Research proof index

Structural form of linked simulation through an explicit state frame.

The quotient of a Hausdorff second countable unimodular group by a central normal closed subgroup is still unimodular.

Serre derivative of E₆: serre_D 6 E₆ = - 2⁻¹ * E₄². Uses the dimension argument: 1. serre_D 6 E₆ is weight-8 slash-invariant (by serre_D_slash_invariant) 2. Weight-8 modular forms are 1-dimensional, spanned by E₄² 3. Constant term is -1/2 (from D E₆ → 0, E₂ → 1, E₆ → 1)

The RandomQuery oracle reduction is perfectly complete.

Let $1<q\le2$ and let $q'$ be its Hölder conjugate. For measurable sets $F,G\subseteq\mathbb{R}$ and measurable $f$ with $\lVert f(x)\rVert\le\mathbf{1}_F(x)$, the real-line Carleson operator satisfies

$\int_G^+ T f(x)\,dx \le C(q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

A fibring identity in the $\tau$-minimizer setup. Let $X_1',X_2'$ be independent copies of $X_1,X_2$ and put $k=d[X_1;X_2]$. Then $d[X_1+X_2';X_2+X_1']+d[X_1\mid X_1+X_2';X_2\mid X_2+X_1']+I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']=2k$.