IPS Problem 1: partial results and where they stop

Partial results on Problem 1 of Impagliazzo, Paturi and Saks (SIAM J. Comput. 1997), which asksfor the best exponent r in their restriction lemma for threshold circuits; r = 2 gives theirconstant θ = 1+√2 in the n^(1+c·θ^(-d)) wire lower bound for parity. For gates with weights in{0,±1} we give an alteration lemma, the bound r ≤ 1 + ½·log_δ K for fan-in at most K, and, forlarge fan-in, a restriction that makes every gate constant while leaving Ω(n/t) variables free(t = maximum degree), via Bang's lemma and discrepancy rounding. None of this improves θ; thenote states where the methods stop and lists open questions. Appendix A: full proofs.Appendix B: Python code and data. Prepared with AI assistance (Claude, Anthropic); the proofs were checked by automated reviewers, not yet by a human expert.