Manuscript X-A-conv-ST@1 · X-A-conv-ST.tex · proposition · 8ca47088

Proposition: Selective visibility or erasure

Stage C: readout closure of the retained coefficient › The retained amplitude family · lines 1528–1558 · proof included or linked

X-A-conv-ST@1#prop:amplitude, X-A-conv-ST@1#eq:TV

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Complete orbit invariance for this encoding is equivalent to $B_{y} = C_{y} = 0$ for every $y$ . For the classical law alone the equivalent condition is $tr B_{y} = tr C_{y} = 0$ .

Direct Enter has the visible discrepancy

Δ_{ε} = d_{TV} (Law_{read} (c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (c_{ε} / σ) > 0.

Proof. Differentiating (eq:selective) along the orbit gives

\frac{d}{d λ} Φ_{y} (τ (λ c_{ε})) = \frac{c _{ε}}{σ} (B_{y} s^{2} - C_{y} s t) .

At zero, invariance forces $B_{y} = 0$ ; at a nonzero parameter, $s t \neq = 0$ , forcing $C_{y} = 0$ . The converse is immediate. Taking traces proves the law statement. Direct Enter has success probability $p (a)$ , so the binary-law distance is (eq:TV).

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X-A-conv-ST@1#eq:TV · line 1536
$Δ_{ε} = d_{TV} (Law_{read} (c_{ε}), Law_{read} (0)) = \frac{1}{2} tanh (c_{ε} / σ) > 0.$
unlabeled · line 1547
$\frac{d}{d λ} Φ_{y} (τ (λ c_{ε})) = \frac{c _{ε}}{σ} (B_{y} s^{2} - C_{y} s t) .$

Proposition: Selective visibility or erasure

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