Definition
Admissibility
Admissibility asks whether an object or comparison satisfies the requirements of the particular construction in which it is being used.
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Definition
Finite Refinement Proposal
A finite refinement proposal specifies what a finite construction is being asked to realize.
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Well-Typed Proposal
A proposal is well typed when every component has the right declared source, target, and conditions of use.
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Realization-Complete Proposal
A proposal is realization complete when it contains enough information to ask every required physical-realization question.
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Formal Equivalence
Formal equivalence means that two proposals specify the same finite problem after permitted changes of presentation.
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Thick Wilson Candidate
A thick Wilson candidate is a proposed positive-thickness realization of a finite Wilson-type construction.
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Finite Physical Realization Predicate
The finite physical realization predicate is the full test a candidate must pass to count as physically realized.
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Admissible Realization Fiber
The admissible realization fiber is the collection of all certified physical realizations of one formal proposal.
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Finite Physical Realization
A finite physical realization is an actual member of the admissible realization fiber.
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Complete Defect Ledger
A complete defect ledger detects exactly when all required realization conditions hold.
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Finite Physical-Cohomological-Coercive Certificate
A finite physical–cohomological–coercive certificate records a realized package, its obstruction data, and a proved positive excitation bound.
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Finite Gauge-Presentation Cover
A finite gauge-presentation cover organizes local descriptions of the same proposed gauge object.
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Finite Spacetime Cover
A finite spacetime cover organizes the regions on which a candidate’s or realization’s data are supported.
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Rectangular Compatibility
Rectangular compatibility means that a gauge comparison and a regional restriction can both be posed on the same valid data.
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Finite Gauge-Spacetime Index
The finite gauge–spacetime index keeps compatible gauge and regional tuples together while retaining their two separate roles.
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Bigraded Finite Realization Carrier
A bigraded finite realization carrier places the realization data over both the gauge and spacetime indices.
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Finite Yang-Mills Physical Package
The finite Yang–Mills physical package collects the realized geometry, operators, sectors, ground, local algebras, constraints, and comparison data.
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Detector Readout
Detector readout is the compression of a regional operator to the chosen detector carrier.
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Finite Admissible Region System
A finite admissible region system specifies which regions and supported inclusions the realized construction actually permits.
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Fiberwise Local Algebra
A fiberwise local algebra is the algebra of admitted observables supported in one region of one physical realization.
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Finite Gauss Certificate
A finite Gauss certificate records the constraint-defined physical space and how observables act on it.
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Finite Ward Profile
A finite Ward profile records the symmetry variations and state identities actually verified for the finite package.
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Ward Seed over the Two-Nerve Carrier
A Ward seed over the two-nerve carrier tracks the Ward data under both gauge changes and regional restrictions.
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Finite AQFT Probe Seed
A finite AQFT probe seed organizes the finite data needed to ask algebraic quantum-field-theory questions.
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Representation-Family Slot
The representation-family slot records the permitted state and representation realizations without prematurely choosing one.
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Strict Defect Coefficient System
A strict defect coefficient system provides consistently composable additive coordinates for measuring comparison failures.
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Filling Obstruction
The filling obstruction tests whether a prescribed mismatch can be filled by an allowed correction.
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Internal Coefficient-Band Class
The internal coefficient-band class records the obstruction carried by reassociating coefficient-line compositions.
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Finite Coherence Type
Finite coherence type distinguishes coefficient systems that can be strictified from those with an intrinsic twist.
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Selector-Comparison Certificate
A selector-comparison certificate supplies the coherent comparison data needed to turn selected fillings into band loops.
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Banded Tensor-Selector Class
The banded tensor-selector class is the degree-two class obtained by evaluating the band’s degree-three class along the certified selected loops.
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Plaquette-Local Physical Hamiltonian Seed
A plaquette-local physical Hamiltonian seed combines electric edge terms, magnetic loop terms, and seam or boundary terms on a finite physical carrier.
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Hamiltonian Descent Certificate on the Two-Nerve
A Hamiltonian descent certificate accounts for how the full Hamiltonian changes under gauge comparisons and regional restrictions.
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Exact Complete-Ground Projection
The exact complete-ground projection projects onto every ground state of the full Hamiltonian.
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Coercive Cone
The coercive cone is the set of two-block lower-bound data that certify a chosen floor κ after the coupling between blocks is included.
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Seam Reserve Vector
The seam reserve vector records whether a comparison has enough capacity to absorb degradation and coupling at a seam.
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Spectrally Legal Finite Surgery
A spectrally legal finite surgery is a controlled change of a certified finite package with explicit comparison estimates.
