Substantial investigations that generated reductions, computations, or proof infrastructure—but no complete proof of the famous target problem.
The campaign isolated the real obstruction as a pointwise, lower-bound-compatible Type-II/Type-III estimate capable of separating primes from semiprimes—the sieve parity problem.
Concrete progress included:
Σ Λ(n) μ(N−n) as a parity-breaking target;1+4/3 bridge reducing the residual factor structure;(1/4,3/8,3/8) and modulus scale N^(5/8);Status: no unconditional estimate strong enough to close the bridge was proved. Goldbach remains open.
Dmono(22)The exact finite campaign established
33 ≤ Dmono(22) ≤ 34.
Boundary regimes and hundreds of canonical refinement cases were eliminated, no 34-point witness was found, and no terminal completed branch returned anything other than UNSAT. The remaining central symmetric cases are the hardest.
Status: no complete proof that 34 is impossible and no 34-point construction.
Multiple algebraic, modular, SAT, and local-improvement ideas were explored. No independently checkable optimal or target-size configuration was produced, and no upper-bound proof settled the instance.
The work explored Lyapunov functions, residue-class descent, entropy drift, and ways to strengthen “almost all” results into a pointwise descent mechanism. No global monotone invariant or complete termination proof was found.
The campaign compared circuit lower bounds, proof complexity, algebraic methods, meta-complexity, natural-proofs barriers, relativization, and formalization possibilities. It produced no separation theorem.
Potential SAT/graph-generation routes to a 6-chromatic unit-distance graph and improved colorings were mapped, but no new unit-distance graph or coloring bound was certified.
The work examined fixed points, cycles, digit-count analogues of 6174, and possible finite-state classifications. No historically new full classification was established.
| Problem | Recorded status |
|---|---|
| WOWII 133 | One branch proved; residual case open. |
| WOWII 314 | Paper classification candidate; top-level Lean bridge missing. |
| Full checkerboard all-slope limit | Four-direction work complete at draft level; all-slope lower bound open. |
| Erdős 885 | Deep arithmetic-geometry progress; rational-point classification incomplete. |
| Erdős 366 | 9,718,456 3-full candidates checked without a witness; open. |