Manuscript Oai119-A@1 · Oai119-A.tex · lemma · fdff7292

Lemma: Exact classical identities

Stage 1: the five proof interfaces › Gate 3: dimension induction and classical semigroup · lines 583–615 · proof included or linked

Oai119-A@1#lem:semigroup, Oai119-A@1#eq:productionsplit

Back to manuscript

Let $P_{t} = T_{e^{- t}}$ . Then

P_{t} = e^{- t N}, \frac{d}{d t} I (P_{t} g) = - D (P_{t} g) .

If $g_{+}, g_{-}$ are the two sections in one coordinate,

D (g) = \frac{D ( g _{+} ) + D ( g _{-} )}{2} + E c (g_{+}, g_{-}) .

Proof. For $A \subseteq {1, \dots, n}$ , the Walsh character $χ_{A} (x) = \prod_{i \in A} x_{i}$ satisfies

T_{ρ} χ_{A} = ρ^{∣ A ∣} χ_{A}, N χ_{A} = ∣ A ∣ χ_{A} .

These characters form a basis, proving the semigroup identity.

The mean is conserved, $h^{'} = - atanh$ , and $\partial_{t} P_{t} g = - N P_{t} g$ . Differentiation of (eq:softinfo) gives the production identity. Separating the last-coordinate term in $N$ gives (eq:productionsplit), with the factor $1/4$ in (eq:joiningcost).

Displayed formulas in this passage

Oai119-A@1#eq:productionsplit · line 591
$D (g) = \frac{D ( g _{+} ) + D ( g _{-} )}{2} + E c (g_{+}, g_{-}) .$
unlabeled · line 586
$P_{t} = e^{- t N}, \frac{d}{d t} I (P_{t} g) = - D (P_{t} g) .$
unlabeled · line 603
$T_{ρ} χ_{A} = ρ^{∣ A ∣} χ_{A}, N χ_{A} = ∣ A ∣ χ_{A} .$

Lemma: Exact classical identities

Displayed formulas in this passage

Nearby in this section