生长式数学:NP问题的闭式构造生成与高维本体

Grounded in the Structure-World (SW) worldview, this paper extends growth-style mathematics to NP problems: starting from parameterized axiom seeds, constructive constraint self-growth releases folded intermediate dimensions, turning under-determined problems into well-determined ones from which closed-form decision functions emerge. The ontological basis is projection necessity: a high-dimensional ontology must project into reality, hence every NP problem family admits low-dimensional quotient projections. Empirical validation on three independent families: (i) the pigeonhole principle, with closed-form proof length D(n) = (e − 1/6)(n+1)! (theorem level; we prove the exact identity ⌊(e − 1/6)(n+1)!⌋ = D(n) + n + 2, with F(9)=2784 verified step-by-step: prediction equals measurement); the pure-resolution purified variant converges to the purification constant (2e−7/3)/(e−1/6) ≈ 1.2162 with closed form D′(n) = ⌊(2e−7/3)(n+1)!⌋ − (2n+3); (ii) Tseitin families, with satisfiable charge fraction 2^(−c); (iii) all-different families, with phase-transition law P(SAT) ≈ exp(−2n(1−p)^n) (4800 instances). A quotient-projection search further shows that PHP admits no low-dimensional projection under resolution coordinates (consistent with Haken’s lower bound) yet does so under matching coordinates — a unified experimental characterization of the known resolution/counting separation. This is Paper III of the growth-style mathematics series (Paper I: mathematical theory of automatic theorem growth; Paper II: the decline of anthropocentrism).