Algorithmic WQO and the Semantic-Monotonicity Gap: Why Graph-Minor-Style Finite Obstructions Do Not Transfer Directly to P vs NP

Round 22 left behind a Graph-Minor-style dream: find a well-quasi-order for P-normal-form algorithms that makes SAT correctness/failure monotone, and the finite forbidden-set theorem could be replicated. This round tests six candidate orders and lands a key correction — a WQO itself is actually cheap to obtain: for program text over a finite alphabet, Higman lemma directly gives a subsequence WQO; for program syntax trees, Kruskal tree theorem gives a homeomorphic-embedding WQO, and this isn't a paper hypothesis either — program-transformation techniques like supercompilation and partial evaluation have long used it as a termination whistle. What actually gets stuck is the second condition: a natural syntactic WQO almost never gives SAT correctness the monotonicity that's needed — just insert one line, if trigger(x): return 1, into the program, and it's still a subsequence syntactically, yet the semantics may flip entirely; conversely, swapping the order for semantic equivalence or language containment immediately grows an infinite antichain on singleton languages or on distinct Boolean functions, losing the WQO property. This round formally distills this tension into the WQO--Semantic Alignment Barrier (WSAB) and the more operational Order Alignment Trilemma: structural WQO, semantic monotonicity, and non-circular resource-relevance form three corners, and a natural candidate usually gets only two of the three at once. The most important positive result is formally separating Termination WQO (which controls the unfolding trajectory) from Hardness WQO (which gives the correctness set a finite basis) — supercompilation uses the former, and it can't automatically be used as the latter; while the derivation trees of a complete P grammar in the Bellantoni–Cook/Cobham style can indeed carry a Kruskal-style WQO, so the point of failure has now narrowed precisely from “no order exists” to “lacking an order that lets the semantic property close.”