Mathine — Evidence In, Trust Out

P versus NP as a Solution Path Problem: A KEPLER Perspective

Mathine: Constructive Closure Route Machine
Link: https://doi.org/10.5281/zenodo.18896911

P versus NP is usually framed as a question of final theoremhood: either polynomial-time decision procedures exist for all polynomially verifiable problems, or they do not. This paper keeps that mathematical burden intact, but argues that it is not the only useful way to organize research progress.

The central move is to treat P versus NP also as a solution path problem. Under this view, the real difficulty is not only whether a final theorem exists, but how candidate advances travel across a chain of increasingly demanding fields: solver success, replayable transport, bounded closure artifacts, subfield decomposition, barrier analysis, conditional candidate deciders, and finally theorem-grade closure.

The paper revises the earlier KEPLER framing in an important way: it makes explicit that the present program is not neutral between the two classical outcomes. It is oriented toward discovering evidence consistent with constructive routes to P = NP, while remaining fail-closed about theorem-grade promotion. That gives the program direction without allowing optimism to masquerade as proof.

A second addition is the symbolic proposal P + NP = 𝟙. The paper does not present this as a formal complexity-theoretic theorem. Instead, it treats it as a co-evolutionary reasoning-field hypothesis and an epistemic ledger for tracking what is truly closed, what is only locally working, and what remains open under declared regimes. In that role, it helps organize progress without replacing the formal burden of proof.

KEPLER is then used to turn this constructive architecture into an explicit travel map: fields, border crossings, invariants, admissible transformations, receipts, and fail-closed promotion gates. This reframes many apparent failures. A line of work may be locally valuable yet still fail to cross the route into universal theorem-grade claims, not because it is useless, but because it has not yet survived the required transport.

The overall picture is deliberately balanced: more constructive than a purely barrier-centric narrative, but more disciplined than symbolic optimism on its own. The contribution is not a proof of P = NP, but a clearer and more faithful map for how a genuinely P = NP-seeking program can be organized, evaluated, and kept scientifically defensible.