Manuscript II@2 · II.tex · section · 50d64def

Descent, currents, and records are separate questions

Certificate patterns that do not transfer › Descent, currents, and records are separate questions · lines 1886–1930

The supplied III record treats a gauge-image membership problem and descent on a coupled equation-space quotient CheungIII. A closed-range condition there is not a contact-matrix test here. Formal compensation is not automatically descent on an unchanged equation space.

The supplied V record distinguishes a raw current, a compact representative, and a residual witness CheungV. Its scalar-response cubic is not the Einstein cubic of Lemma lem:diagonal. No current-classification result from V enters the Ward reduction.

Measurement uniqueness, including Cheung's Enter characterization, does not construct the coherent-matching interface that would turn a tree coefficient into a record coefficient CheungEnter,CheungI. Likewise, the represented models in IV have their own generators, domains, and calibration estimates CheungIV.

To state the limited record implication explicitly, suppose two proposed output descriptions satisfy

⟨ f ∣ ρ_{ℓ} (g) ∣ i ⟩ = i g^{2} t_{ℓ} + O_{form} (g^{3})

with the same code vectors, phase reference, and probes. For

Y = - i ∣ i ⟩ ⟨ f ∣ + i ∣ f ⟩ ⟨ i ∣, F_{acc} = (I + Y) /2,

where $Y = 0$ on the code complement,

p_{acc} (ρ) = \frac{1}{2} + Im ⟨ f ∣ ρ ∣ i ⟩ .

The difference of order-two acceptance coefficients is therefore $Re (t_{2} - t_{1})$ . This follows by taking the trace of the effect. It does not supply the hypothesized output states or an instrument realizing them.

Displayed formulas in this passage

unlabeled · line 1910
$⟨ f ∣ ρ_{ℓ} (g) ∣ i ⟩ = i g^{2} t_{ℓ} + O_{form} (g^{3})$
unlabeled · line 1916
$Y = - i ∣ i ⟩ ⟨ f ∣ + i ∣ f ⟩ ⟨ i ∣, F_{acc} = (I + Y) /2,$
unlabeled · line 1921
$p_{acc} (ρ) = \frac{1}{2} + Im ⟨ f ∣ ρ ∣ i ⟩ .$