FINDING: The Polymath project's most mathematically substantive output is the retrospective on bounded prime gaps, formalizing \(H_m := \liminf_{n \to \infty} (p_{n+m} - p_n)\), with the twin prime conjecture equivalent to \(H_1 = 2\). | MATH: \(H_m\) definition; Zhang's breakthrough implies \(H_1 < 7 \times 10^7\), later Polymath-reduced to \(H_1 \le 246\) (unconditionally) and \(H_1 \le 6\) under Elliott–Halberstam. No new constants or ratios emerge from the search results themselves. | CONNECTION: None direct. Prime gaps are irregular; no Fibonacci, golden ratio, or base-60 structure appears in the cited material. The "Golden Square" episode title is a red herring — it refers to a cryptocurrency/economic concept, not mathematical geometry. | DEPTH: 4 — The collaborative methodology is profound for process, but the specific findings here are incremental refinements of Zhang's bound, not a new structural revelation. The \(H_m\) formalism is elegant but not harmonic. --- FINDING: Ter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
No quasiperfect number ($σ(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $ω\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \mid n$ and $ω= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \nmid n$ and $ω= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $ω\le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $ω(n) \ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.