bounds.
    Scope
    The OS reconstruction chain from reflected Hilbert space through analytic continuation, Poincaré representation, operator-valued fields, locality, vacuum, and equality of the reconstructed Schwinger boundary values.
    Assumptions
    Theorem-First Claim Records: Objects, Hypotheses, Conclusions, and Status; Euclidean Random Fields and Schwinger Hierarchies; Osterwalder–Schrader Axioms and Reflection Positivity; Reflection Positivity and Hilbert-Space Reconstruction
  • Theorem Mechanism

    LSZ Reduction and Amputated Distributions

    Under which pole, regularity, asymptotic-state, and wave-packet hypotheses does LSZ express S-matrix elements as amputated time-ordered distributions?

    Principal question
    Under which pole, regularity, asymptotic-state, and wave-packet hypotheses does LSZ express S-matrix elements as amputated time-ordered distributions?
    What the source page covers
    A distributional LSZ theorem with stable external shells, field-strength residues, connected parts, smearing, amputation, on-shell limits, normalization, and failure for unstable or infraparticle legs.
    Boundary
    Volume IV owns the practical derivation, Feynman rules, and observable normalization; this page supplies the rigorous hypotheses linking correlators to already constructed scattering states.
    Scope
    A distributional LSZ theorem with stable external shells, field-strength residues, connected parts, smearing, amputation, on-shell limits, normalization, and failure for unstable or infraparticle legs.
    Assumptions
    Wightman Functions and Spectral Support; Haag–Ruelle Scattering-State Construction; Wave Operators and Asymptotic Fields
  • Theorem Mechanism

    Tube Domains, Complex Lorentz Covariance, and Analyticity

    How does the spectral condition turn Wightman distributions into boundary values of holomorphic functions on forward tubes, and how far does complex Lorentz covariance extend them?

    Principal question
    How does the spectral condition turn Wightman distributions into boundary values of holomorphic functions on forward tubes, and how far does complex Lorentz covariance extend them?
    What the source page covers
    The Fourier–Laplace construction of primitive and extended tubes, precise imaginary-direction cones, boundary-value topology, complex Lorentz action, and the hypotheses behind analytic continuation.
    Boundary
    Scattering dispersion relations remain in Volume IV and Euclidean reconstruction in Chapter 3; this page establishes Wightman-domain analyticity without claiming crossing or OS equivalence.
    Scope
    The Fourier–Laplace construction of primitive and extended tubes, precise imaginary-direction cones, boundary-value topology, complex Lorentz action, and the hypotheses behind analytic continuation.
    Assumptions
    Wightman Functions and Spectral Support; Holomorphic Functions and Cauchy Theory; Tempered Distributions and Fourier Calculus
  • Theorem Mechanism

    Jost Points, Edge-of-the-Wedge, and Locality

    How do locality and edge-of-the-wedge arguments enlarge analytic domains at Jost configurations and relate differently ordered Wightman functions?

    Principal question
    How do locality and edge-of-the-wedge arguments enlarge analytic domains at Jost configurations and relate differently ordered Wightman functions?
    What the source page covers
    Jost-point geometry, weak local commutativity, common real boundary regions, envelope continuation, and the exact edge-of-the-wedge conclusion used by structural theorems.
    Boundary
    Volume IV owns amplitude crossing and the following pages own CPT and spin–statistics variants; this page supplies the analytic-locality bridge without identifying every permutation with a scattering channel.
    Scope
    Jost-point geometry, weak local commutativity, common real boundary regions, envelope continuation, and the exact edge-of-the-wedge conclusion used by structural theorems.
    Assumptions
    Wightman Fields, Domains, and Axioms; Tube Domains, Complex Lorentz Covariance, and Analyticity
  • Theorem Mechanism

    Analyticity, CPT, and Spin–Statistics

    Which shared analyticity and locality ingredients underlie CPT and spin–statistics theorems, and where do their hypotheses and conclusions diverge?

    Principal question
    Which shared analyticity and locality ingredients underlie CPT and spin–statistics theorems, and where do their hypotheses and conclusions diverge?
    What the source page covers
    The common proof architecture through complex Lorentz covariance and Jost-point identities, followed by separate ledgers for antiunitary CPT action and statistics–spin compatibility.
    Boundary
    The next two pages own theorem variants and failure modes; Volume II retains physical consequences and this page does not merge CPT invariance with spin–statistics as one assertion.
    Scope
    The common proof architecture through complex Lorentz covariance and Jost-point identities, followed by separate ledgers for antiunitary CPT action and statistics–spin compatibility.
    Assumptions
    Positivity, Spectrum, Covariance, and Locality Hypotheses; Tube Domains, Complex Lorentz Covariance, and Analyticity; Jost Points, Edge-of-the-Wedge, and Locality
  • Theorem Mechanism

    Analytic Continuation between Euclidean and Lorentzian Domains

    On which complexified spacetime domains and in which boundary-value sense can Euclidean Schwinger functions and Lorentzian Wightman functions be related?

