Arithmetic and Structural Barriers in Cauchy-Packet SAT Contraction Exact Two-Adic Valuations, Packed-Expander Mode Rank, and Crossing-Row Saturation

We establish three arithmetic and structural results about the mode responses arising in theCauchy-packet contraction framework of the October 2026 OpenAI exact SAT-counting manuscript.First, for Mersenne packet width q = 2r − 1 and arbitrary odd two-adic parameters, we determineevery target-mode coefficient valuation exactly by a multinomial valuation; a trailing-one extensionand an integral moment inverse sharpen the associated precision budget. Second, we give adeterministic construction of alternative legal balanced packings of bounded-degree expander-based tensor networks whose scalar two-adic mode-response flattenings retain 2Ω(M/√u) nonzeroSmith factors across balanced tile-block cuts. This obstructs one class of uniformly narrowlinear tensor-train representations, but does not establish a complexity lower bound or cover thesource manuscript’s prescribed SAT packing. Third, the published variance certificate alone forcesH ≥ (63/64)M − √2DarrM ∆/64 = Ω(M ) raw crossing rows, proving that a direct 2H packetexpansion cannot furnish the desired general exponent improvement. Further results include apolynomial-space poly (s)2O(L) local coefficient extractor, exact binary packet separation, andcounting-guided target avoidance with explicit verifier-size conditions. No improvement to theoverall general-SAT time bound, independent refutation of ETH, or resolution of P versus N P isclaimed.