Do multilingual translation models organize meaning around a shared conceptual space, or do they primarily encode language identity and surface form? We probe the encoder geometry of Meta's 600M-parameter NLLB-200 model across 135 languages and 101 Swadesh concepts in contextualized carrier-frame and decontextualized bare-word regimes against null and confound controls. First, two-way variance partitioning reveals an additive language-shift plus conceptual-manifold geometry: language shifts account for 45.7% of variance on bare words vs. 13.7% in context, and mean-centering improves the between-to-within-concept distance ratio (vs. a 0.99 permutation null) by 2.14x on bare words (1.15 to 2.46; 2.27x across disjoint scripts) and 1.19x in context (2.25 to 2.69), with conceptual separation peaking at Layer 10 (mutual k-NN overlap jumping 4.35x). Second, semantic category explains 25.4% of Swadesh convergence variance ($F(7,93)=4.53$, $p=2.21\times 10^{-4}$; partial $F=7.74$ controlling for fertility and length)---over 20x orthography or phonology ($R^2\le 0.012$)---while the advantage of Swadesh over non-loanword concrete nouns ($d=0.54$--$0.69$) is mediated by Zipfian subword compactness ($p=7.43\times 10^{-7}$). Third, language distances track ASJP NLD and LDND phonetic distances within subfamilies under block-permutation and cross-script controls ($ρ_s=0.65$--$0.67$, block $p=10^{-4}$; global bare-word Mantel $ρ_s=0.24$, $p=0.001$), and CLICS$^3$ colexified concepts lie closer across string-distinct, within-category ($ρ_s=0.32$), and bare-word ($d=1.79$) controls ($d=1.43$, $p=1.55\times 10^{-33}$). Finally, basic color terms recover the Berlin and Kay hierarchy ($ρ_s=0.91$--$0.93$, $p\le 2.18\times 10^{-4}$), and semantic offsets exceed distance-matched controls ($p=3.55\times 10^{-5}$, $d=1.06$). We release InterpretCognates, an interactive toolkit and reproducible suite.
Every semi-magic square of order $n$ satisfies the axial moment identities $\sum_{i,j}s_{ij}j^m=\sum_{i,j}s_{ij}i^m=M(n)\sum_jj^m$ for all $m\ge1$, where $M(n)=n(n^2+1)/2$. For real matrices the converse requires $1\le m\le n-1$, whereas for arrangements of $\{1,\dots,n^2\}$ fewer moments suffice once $n\ge14$. We define the permutation threshold $k^*(n)$ as the least $k$ such that every arrangement satisfying these identities for $1\le m\le k$ is semi-magic. In terms of the column-sum deviations $γ_j$, the degree-$k$ conditions state that $\sum_jγ_jt^j$ is divisible by $(t-1)^{k+1}$, linking $k^*(n)$ to the Prouhet--Tarry--Escott problem. While Schur's bound on real zeros gives $k^*(n)\le\sqrt{2(n-1)\log(n^4/4)}$, we prove the converse realisability bound $k^*(n)\ge(\sqrt2-o(1))\sqrt n$ (and $k^*(n)\ge(\sqrt6-o(1))\sqrt n$ when $4\mid n$) by perturbing semi-magic squares with complementary row pairs via horizontal swaps. We also show that $k^*(n)=n-1$ for $3\le n\le13$, $k^*(n)=n-2$ for $14\le n\le17$, and $k^*(n)\le n-3$ for $n\ge18$, and determine $k^*(n)$ for $18\le n\le40$ with computer assistance. Finally, at the $k=1$ level of the hierarchy, we specialise the Mantel--Hubert quadratic assignment variance formula to obtain exact means, variances, and limit laws under a uniform random arrangement for the full line-sum energy $E_{\mathrm{full}}$, for the four-term energy $E_4$ combining the first moments and the two diagonals, and for Loly's moment of inertia, and find that $|L_4|=14{,}639{,}440$ arrangements of order $4$ satisfy $E_4=0$.
Given an arrangement of $\{1,\dots,n^2\}$ in an $n\times n$ grid, we lift each cell to $\mathbb{R}^3$ by centring its grid coordinates $(x,y)$ at the origin and taking its centred entry $z = s_{ij} - (n^2+1)/2$ as a vertical height. The convex hull of these $n^2$ points is the gem of the arrangement, whose lower and upper boundaries are the regular subdivisions of the grid induced by $\pm z$. While random arrangements and generic linear magic squares have gems whose vertex counts grow with $n$, our main theorem shows that the classical Siamese (de la Loubère) magic square of every odd order $n\ge5$ has $f$-vector $(12,26,16)$, isometry group $\{\pm\mathrm{id}\}$, and only the twelve vertices forced by the boundary of the grid. The proof shears the lift onto five horizontal terraces and reduces the hull calculation to a finite table of polynomials in $m=(n-1)/2$; the same method gives $f$-vector $(12,24,14)$ for the base-$n$ digit-exchanged square. Along any family of linear squares with fixed steps the $f$-vector is eventually periodic in $n$, whereas over all linear magic squares of prime order $n$ the maximum vertex count grows as $Θ(\ln n)$. Whenever the heights have vanishing first moments, a short principal-axes argument shows that the point set has a non-trivial isometry if and only if the arrangement is self-complementary up to dihedral symmetry. An exact census of the $880$ magic squares of order four yields $616$ congruence classes and $352$ combinatorial types of gems, and shows that while vertex and facet counts are nearly insensitive to the magic line-sum conditions, the volume and solid centroid shift systematically.