The First Surviving Diagonal Type g = p²q: the Seven-State System, Four Unresolved Classes and 57 → 42 → 22 Orbits

For g = p²q there are seven even-leg states and 105 relations, expanded and factorised; a scan over p, q ≤ 20,000 (1,264,500 ordered pairs) finds no internal triple; valuation and conjugation arguments reduce the seven-state problem to four unresolved classes {1,6,7}, {2,4,6}, {2,4,7}, {4,6,7}; conjugation reduces the 105 relations to 57 orbits, two prime-norm obstructions plus inherited gates remove 15, leaving 42 orbits (78 raw relations); a 2-adic valuation-pattern gate removes 20 more, leaving 22 orbits (44 of the 105 raw relations).