Research proof index

Double Euclidean Levi-Civita contraction ∑_h (ε4)_{r,s,h} · (ε4)_{t,w,h} = 2 · (δ[r,t]·δ[s,w] - δ[r,w]·δ[s,t]) at the symbol level: contracting two of the four Fin 4 component slots of ε4 with the naive Kronecker pairing leaves two free pairs and the factor 2! = 2. The Lorentz form carries an extra det η = -1.

Full Euclidean Levi-Civita contraction ∑_b (ε4)_b · (ε4)_b = 24 at the symbol level: summing the square of every standard-basis component of ε4 over all four Fin 4 index slots, paired naively (no metric), counts the 4! = 24 permutations. The Lorentz contraction ε^{μνρσ} ε_{μνρσ} lowers one factor with η and equals -24 instead.

For every integer $n\ge0$, the corrected harmonic approximation $H_{n+1}-\log(n+1)-\frac{1}{2(n+1)}$ is a lower bound for the Euler–Mascheroni constant $\gamma$.

Overwriting one coordinate of a uniform function table is measure-preserving. Drawing a value u uniformly from R, then a full function table g : D → R uniformly, and returning Function.update g t u yields the same distribution as drawing the table directly. This is the t-marginal independence of the uniform (product) distribution on D → R:...

Restricting a uniform function table to a subdomain along an injection is uniform. For an injection e : A → B between finite types, drawing a uniform table g : B → R and restricting it along e (i.e. g ∘ e) yields the uniform distribution on A → R. This is the marginalization of the uniform (product) distribution on B → R onto the block of...

Evil Alice produces a close true trace with low probability, since by remaining close she looks like a close oracle.