Operator-product readers and collision extensions

Operator-product readers and collision extensions

Let be the supplied filtered field space, with its fixed finite-dimensional complements, bases and projections. Use an isomorphic realized field-label space with readout and section This is the field-label isomorphism after the source’s physical detector quotient; it is not an inverse on all unreduced frame histories. Set

Searchable transcription

FZ,≤m = q−1F≤m,
ΠZ
m = iΠmq.

Every channel label and cluster geometry is retained. For the actual source reader its stated separated-configuration domain,

Searchable transcription

KZ
m(ξ; v1, . . . , vn) = iKm(ξ; qv1, . . . , qvn).

The same fixed complements are used at every truncation degree, rather than refitted to each test.

Proposition E.1 (Reader and remainder recovery). All algebraic reader identities and tested expansion estimates of the supplied construction are recovered by, with unchanged coefficients and truncation degrees. The full expansion residual is preserved when it is retained as a separate component. Compatibility between finite packets transfers with the same projections.

Searchable transcription

Proof. Applying q to the displayed reader cancels qi = I and recovers Km exactly.
Similarly
qΠZ
m = Πmq, and every finite identity obtained by composing these maps follows by inserting these
inverse identities at each slot. A tested residual is the difference of the original tested cluster and
its finite reader expansion. Each term has the same scalar reading, so the residual and its absolute
bound are identical. Infinite summations use the original convergence warrant, as in Theorem B.1.
No inversion of Πm is used: its discarded tail is retained in the residual. The argument applies
to each original nested cluster and spectator domain without altering its scale ratios or error
exponents.

The collision-extension device is also retained as data. To make its recovery explicit, suppose one stage of the source subtraction is a continuous linear map on its specified test space, and the punctured kernel acts integrably on. Write that stage, including its chosen contact functional, as

Here this expression is used only on the domain and with the combined-integrability bounds supplied by that stage; separate nonsummable pieces are not assigned integrals. For corresponding realized tests let be the test-space isomorphism with its transported topology. Define

Searchable transcription

WZ = L−1WL,
tZ(g) = t(Lg),
CZ(g) = C(Lg).

Then). This is a direct substitution and proves continuity in the transported topology. It preserves the contact choice, not just the punctured restriction.

For the full finite forest of nested subtractions use this identity at each node in the source’s prescribed order. Induction on the number of nodes proves equality of the combined subtracted test and every retained contact contribution. All normal-test spaces, cutofffunctions, density Jacobians, overlap records and finite projections remain fixed. The source’s combined majorant justifies its integrations and gluing; those steps are not replaced by termwise integration of divergent summands. If two extensions differ by a diagonal-supported distribution, their punctured restrictions agree, but their full readouts agree only when that contact distribution agrees on every allowed test. Thus punctured values alone are not a recovery map for the chosen extension.