Weisfeiler–Leman Refinement, PCP for PPAD and 8 More in Theoretical Computer Science

Constructs a single language in deterministic polynomial time whose n-bit membership function requires 2^ω(√n) total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds. (Lean)

Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation. (Lean)

At every fixed deletion probability in (0,1), reconstructing an arbitrary length-n binary string requires n^{Ω(log log n)} independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most n^-ε for fixed ε > 0, both bounds become polynomial in the input and parameter encoding. (Lean)

Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule. (Lean)

Gives a deterministic polynomial-time (1+2/e+ε)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming P ≠ NP, the optimal infimum approximation factor is 1+2/e. (Lean)

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels. (Lean)

Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size. (Lean)

Proves unconditional n^Ω(k) deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete. (Lean)

Over every characteristic-zero field, the (1,1) entry of a product of n independent n× n variable matrices requires n^Θ(√n) gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables. (Lean)

Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length N(log N)^O(1) such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.