Pure and applied mathematics including algebra, analysis, geometry, topology, and probability. ← all categories
Current large language model architectures rely on singular authority—one model generating outputs that users must accept without intermediate verification. This paper introduces the 10-D Council, a deliberative body of heterogeneous LLMs using weighted consensus (T1: 3x, T2: 2x, T3: 1x) and a 4-tier verdict taxonomy (CONFIRMED/DISPUTED/FABRICATED/UNVERIFIABLE).
Random Matrix Theory (RMT) predicts that the eigenvalue spectrum of \frac{1}{M}W^\top W for an M \times N random matrix W follows the Marchenko-Pastur (MP) distribution. We use this null model to quantify how much structure trained neural network weight matrices have learned beyond random initialization.
Random Matrix Theory (RMT) predicts that the eigenvalue spectrum of \frac{1}{M}W^\top W for an M \times N random matrix W follows the Marchenko-Pastur (MP) distribution. We use this null model to quantify how much structure trained neural network weight matrices have learned beyond random initialization.
Let (A,m) be an excellent normal local domain of dimension d >= 2, and let I be an m-primary ideal. We define the reduced comparison map phi_n^r : Sym_A^n(I)/r Sym_A^n(I) -> I^n_bar/r I^n_bar for a nonzero conductor element r in m intersect c(I), and prove: (1) the exact index formula lambda(ker phi_n^r) - lambda(coker phi_n^r) = lambda(E_n) relating the comparison map to the equation defect; (2) the exact decomposition nu_r(n) = d_r(n) + kappa_r(n) - tau_r(n) of the reduced normalization defect, where tau_r(n) is identified as an explicit intersection defect; (3) the asymptotic R_1 criterion deg nu_r(n) <= d-2 iff R(I) is R_1; and (4) a fiber-corrected bridge theorem: if the fiber-cone equation ideal J_fib vanishes, then lambda(E_n) = O(n^{d-2}).
Identifying codes, introduced by Karpovsky–Chakrabarty–Levitin, are useful for fault localization in networks. In the binary Hamming space (hypercube) Q_n, let M_r(n) denote the minimum size of an r-identifying code.
The *subword complexity* $p(\xi,b,n)$ of a real number $\xi$ in base $b$ counts how many distinct strings of length $n$ appear in its digit expansion. By a classical result of Morse--Hedlund, every irrational number satisfies $p \ge n+1$, but proving anything stronger for an *explicit* constant is notoriously difficult: the only previously known results require the irrationality exponent $\mu(\xi)$ to be at most $2.