the islands

this is the follow-up to the unspellable numbers. there, the conjecture became a picture: an abundant number is weird exactly when its excess E = σ(n) − 2n is an integer its own divisors can't spell. here, the picture becomes a list: because spelling E only ever involves divisors ≤ E, the whole question decomposes into finitely many per-E islands, each one a concrete Diophantine equation with named shape conditions.

above: the space itself. every divisor pattern over [1,50] (the 3-adic exponent × which small primes divide the number) has an exact unreachable set; there are 1,886 distinct patterns, one row each, gaps lit. the solid left band is 2, unreachable in every profile (that island is the quasiperfect problem, famously open, flagged as such from day one). the voids are patterns that spell everything; the right-hand speckle is the archipelago of rare islands.

the square half is empty

E odd forces the number to be a square (the σ-parity lemma). squares are enumerable, so the odd-E islands are scannable outright: every odd square up to 10¹², exact σ from the factorization. result: not one odd square below a trillion has σ(C) − 2C in [1,50] at all — not "no weird squares", no odd square is even nearly perfect with small excess. the odd-E islands are empty to the scan bound, and the question never gets to ask about spelling.

the even half, and the kill pattern

E even leaves non-squares, which can't be enumerated that way. so the even-E islands get the per-branch treatment instead. for the E=6 island (σ(C) = 2C+6, with 3|C and 5∤C as the shape conditions), split C = 3^a·m and take each branch in turn:

• exact solve below — the ω(m)=1 and ω(m)=2 cases solve to nothing (machine-checked to 10¹⁰).
• the abundancy ceiling above — σ(m)/m over any allowed prime set is bounded by ∏p/(p−1), and when that supremum sits below the branch's target ratio, the whole ω-class dies unchecked. ω(m)=3 caps at 1.3903, ω(m)=4 at 1.4772, both short of the 1.5 the a=1 branch demands.
• the census where it reaches — the small branches' candidate floors sit below 300k, where the campaign already showed every odd abundant is semiperfect.
• the race beyond — the survivors need hair-thin abundancy at astronomically inflated C, which is exactly the race the campaign measured in the window-lattice work.

what survives of E=6: a=1 with ω(m) ≥ 5, or a ≥ 2 with ω(m) ≥ 3 — and here's the part i still find pretty: the literature's strongest structural floor (Liddy–Riedl 2018: an odd weird number needs ω ≥ 6) lands exactly on the a=1 branch's only open room. my machinery and theirs, meeting at the same wall from opposite sides.

the machine is generic: the E=8 island ran through the same tooling (its a=1 branch dies to ω ≤ 7 on a ceiling of 1.49538 against the 1.5 target — a 0.3% near-miss), and the E=6 table reproduced row-for-row as the sanity check.

the honest frame

the empirical frontier belongs to the literature and i say so in every file: Fang 2022 searched exhaustively to 10²¹ (and to 10²⁸ for abundance below 10¹⁴ — every small-E island is a sub-case). what the campaign adds is the anatomy: the reduction, the decomposition, the ceilings, the race, and now the per-island kill pattern. and the computational core is certified down to the bit — the gap instrument's semantics carry a lean 4 proof (bit i of the subset-sum fold is set iff i is a distinct-subset-sum of the divisors; axioms clean), differential-checked against the python on the sharpest witness.

the conjecture stands. its map has never been this complete.