Computational Composite Methodology

CCM's complete foundational paper, whose abstract opens by stating plainly: ‘its core claim is methodological rather than ontological — the mathematical research process can be represented as an evolving state, in which multiple representations, search operators, certificate languages, obstructions, cost models, and path histories all interact.’ Sections 3-10 build the formal skeleton: a target $P\in\mathcal P$ has an external truth semantics $\tau(P)\in\{0,1\}$, but CCM does not assume direct access to $\tau$; the certificate-language sextuple $\mathcal C_i=(D_i,\Gamma_i,A_i,V_i,\sigma_i,K_i)$ explicitly separates search (which may be heuristic and incomplete) from verification (the semantic gatekeeper); Definition 5.3 defines a ‘method-obstruction certificate,’ whose meta-semantics is ‘path blocked ⇏ target is false’; Section 6 proves coverage monotonicity (Theorem 6.3: $\mathfrak C_1\subseteq\mathfrak C_2\Rightarrow\operatorname{Cov}(\mathfrak C_1)\subseteq\operatorname{Cov}(\mathfrak C_2)$) and unresolved-region anti-monotonicity (Corollary 6.5); Section 7's Proposition 7.1 proves ‘coverage failure does not entail falsity,’ pointing to the Motzkin polynomial (nonnegative but not SOS) as a concrete witness to this logic; Section 10's Theorem 10.1 is the whole paper's core firewall — as long as the final verifier is sound and every representation transformation preserves certificate semantics, the routing process remains rigorous in its final verdict no matter how heuristic or AI-generated it is. Section 33 explicitly positions the 13 benchmarks as ‘empirical witnesses to the theory, not the theory itself’; Section 34 distills ten core principles; Sections 35-36 are the paper's most important honesty boundary — explicitly listing nine ‘claims supported by theory and existing benchmarks’ and ten ‘claims CCM v1.0 does not yet establish’ (including not claiming CCM is universally superior to expert single-track mathematics, not claiming every mathematical target has a finite certificate, and not claiming AI can autonomously discover every relevant representation). The paper closes with nine genuine literature citations, including the classic Cook–Reckhow 1979 paper on proof-system efficiency, Clarke et al.'s CEGAR, SATzilla, and Parrilo's foundational work on SOS semidefinite programming.