The Parity Collapse and Entropy Drain: First Bounds on Erdős's Discrepancy Threshold N(k,2)

Let N(k,ℓ) denote the smallest N such that every f:{1,...,N}→{-1,+1} admits a k-term arithmetic progression P with |Σf(n)| ≥ ℓ. Erdős and Graham noted that "no decent bound" was known for N(k,2). We observe that for even k, a parity argument reduces the problem to Spencer's solved case N(k,1), giving N(k,2) = 2^t(k-1)+1 where k = 2^t·m with m odd. For odd k ≤ 11, we determine the first exact values: N(3,2)=9, N(5,2)=22, N(7,2)=49, N(9,2)=65, and N(11,2)=112. All values satisfy N(k,2) ≤ k², and we conjecture this bound holds for all k ≥ 2. Per-position entropy of the feasible coloring set is monotonically non-increasing for every tested odd k, suggesting an entropy-decrement route to N(k,2) = O(k²).