Mathematical registry
Source-bound theorem records with hypotheses, dependencies, exports, and status.
Theorems, propositions, and corollaries: hypotheses, conclusions, examples, and claim limits. Source designations and manuscript status remain visible.
When certified physical continuation supplies coherent reindexing functors on a selected admissible branch, the realizations and their comparisons form a Grothendieck fibration over its refinement category. The lifting property packages continuation already established by the source theory.
Read statement and scope →If the distinguished source vector is a Dirichlet zero mode and the carrier transports it to the distinguished target vector, then the target vector is also a Dirichlet zero mode.
Read statement and scope →The reconstruction must preserve the declared ground and spectral structure through its limiting comparisons. Published comparison–Mosco results explain preservation of a coercive lower bound and the exact thin ground; the registry’s broader no-leakage title still needs its precise theorem and source binding.
Read statement and scope →For a sharply covariant spectral system, a positive height exclusion holds along every admitted symmetry orbit. It therefore yields an invariant lower bound for the orbit-floor operator, with the same squared-threshold constant.
Read statement and scope →Two common-refinement-equivalent raw presentations may each be individually certified while inducing different carrier maps. Certification therefore does not imply presentation independence.
Read statement and scope →A unital completely positive, trace-preserving observable map can fail GLC relative to a separately selected carrier map:
Read statement and scope →Composing certified carrier maps preserves the distinguished vectors and Dirichlet covariance. If the observable maps also compose on the transport core, GLC holds for the composite.
Read statement and scope →Nested finite von Neumann algebras with trace-preserving conditional expectations and compatible depolarizing Dirichlet semigroups satisfy GLC, Dirichlet covariance, raw composition, and common-refinement invariance simultaneously.
Read statement and scope →GLC and preservation of the distinguished vector do not imply Dirichlet covariance.
Read statement and scope →Two strongly continuous unitary groups on the same Hilbert space are equal if they implement the same dynamics on a represented algebra and fix the same cyclic vector. Their self-adjoint generators then agree, including their maximal domains.
Read statement and scope →A certified Dirichlet realization with a normalized strictly positive rank-one ground admits a ground-state transform whose semigroup preserves positivity and constants and is symmetric in the ground-weighted measure.
Read statement and scope →Before sharp detector compression, the finite electric–magnetic form is a closed symmetric Dirichlet form on the compact configuration space. Restricting it to gauge-invariant functions preserves that structure.
Read statement and scope →When the certified ground-state bridge preserves the configured positive floor, that same constant bounds variance by Dirichlet energy and gives an exponential rate of return to equilibrium in the ground-weighted L² norm.
Read statement and scope →The selected certified physical realizations, their full-package comparisons, and invertible comparison modifications form a bicategory. Composition retains the prescribed physical reassociators and defect data, and projects coherently to the selected transport base.
Read statement and scope →A seam form can be added to the electric–magnetic Dirichlet form while preserving the Dirichlet property if it is gauge invariant, satisfies the normal-contraction inequality, has the required common domain/core, and yields a closed form sum. Nonnegativity alone is insufficient.
Read statement and scope →For an
The homotopy long exact sequence of the reconstruction fiber simultaneously records existence, ambiguity, and hidden symmetry. This is the formal bridge between “the witnesses fail” and the precise homotopical way in which they fail.
Read statement and scope →An inclusion of witness systems induces a comparison of reconstruction objects and fibers. There is no general monotonicity theorem based only on the number of witnesses.
Read statement and scope →Dold–Kan realization identifies the truncated shifted relative complex with the reconstruction 2-groupoid. For a liftable field,
Read statement and scope →The class in
A missing tetrahedral coherence supports a nonzero
Deleting a dominated patch star can preserve reconstruction. An explicit chain contraction makes the relative defect complex acyclic. This is the paper's sparse positive certificate.
Read statement and scope →If every reachable stage of a finite completion tower has a contractible filler space, the initial admitted data reconstruct the full nonabelian field. The theorem is pointwise: it separates completion of one chosen partial field from a uniform equivalence for all fields.