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Finite Analytic Shadow
The finite analytic shadow retains the operator and quantitative comparison information of a package while forgetting its cohomological coordinates.
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Banded 2-Transport Certificate
A banded 2-transport certificate makes coefficient-line transport coherent along physical passages and their comparison surfaces.
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Calibrated Mixed Interaction Shadow
The calibrated mixed interaction shadow is the phase comparing an actual mixed surface with its independently certified calibration.
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Finite Admissible Decoupling
Finite admissible decoupling means that the actual mixed comparison factors through independently certified marginal comparisons with the required relative higher comparison.
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Hamiltonian-Bearing Admissible Comparison
A Hamiltonian-bearing admissible comparison transports the energy structure as well as the carrier and observable algebra.
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Admissible Grassmannian Transport Category
The admissible Grassmannian transport category organizes allowed projection-selected subspaces and the certified passages between them.
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Global Admissible Realization
A global admissible realization is a coherent choice of local realizations and comparisons throughout the selected indexing category.
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Admissible Section Module
An admissible section module is the family of module fibers belonging to a coherent sector, together with their transport and composition data.
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Representable Physical Section
A representable physical section is a section module that has passed the additional requirements for a physical Hilbert-space realization.
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Restricted Admissible Grassmannian
The restricted admissible Grassmannian keeps the admissible projections whose difference from a chosen reference projection lies in a declared operator ideal.
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Total Comparison Form
The total comparison form measures the combined defect of a proposed comparison in the declared comparison space.
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Genuine Admissible Transport
Genuine admissible transport is a represented comparison that meets all the support, algebra, physical-form, reference, and coherence conditions on the effective branch.
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Spatially Effective Selected Correspondence
A selected correspondence is spatially effective when a concrete corner operator realizes it as the graph of an onto algebra isomorphism.
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Local Linking Defect
The local linking defect measures how far a selected local correspondence is from having the required graph realization.
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Sector-Faithful Central Transport
Sector-faithful central transport preserves the marked decomposition into physical comparison sectors.
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Uniformly Separated Central Marking
A uniformly separated central marking keeps distinct branch labels a definite spectral distance apart along the certified refinement tail.
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Pre-Spectral Branch Retention
Pre-spectral branch retention means that every marked source packet reaches exactly one allowed represented destination without losing rank.
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Asymptotically Central Source-Native Marker
An asymptotically central source-native marker is a branch-label operator whose commutators with the determining local observables tend to zero.
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Separating Marker Subfamily
A separating marker subfamily distinguishes all physical branch classes that the complete marker family distinguishes.
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Relative Linking Obstruction
The relative linking obstruction tests whether selected local source–destination spatializers can fit into one coherent graph comparison.
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Effective Twisted Spatialization
Effective twisted spatialization realizes the selected correspondence coherently while working in its prescribed twisted coefficient system.
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Finite-Window Zero-Cost Normalizer
A finite-window zero-cost normalizer is an internal unitary change that preserves all the declared comparison data on that window.
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Transport and Migration Spaces
Transport and migration spaces separate zero-defect comparisons that retain the source sector from those reaching inequivalent admissible sectors.
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Represented Destination Family
The represented destination family lists the effective destination sectors actually carried by the nonzero marked supports.
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Compactness Modulo Admissible Sector Transport
Compactness modulo admissible sector transport means bounded-energy comparisons can be made convergent by permitted retargeting to represented destinations.
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Branch Certificate
A branch certificate states the concrete data, witnesses, estimates, and refinement tail on which a hypothesis has been verified.
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Coherent Certificate
A coherent certificate is a family of certificates whose witnesses agree across the retained faces and compositions.
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Derived Certificate Package
A derived certificate package assembles new certified conclusions from compatible primitive certificates and explicit theorem implications.
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Finite Physical Yang-Mills Cargo
Finite physical Yang–Mills cargo arranges an already certified finite realization according to what continuation must carry forward.
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Certified Admissible Continuation
Certified admissible continuation is a full-package comparison across a specified admitted passage, with an explicit account of each retained coordinate.
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Orbit-Covariant Spectral System
An orbit-covariant spectral system is spectral data equipped with a compatible symmetry action and a nonnegative height function.
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Orbit Floor
The orbit floor is the lowest value a height function approaches anywhere along a symmetry orbit.
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Thin Dynamical Packet
A thin dynamical packet collects a closed excitation form, its operator, its exact ground, and a positive lower bound before invariant spectral identification.
