Manuscript Oai279-A@1 · Oai279-A.tex · section · a1abc73f

Arithmetic support values at the frozen input

Stage 6: the explicit circuit for the frozen input › Arithmetic support values at the frozen input · lines 1075–1099

Oai279-A@1#eq:frozen-pstar

At $ℓ_{0} = 4$ , the source's padding still gives $n = 128$ . Consequently,

K = 2^{35}, L = 2^{70}, W = 2^{42} + 128,

and the prescribed known probability would be

p_{*} = \frac{( 2 ^{128} - 1 ) ^{2^{70}}}{2 ^{2^{77}}}, 1 - 2^{- 58} \leq p_{*} < 1.

These are exact arithmetic identities, not the resource ledger or success probability of $U_{4}$ . The finite certificate does not claim to have allocated the source process's $L$ transition records or evaluated its huge numerator.

Similarly, the order example $a = 5$ modulo $12$ supplies $φ (2) λ (2) = 2^{- 11}$ only as arithmetic support for understanding the proposed order-trial statement. The circuit $U_{4}$ does not call that trial.

Displayed formulas in this passage

Oai279-A@1#eq:frozen-pstar · line 1083
$p_{*} = \frac{( 2 ^{128} - 1 ) ^{2^{70}}}{2 ^{2^{77}}}, 1 - 2^{- 58} \leq p_{*} < 1.$
unlabeled · line 1079
$K = 2^{35}, L = 2^{70}, W = 2^{42} + 128,$