P05 / CANDIDATE THEOREMA finite search became an all-n construction.
P05 asks whether every length n ≥ 17 admits an incongruent restricted disjoint covering system whose minimum modulus is at least 3. The current proof draft answers yes by combining finite verified bases with recursively closed N3 and M3 families. The construction separates lifted modulus families by 3-adic valuation. The main theorem is labeled apparently novel pending expert review—not solved or published.
P05 / VERIFICATIONThe finite burden is explicit and reproducible.
The proof depends on 351 base certificates: P for 17–108, N3 for 41–81, and M3 for 109–326. Clean-room regeneration and multiple checkers accepted the required bases; generated covers were swept through n=100,000 and spot-checked to 10⁸. A primary-source audit through September 25 found no earlier statement of the main all-n theorem, while correctly identifying partial prior art for the stronger all-moduli-divisible-by-3 family.
P02 / ACTIVE SEARCHBroad mining gave way to structure.
P02 asks for a rational Diophantine septuple. Phase 1 built independent exact verifiers and certified extension machinery. Phase 2 accumulated 2,224 verified sextuples from 6,672 certified mining units without producing a new almost-septuple or septuple candidate, so broad mining was stopped. The active route now targets elliptic-curve hub compatibility and parametric families instead of spending compute on a low-yield search.
P04 / OPEN FRONTIERThe negative frontier moved, but the problem remains open.
P04 asks for an exact congruence cover in which every class hits at least three times. Two independent code bases find no example for n ≤ 108; published prior work already covered n ≤ 105, making 106–108 the new cross-checked finite extension. Structural proof drafts show increasingly strong constraints, including a computer-assisted result forcing some modulus ≤ n/6. None of this proves global nonexistence.
P06 / ACTIVE FOUNDATIONThe next discovery target is a bijection, not a brute-force witness.
P06 targets an explicit uniform bijection between Gog and Magog trapezoids. A current prior-art audit confirms the general problem remains open and found no known k=3 bijection; k=1 and k=2 are the solved baselines. The research program starts by reproducing known maps and exact counts, then compares statistics and searches for low-description-length invertible rules that generalize beyond the training sizes.
FIELD NOTE / METHODVerification—not model confidence—is the gatekeeper.
The project deliberately favors problems with falsifiable finite attack surfaces. Exact solvers, independent verifiers, mutation tests, proof critics and fresh literature checks are used to separate an interesting computational signal from a publishable mathematical claim.