Ceilings and Impossibility Results for Finite Witnesses of Sub-n log n Integer Multiplication

OpenAI's manuscript Integer multiplication below n log n reduces bounds T(n) = O(n (log n)^(1-κ)) to finite, exactly certified constructions, and a public effort around Douglas Colkitt's repository has raised the certified κ through a sequence of such witnesses. Current witnesses use paired cubes on three-stage Cayley covers. We prove ceilings and impossibility results inside this framework. A visit ceiling bounds the saving of a side by its total local displacement. For the complex word of PR #144, and assuming that its compiled children stay within visits (hypothesis H1) and that its visits are monotone, no change of cover, sharing, padding or exact compilation that keeps its visits takes the saving above 2^-10; this is specific to that word, and the pooled ceilings of our own round-10 complex words exceed 2^-10 in three of the four cases computed. Meet-capacity and cut-load theorems bound the share of centre demand that producers without auxiliary registers, or with relays, can serve; with one chain per port the share is at most 1/4 + O(1/h). We prove copy floors for addition modules, show that, for paired cubes with p ≥ 5 and in the frame-0 copy model, their star centres are optimal in three classes of centre families, show that in characteristic 0 or 2 every motif saves at most 1 - log_m 2, strictly for m ≥ 3, and that same-characteristic codes never gain, prove a rank inequality for relations of rank-one moves whose pairwise products vanish, show by exact computation that a characteristic-3 Cayley deck has no 3-adic lift respecting its sign symmetry L(-J) = -L(J), and show that several changes to the recursion cost model cannot help. Each result is stated with its hypotheses.