Manuscript I-B-VP-0@1 · I-B-VP-0-v1.tex · section · 8a9d0682

Finite reports are not continuum acquisition

Companion interfaces and reporting discipline › Finite reports are not continuum acquisition · lines 1148–1189

A finite report may contain the inputs $(α, β, r)$ , the current convention, a formula identifier, an approximation, and its error status. The I-B/En09-A destructive supplied-report interface is

M (X) = j \sum ∣ j, w_{j} ⟩ ⟨ j, w_{j} ∣_{R} \otimes D_{j} X D_{j}^{†}, D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A} .

It is completely positive and trace preserving because

j \sum D_{j}^{†} D_{j} = I_{C A} .

Expanding $X$ in the ${τ_{j}}$ basis gives

Tr_{R} M (X) = Tr_{C} X,

also after an untouched reference is adjoined CheungIB,CheungEn09,CheungO1ST. The output contains the surviving $A$ , not the consumed quantum clock $C$ . It reports installed words; it does not evaluate an unknown vacuum current by measurement.

The finite-test coverage distinction of Oai282-A CheungOai282A also applies at the logical level. Finitely many linear evaluations cannot determine a general distribution in an infinite-dimensional class. For currents, an explicit infinite-dimensional test class consists of

j^{μ} = \partial_{ν} A^{ν μ}, A^{ν μ} = - A^{μν} \in C_{c}^{\infty}

in a regular flat chart. These are conserved, and disjoint compact supports give arbitrarily large linearly independent families. Finite evaluation maps have nontrivial kernels there. This example is not asserted to be a difference of two Friedrichs-state currents. It shows why finite tests alone cannot prove a general continuum identity.

Displayed formulas in this passage

unlabeled · line 1154
$M (X) = j \sum ∣ j, w_{j} ⟩ ⟨ j, w_{j} ∣_{R} \otimes D_{j} X D_{j}^{†}, D_{j} = ⟨ τ_{j} ∣_{C} \otimes I_{A} .$
unlabeled · line 1161
$j \sum D_{j}^{†} D_{j} = I_{C A} .$
unlabeled · line 1165
$Tr_{R} M (X) = Tr_{C} X,$
unlabeled · line 1179
$j^{μ} = \partial_{ν} A^{ν μ}, A^{ν μ} = - A^{μν} \in C_{c}^{\infty}$