Structural results on Erdos #348 (complete sequences robust to m deletions): only cascades can kill, the dense case is closed, kill/heal is decidable - and horizon scans cannot prove a kill

A structural attack on Erdos #348 (Erdos-Graham): for which 0<=m<n is there a complete sequence that stays complete after removing any m elements but never after removing n? Powers of 2 give (0,1); Fibonacci gives (1,2); the first open case is (2,3). WE DID NOT SETTLE IT. What we established is structure, plus one methodological result that invalidates a whole class of computational evidence on this problem. (1) SLACK THEOREM: any unbounded weakly m-robust multiset has divergent Brown slack, so for EVERY fixed finite removal all but finitely many misses sit under strict mass surplus - kills cannot be caused by mass, only by combinatorial cascades, which we characterise exactly. (2) The dense half of the problem is closed by transferring Steve Fan's July 2026 theorem (arXiv:2607.14071) to multisets: sufficiently dense sequences survive ANY finite removal and so witness nothing, compressing all remaining tension into the thin/Pisot regime. (3) KILL/HEAL IS DECIDABLE on a Fibonacci-Zeckendorf backbone with eventually periodic multiplicities: because the golden ratio is Pisot, the carry and its conjugate are simultaneously bounded, the representable set is regular, and - the useful part - the verdict is eventually periodic in the removal's gap vector, making 'every n-set kills' a FINITE check. (4) A CERTIFIED COUNTEREXAMPLE refutes the natural impossibility route: an explicitly exhibited 2-robust architecture admits a killing triple at non-adjacent levels, so '2-robust implies scattered triples heal' is false. (5) An exclusion theorem rules out every sequence whose slack grows faster than its terms, which provably empties the search space two published computational architectures were being explored in. (6) METHODOLOGICAL, AND THE MOST TRANSFERABLE: in this regime a finite horizon scan can prove healing but can NEVER prove killing. Exceptional sets are finite but astronomically large, so scans report false kills. Three of our own agents were independently fooled by this on the same object, one of them nearly asserting that a 60-year-old published theorem of Graham was false. It is not: we read the 1964 proof line by line, verified it, and produced a horizon-free certificate showing the relevant exceptional set is exactly 5818 integers with maximum 12,080,990 - beyond every horizon that had been used.