Proof of the P=NP Problem

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This manuscript is currently in the working draft stage and has not yet been finalized for publication. We appreciate your continued interest as this research progresses.

[Abstract]

This paper derives P=NP by assuming the truth of the Riemann Hypothesis (RH) and the rational coefficient Hodge Conjecture (RHC; simultaneously including Standard Conjectures B, C, D, I), integrating two pillars: geometric construction and computational complexity theory.

First Pillar (HCR in P): Using the idempotent projector system {Πk} constructed by RHC and the algebraic correspondence of the Hard–Lefschetz inverse, we effectively implement Algorithms E1–E5, a deterministic polynomial-time procedure for the Hodge Class Realisation (HCR) problem. The projectors are bounded by deg Πk ≤ deg X and log|coeff(Πk)| = O(polylog deg X). By employing spectral bounds of the self-adjoint weight operator side obtained from RH, we polynomially control iterations of linear algebra loops (Gaussian elimination, etc.). The overall runtime is explicitly
T_total = O((b_{2p} + ρ) p^3)
(polynomial in input bit length |x|), and the witness length and verification can be implemented in NC^2.

Second Pillar (SAT ≤P OCSP ≤P HCR): We introduce the geometric NP-complete problem Operator–CSP (OCSP), defined by unary and binary identities with projectors as variables. By templating constraints such as idempotence, self-adjointness, trace, and orthogonal sum, we construct (i) SAT ≤P OCSP (linear size increase) and (ii) OCSP ≤P HCR (polynomial kernel), thereby establishing NP-completeness of OCSP and the one-way reduction chain SAT ≤P OCSP ≤P HCR. Common denominator unification (lcm) and CRT splitting, Tseitin transformation variable/clause counts, and fixed-width bitstring encoding standards (size function |x|) are ledgered together, systematically ensuring polynomial closure of reduction length and runtime bounds.

By composing these two pillars, it follows that SAT ∈ P, and by Cook–Levin we conclude P=NP. This work combines the analytic spectral control provided by RH with the algebraic controllability ensured by RHC, presenting a deterministic polynomial-time framework for solving NP-complete problems grounded in the low-degree, low-bitlength nature of geometric projectors.