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Selected Transport Base
The selected transport base specifies the contexts and changes of context for which the source supplies a meaningful physical comparison problem.
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Liftable Locus
The liftable locus of a base arrow consists of source realizations that have at least one certified physical passage over that arrow.
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Raw Certified Realization
A raw certified realization keeps the substantive reconstruction data together with the chosen certificate and auxiliary presentation choices.
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Raw Certified Comparison
A raw certified comparison records both the admitted base arrow and the full physical comparison over it.
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Certificate Refinement Class
The certificate refinement class consists of changes that refine how a fixed intrinsic datum or passage is certified.
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Enhancement Change Class
The enhancement change class consists of changes to auxiliary anchors, roots, or Cauchy presentation choices.
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Substantive Physical Comparison
A substantive physical comparison is the actual continuation information retained after justified changes of certification and auxiliary presentation are removed.
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Configured Persistent Floor
The configured persistent floor is the common lower bound that survives across an entire certified branch.
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Gauge-Invariant Markov Ancestor
The gauge-invariant Markov ancestor is the electric–magnetic Dirichlet-form structure before sharp detector compression.
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Ground-State Markovization
Ground-state Markovization converts an independently certified Dirichlet realization with a strictly positive one-dimensional ground into conservative Markov evolution.
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Markov Coarse-Graining Shadow
A Markov coarse-graining shadow is a finite-rank, gauge-compatible averaging operation that preserves positivity and constants.
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Measurement Markov Interface
The Measurement Markov Interface exports a certified relaxation system in a form that downstream measurement research can use.
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Full descent diagram
For a cover
Definition
Admitted witness system
An admitted witness system is a small indexing category
Definition
Witness reconstruction object
The object reconstructed from the admitted witness subdiagram; it records exactly the local data and comparisons that have been retained.
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Reconstruction fiber
The space of full reconstructions compatible with specified admitted data. Its emptiness, components, and higher homotopy describe obstruction, ambiguity, and invisible automorphisms.
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Complete witness system
A witness system is complete for the theory when
Definition
Truncated limit-initiality
For an
Definition
Relative defect complex
For a restriction of abelian cochain complexes
Definition
Abelian globalization obstruction
A relative cohomology class that detects whether admitted abelian cocycle data extend to the full descent diagram.
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Completion tower and filler space
A staged reconstruction in which each added coherence requirement has a space of possible fillers.
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Transported-action obstruction
A chosen
Definition
Witnesswise-complete refinement
A refinement is witnesswise complete when it preserves both the full reconstruction object and the admitted witness reconstruction object by equivalences, compatibly with the comparison maps.
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Reconstruction equivalence
Two witness systems are reconstruction-equivalent for a diagram when their admitted homotopy limits are equivalent compatibly with the maps from full descent.
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Ward-compatible BV diagram
A Ward-compatible local BV diagram assigns a cochain complex with differential
Definition
Yang–Mills equation-bundle descent
The Yang–Mills equation is represented as a section of an equation bundle over the stack of connections. The solution stack is the derived equalizer of the Euler–Lagrange section and the zero section in that slice.
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Dirichlet Passage Pencil
The Dirichlet passage pencil is the finite positive matrix problem that measures the least ordinary-gluing energy carried by a selected charged passage after every allowed interior relaxation and every common Dirichlet zero mode have been removed.
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Energetic Confinement on a Selected Branch
Energetic confinement on a selected branch means that separating the two nontrivial center sectors by distance
Definition
Relative Center Class
The relative center class records the failure of a marked external
Definition
Relative Seam Defect
The relative seam defect is the ordinary gluing jump restricted to boundary data that already satisfy the source-twisted matching law.
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Renormalization Curvature
Renormalization curvature measures the failure of refinement and charged passage transport to commute on the selected branch.
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Seam Reserve
The seam reserve is the positive fraction of the local Dirichlet passage cost that remains after partition, boundary-comparison, and catalog-representative errors are subtracted.
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Coherent Selected Branch
A coherent selected branch is a compatible family of physical choices—carriers, source and fusion sectors, comparison maps, associators, form domains, complete-ground lineage, and Hamiltonian-descent controls—on which the later analytic statements are actually defined.
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Certified Raw Arrow
A certified raw arrow is a raw refinement whose observable and carrier maps satisfy GLC, whose carrier preserves the distinguished vector, and whose carrier is Dirichlet-covariant.
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Common-Refinement Equivalence
Two raw refinement presentations are common-refinement equivalent when they are related by admitted refinement moves inside a source- and target-preserving categorical congruence.
