Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: A fixed-decoder interval

Dimension-explicit recovery certificates › The imported approximate conversion · lines 1742–1772 · proof included or linked

Hawk04-B@2#prop:interval

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Let

L_{m} = M_{N} + m Λ + 2 G m + 1 .

Suppose a fixed finite decoder has a validated point certificate $r_{dec} (t_{0}) \leq r_{0}$ . Then its continuum extension obeys, on the chosen finite interval,

d_{⋄} (D N_{t}, id_{C}) \leq min {1, d r_{0} + L_{m} ∣ t - t_{0} ∣ + b_{reg}} .

If $L_{m} > 0$ and $d r_{0} + b_{reg} < ε$ , it has a nonempty certified interval

∣ t - t_{0} ∣ < \frac{ε - d r _{0} - b _{reg}}{L _{m}},

intersected with the observation interval. If $L_{m} = 0$ , the proxy channel is time independent.

Proof. The norm of the finite active Hamiltonian is at most $L_{m}$ . Duhamel's formula and (eq:isobound) give proxy channel distance at most $L_{m} ∣ t - t_{0} ∣$ . Apply the fixed decoder after the point certificate and add the continuum comparison.

Displayed formulas in this passage

unlabeled · line 1745
$L_{m} = M_{N} + m Λ + 2 G m + 1 .$
unlabeled · line 1751
$d_{⋄} (D N_{t}, id_{C}) \leq min {1, d r_{0} + L_{m} ∣ t - t_{0} ∣ + b_{reg}} .$
unlabeled · line 1758
$∣ t - t_{0} ∣ < \frac{ε - d r _{0} - b _{reg}}{L _{m}},$

Proposition: A fixed-decoder interval

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