Read statement and scope →For the split crossed module
A witnesswise-complete refinement preserves the reconstruction fiber. In the abelian model it also preserves the relative defect complex up to quasi-isomorphism.
Read statement and scope →Under barycentric subdivision, the original tetrahedron becomes 24 refined tetrahedra. Their defects satisfy one alternating-product invariant:
Read statement and scope →Given a coherent diagram of already-constructed local BV complexes, admitted totalization reconstructs the BV complex when the relevant comparison is an equivalence. Observable-algebra reconstruction additionally requires multiplicative totalization and multiplicative restriction maps.
Read statement and scope →Local
If connection descent, equation-bundle descent, both sections, the relevant slice limits, and preservation of the finite homotopy limit are supplied, then the Lorentzian Yang–Mills solution stack descends as a derived equalizer.
Read statement and scope →Equivalent witness presentations can have different numbers of objects and arrows. Minimal reconstruction complexity cannot be defined by raw witness count alone.
Read statement and scope →A particular diagram may admit a finite reconstruction core even when no one finite subcategory reconstructs every diagram of the relevant truncation level.
Read statement and scope →A connected filler space may retain a nontrivial automorphism group. Existence and connectedness do not imply the contractibility required for reconstruction.
Read statement and scope →Arbitrary screening data built from the
The two nontrivial
Persistent adjoint-unscreened center charge, a coherent selected carrier, a uniform positive passage floor, oriented Stokes propagation, linear separator packing, Morita noncollapse, and controlled refinement imply a linear lower bound for static-source separation energy on the selected branch.
Read statement and scope →A normalized selected passage block has a strictly positive Dirichlet passage eigenvalue exactly when its selected ordinary-boundary realization remains injective after quotienting the common Dirichlet zero modes.
Read statement and scope →Under the confinement-reduction hypotheses, the minimum energy in either nontrivial center sector diverges as the source separation tends to infinity, and pure adjoint screening does not change that conclusion.
Read statement and scope →Across a source-free corridor, the center label and signed conormal response propagate with orientation, while the positive quadratic passage costs remain positive and add over disjoint collars.
Read statement and scope →When the raw
GLC, preservation of the distinguished vector, Dirichlet covariance, raw functoriality, and common-refinement invariance produce fixed-sector transport, a raw certified functor, and its unique descent to the quotient scale category.
Read statement and scope →A certified raw arrow simultaneously realizes the observable map on the declared carrier and preserves the specified Dirichlet fixed sector.
Read statement and scope →If common-refinement-equivalent raw arrows induce the same carrier map, the raw functor descends uniquely to
Read statement and scope →A bounded Dirichlet-covariant carrier map sends the source fixed sector into the target fixed sector:
Read statement and scope →Local GLC, local Dirichlet covariance, isotony compatibility, localization-map composition, carrier composition, and common-refinement agreement give certified transport of local fixed sectors and descent of the local scale system.
Read statement and scope →Certified carrier maps satisfying the identity and composition laws define
Read statement and scope →The manuscript uses standard finite-index equivalences as imported infrastructure. The scalar Jones-index formula is stated in the factor case.
Read statement and scope →The Jones projections
Here group index supplies exactly the infinite orthonormal family required by Theorem 4.5. The result concerns the multiplication face on the fusion carrier.
Read statement and scope →The algebraic multiplication identities still hold. The dual operator-valued weight still realizes the face on its finite ideal. This is an obstruction to the specified ordinary Hilbert-operator realization.
Read statement and scope →Cauchy-Schwarz gives the bound; equal coefficients attain it. The projections are orthonormal and the trace is normalized.
Read statement and scope →This is an algebraic result. Commuting quotient maps do not guarantee analytic admissibility of every corner.
Read statement and scope →A realization survives these reductions. The theorem makes no upward or directed-totalization assertion.