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Dirichlet Covariance
A carrier map is Dirichlet-covariant when it intertwines the scale-dependent Markov semigroups:
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Ground-Lineage Compatibility
Ground-lineage compatibility says that the observable coarse-graining map and the selected Hilbert-space carrier map act identically on vectors generated from the distinguished carrier state:
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Dirichlet Carrier Category
Definition
Local Ground-Lineage Compatibility
For a region
Definition
Fiber / Fibration Distinction
A fiber collects realizations over one specified input. A fibration also supplies coherent ways to move realizations along changes of input.
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Admissible Transport
Admissible transport first meant scale-comparison transport encoded by a 2-lax natural transformation; later GAT formulations organize more general certified physical comparisons.
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Vacuum Lineage
A vacuum lineage identifies complete ground sectors across changing carriers through certified, coherently composable comparisons.
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Maximal nonclosability
Every target value can occur as a graph limit over source vectors tending to zero. A closed single-valued operator cannot have this behavior.
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Expectation-orthonormal unitary family
Orthogonality is measured by the conditional expectation. This is the precise multiplicity hypothesis in the general nonclosability theorem.
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Descent defect
The defect records outputs that remain when the chosen realization makes the inputs disappear. If
Definition
Source-window system
A window specifies how much source information has been exposed. The allowed enlargements are part of the problem.
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Continuation context
It specifies which operations the realization must support. This varies independently of the source window.
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Context kernel and core carrier
The quotient identifies exactly those source differences invisible to every demanded continuation. It is the coarsest sufficient algebraic carrier; choosing an analytic completion is a further step.
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Analytic realization and admissibility
Admissibility is relative to these declared carriers, topologies, domains, and allowed maps. It does not simply mean that an algebraic formula exists.
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Source-restriction stability
Shrinking a source must respect both the carrier class and the allowed operator class.
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Rectangular obstruction
Each separate enlargement can succeed while their combination fails. Under restriction stability the fourth, weakest corner is also admissible.
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Directed source rectangular obstruction
The total source and its completion are part of the declaration. Finite-stage success does not determine the answer there.
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Directed graph defect
The graph defect tests what survives when finite source vectors vanish in the total norm. It is zero exactly when the induced operator is closable.
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Graph-controlled witness
The witness cannot grow without bound along sequences that are Cauchy in the continuation graph norm.
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Closably totalizable witness
Its graph closure must remain single-valued; boundedness of each finite restriction does not establish this.
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Relative analytic survival
Both frameworks and their admissibility conditions must be named. This is a comparison of realizations of an operation.
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Context-tight realization
Reduction to the demanded context has stabilized. The reduction calculus, including its recognition axiom, is an explicit assumption.
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Matching family
Local realizations agree on every shared continuation demand.
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Effective matching family
Compatibility becomes effective only when a permitted global tight realization actually exists.
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Admissible amalgam
The amalgam carries all demanded information in one permitted object. It need not already be tight.
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Recognition
Recognition is the uniqueness part of descent. Existence of an amalgam is a separate requirement.
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Pro-realization
A coherent family of finite carriers need not be represented by one carrier in the prescribed analytic class.
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Uniform finite relative coercivity
The constant is independent of the finite context. Separate finite constants do not supply the total certificate.
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Finite, marginal, and joint totalizability
The two marginal conditions test different limits. Neither definition incorporates the doubly totalized corner.
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Mixed incidence
It measures what the new source increment adds to a given reconstruction component. This is an algebraic quotient, before analytic closure.
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Relative coercivity of retention and assembly
The retained data are controlled by the source and assembled output together. This is a graph-control condition, not an energy-gap assertion.
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Graph equivalence of certificates
The change of retained representation preserves both graph geometry and the decoded operation.
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Persistent strictifiability
One can choose strictifiers compatible across all refinements.
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Finite compatible strictifiability
This asks for compatible choices on finite diagrams, rather than merely a choice at each isolated stage.
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Certificate-bearing strictifier datum
The relational repair travels with the analytic certificate that proves it preserves the operation.
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Intrinsic mixed distortion
This is the best worst-case two-sided comparison bound within the declared route-preserving equivalence class.
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Mixed relative coercivity
The source variable for this certificate is the output of route
Definition
Uniform mixed relative coercivity
Uniformity is across finite contexts. This controls retained graph data and allows closed unbounded total comparisons.
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Finite reconstruction depth
Depth depends on the chosen presentation and declared continuation problem. It is not asserted to be an intrinsic invariant of every relational object.
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Adequate provenance package
Adequacy is measured against the claims being made. No minimality is required.
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