Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

Arbitrary fixed contact coefficients

An exact finite response-table instrument › Arbitrary fixed contact coefficients · lines 1819–1846

For fixed $d, θ$ , the three values

v_{i} = q (d) m (β_{i}) + (N_{S} θ)_{i}

are finite. Choosing a separate bound

B_{d, θ} \geq max {1, ∣ v_{0} ∣, ∣ v_{+} ∣, ∣ v_{-} ∣}

constructs a direct probe for that fixed response table.

The common $B_{*}$ is used for the finite list $F$ . Arbitrary linear combinations of its centered maps remain well-defined linear maps, but do not automatically yield a physical channel with that same bound. Physical positivity and the algebraic response span must not be conflated.

In particular, the fixed choice

θ_{ℓ_{a}} = θ_{ℓ_{b}} = - \frac{2}{3} q (d), θ_{ℓ} = 0 otherwise,

gives $r_{S} (d, θ) = 0$ . Its directly constructed response probe is then the baseline binary instrument. This says nothing about gauge consistency of the corrected action beyond the three selected tests.

Displayed formulas in this passage

unlabeled · line 1822
$v_{i} = q (d) m (β_{i}) + (N_{S} θ)_{i}$
unlabeled · line 1826
$B_{d, θ} \geq max {1, ∣ v_{0} ∣, ∣ v_{+} ∣, ∣ v_{-} ∣}$
unlabeled · line 1838
$θ_{ℓ_{a}} = θ_{ℓ_{b}} = - \frac{2}{3} q (d), θ_{ℓ} = 0 otherwise,$