Read statement and scope →The full core permits left amplification. On the unamplified span of the witness projections, the outputs lie in the scalar multiples of the identity; do not assign the full-target defect to that smaller operator.
Read statement and scope →This is the explicit directed rectangular obstruction. The chosen canonical completion is essential to the statement.
Read statement and scope →All three conditions are part of the certificate. With unconstrained witness choice, an already closable operator admits the tautological choice
The bounded decoder in Theorem 6.3 already supplies this reverse estimate. The stated closure equality is therefore available in that setting as well.
Read statement and scope →The weight is an admissible realization in its own framework. It is not identified with a closable Hilbert-space operator.
Read statement and scope →Amalgamation supplies existence; recognition supplies uniqueness up to the chosen equivalence. This is a fixed-source, set-valued theorem.
Read statement and scope →This is conditional on the reduction calculus and the existence of an admissible amalgam. Matching alone is insufficient.
Read statement and scope →Orthogonal outputs give norm one; identical unit outputs give norm
The total retention and assembly are assumed to have these analytic properties. Finite-stage certificates alone do not establish them.
Read statement and scope →The theorem combines analytic control and relational persistence. It is a sufficient route with explicit totalization hypotheses; it is not an automatic construction of those totalizations.
Read statement and scope →The proof uses coordinate demands on
The same averages that vanish in source norm still have multiplication value one. This theorem asserts nonclosability on the witness span; full-target maximality concerns the amplified full core in Paper I.
Read statement and scope →Retaining coordinates orthogonally gives the identity on
Closed assembly may be unbounded. The graph carrier makes its decoder bounded and lets the retained graph control the completed operator.
Read statement and scope →In the manuscript,
This is invariance under the declared graph equivalence. An arbitrary algebraic rewriting need not preserve a certificate.
Read statement and scope →Compatibility makes the algebraic limit map well defined; two-sided uniform control lets it extend with an inverse. The Hilbert direct-limit setting uses isometric transition maps.
Read statement and scope →The common concrete realization and preserved mark are essential hypotheses. The proof uses standard von Neumann algebra density arguments.
Read statement and scope →A nontrivial
Compactness and the finite-intersection property give a coherent family. Existence does not give uniqueness or analytic certificate preservation.
Read statement and scope →The graph equivalence in the pair is what connects strictification to analytic preservation.
Read statement and scope →The compatible family preserves both the relational choices and their graph equivalences. Compactness of these spaces is a hypothesis; a norm bound alone does not prove it.
Read statement and scope →This supplies a uniform strictified graph estimate. Analytic totalization remains a separate hypothesis in the next result.
Read statement and scope →The chosen module operator category is part of the hypothesis. This statement does not automatically supply regularity or adjointability for unbounded module operators.
Read statement and scope →For
This is the standard graph criterion applied to the route-generated comparison. It provides a direct test for the defect at the total stage.
Read statement and scope →The order of factors matters. The identity follows from cancellation of these specified finite routes; it does not establish analytic totalization. Trivial directional defects imply equality of the two surface comparisons, not that they equal the identity.
Read statement and scope →The scalar comparisons are invariant under the declared route-preserving similarities. This proves a bounded-class obstruction. Divergence of distortion does not imply nonclosability.
Read statement and scope →The retained route pair generates the comparison graph. The certificate prevents a limit of the form
The hypotheses retain the route comparisons through totalization without first requiring strictification. Their analytic totalizations are explicit assumptions.
Read statement and scope →The explicit choice is
Equality persists on the domain of the closed retained map. This does not identify arbitrary products of independently closed comparison factors.
Read statement and scope →The result reconstructs the common represented route, including its retained intermediate data. Individual closability of comparison factors is insufficient to guarantee equality of their separately formed products.
Read statement and scope →The theorem concerns the represented claims in a specified finite-depth package. It does not prove finite depth for every system, canonical minimality, or infinite-depth totalization. Constants may depend on the level.
Read statement and scope →