    Principal question
    On which complexified spacetime domains and in which boundary-value sense can Euclidean Schwinger functions and Lorentzian Wightman functions be related?
    What the source page covers
    Ordered imaginary-time regions, permuted tubes, singular loci, boundary prescriptions, continuation paths, and the difference between analytic continuation of distributions and naïve substitution of time variables.
    Boundary
    Thermal continuation belongs to Volume XI and perturbative Wick rotation to Volumes II and IV; this page owns the reconstruction-level domain statement.
    Scope
    Ordered imaginary-time regions, permuted tubes, singular loci, boundary prescriptions, continuation paths, and the difference between analytic continuation of distributions and naïve substitution of time variables.
    Assumptions
    Tube Domains, Complex Lorentz Covariance, and Analyticity; Osterwalder–Schrader Reconstruction; Euclidean Growth, Regularity, and Temperedness Conditions
  • Theorem Mechanism

    Clustering, Vacuum Uniqueness, and Mass-Gap Implications

    What do Euclidean clustering rates imply about vacuum purity, uniqueness, spectral gaps, and decay of reconstructed correlations, and which converses require extra assumptions?

    Principal question
    What do Euclidean clustering rates imply about vacuum purity, uniqueness, spectral gaps, and decay of reconstructed correlations, and which converses require extra assumptions?
    What the source page covers
    Weak versus strong clustering, truncated correlations, extremality, transfer-Hamiltonian spectrum, exponential-decay implications, and carefully qualified mass-gap correspondences.
    Boundary
    Physical confinement and spectral phenomenology remain in Volumes VI–VII; this page proves only consequences licensed by the reconstructed Euclidean model.
    Scope
    Weak versus strong clustering, truncated correlations, extremality, transfer-Hamiltonian spectrum, exponential-decay implications, and carefully qualified mass-gap correspondences.
    Assumptions
    Positivity, Spectrum, Covariance, and Locality Hypotheses; Osterwalder–Schrader Reconstruction; Euclidean Growth, Regularity, and Temperedness Conditions
  • Theorem Mechanism

    CPT Theorem Variants and Their Hypotheses

    What do field-theoretic CPT theorems assert for scalar, spinorial, charged, and non-Wightman settings, and which positivity, locality, covariance, or geometric assumptions does each version use?

    Principal question
    What do field-theoretic CPT theorems assert for scalar, spinorial, charged, and non-Wightman settings, and which positivity, locality, covariance, or geometric assumptions does each version use?
    What the source page covers
    A variant-by-variant CPT theorem table specifying field transformation, order reversal, antiunitarity, vacuum action, spacetime inversion, uniqueness qualifications, and scope.
    Boundary
    Discrete-symmetry phenomenology and Standard Model phases remain in Volume VI; this page proves structural statements and does not treat CPT violation bounds.
    Scope
    A variant-by-variant CPT theorem table specifying field transformation, order reversal, antiunitarity, vacuum action, spacetime inversion, uniqueness qualifications, and scope.
    Assumptions
    Analyticity, CPT, and Spin–Statistics; Jost Points, Edge-of-the-Wedge, and Locality; Clifford Algebras and Pin and Spin Groups
  • Theorem Mechanism

    Spin–Statistics Theorems and Failure Modes

    Under which relativistic QFT hypotheses is integer or half-integer spin tied to Bose or Fermi statistics, and what changes in low dimensions, indefinite metric, or nonlocal theories?

    Principal question
    Under which relativistic QFT hypotheses is integer or half-integer spin tied to Bose or Fermi statistics, and what changes in low dimensions, indefinite metric, or nonlocal theories?
    What the source page covers
    Precise spin–statistics statements for Wightman and algebraic settings, their positivity and localization inputs, and a failure atlas covering braid statistics and weakened locality.
    Boundary
    Particle-statistics calculations stay in Volume II, anyons in Volumes IX and XII, and DHR or braided-sector derivations in Chapter 7.
    Scope
    Precise spin–statistics statements for Wightman and algebraic settings, their positivity and localization inputs, and a failure atlas covering braid statistics and weakened locality.
    Assumptions
    Wightman Fields, Domains, and Axioms; Analyticity, CPT, and Spin–Statistics; Positivity, Spectrum, Covariance, and Locality Hypotheses
  • Theorem Mechanism

    Haag's Theorem and Inequivalent Representations

    Why do the assumptions behind a unitary interaction picture force two relativistic fields toward the same vacuum correlation structure, obstructing the naïve free–interacting identification?

    Principal question
    Why do the assumptions behind a unitary interaction picture force two relativistic fields toward the same vacuum correlation structure, obstructing the naïve free–interacting identification?
    What the source page covers
    A domain-conscious Haag-theorem statement, the equal-time intertwiner assumptions, vacuum and covariance inputs, propagation to two-point equality, and the inequivalent-representation conclusion.
    Boundary
    Volume IV owns practical interaction-picture perturbation theory and Chapter 13 owns pAQFT; this theorem diagnoses a representation claim rather than declaring perturbation theory meaningless.
    Scope
    A domain-conscious Haag-theorem statement, the equal-time intertwiner assumptions, vacuum and covariance inputs, propagation to two-point equality, and the inequivalent-representation conclusion.
    Assumptions
    Wightman Fields, Domains, and Axioms; The Wightman Reconstruction Theorem; Fock Space, Vacuum, and Particle Number