Reflection quotients and field recovery

Reflection quotients and field recovery

Fix the common continuum hierarchy produced by the supplied construction, with its actual phase and test labels. Let be its algebraic future-word space and its positive semidefinite reflection form. Take to be the linear span of the corresponding realized words, with their complete labels and common-witness records, and let read those words. The identity-frame word section satisfies

Searchable transcription

qZ(v, w) = q(pv, pw),
N = {u : q(u, u) = 0},
NZ = {v : qZ(v, v) = 0}.

These word maps are distinct from the finite cochain maps of the tangent chapter.

Theorem C.1 (Unitary recovery of the reflection space). The map is a well-defined isometric bijection. It extends uniquely to a unitary U. The empty word is preserved. Bounded operators intertwined on these dense quotients extend by UU , preserving norms and adjoints.

Proof. For any positive semidefinite form the Cauchy–Schwarz inequality follows by requiring 0 for every complex. Thus a zero-norm vector is orthogonal to every vector. The defining equality gives and equality of quotient inner products, proving well-definedness and injectivity. Every [] has preimage [], proving onto recovery before completion. The isometry therefore extends to the completions, with dense and closed range, hence onto. The empty word maps to itself. Conjugation by this unitary proves the bounded-operator assertions, including the adjoint identity by the inner-product definition.

Unbounded fields require the supplied common cyclic domain and its actual null-preservation proof. Once a target field Φ() is defined there, set

Searchable transcription

DZ = U −1D,
ΦZ(f)v = U −1Φ(f)U v
(v ∈DZ).

This specifies its domain as well as its values. Products on the invariant cyclic domain, conjugatelabel adjoint pairings and locality identities are recovered by inserting U U between successive fields. No bounded extension of a field to the entire Hilbert space is inferred. If the target field is closable, taking convergent graph sequences proves that its lifted graph closure is the unitary image of the original graph closure.

The same conjugation transports the constructed strongly continuous spacetime action and heat semigroup: continuity follows fromU is the source’s momentum projection-valued measure, then UU is countably additive in the strong topology, has the same support restrictions and recovers its scalar spectral pairings. In particular the ground projection and its rank are preserved. A rank-one vacuum projection does not imply a positive lower bound on the nonzero energy spectrum.

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Boundary and reflection spaces use their respective unitary recovery maps.
A constructed
attachment J is lifted as JZ = U −1
ref JUbdry. This preserves its isometry property and, when the
supplied density argument proves onto range, its surjectivity. Isometry alone supplies no surjectivity
argument. Tested boundary distributions remain those of the same hierarchy, with their original
gap bounds and derivative requirements, by Theorem B.1.