Finding the Path Home: A Geometric Exploration of P = NP

This paper introduces a mechanism that directly addresses cross-clause interference through a multi-scale descent process with three components:

(i) A potential function with clause-local decomposition. We define a rational-valued potential Φ(s) = U(s) + F(s)/B over 3-SAT states, where U counts unsatisfied clauses, F is a clause-local tie-breaking term with distinct weights, and B = n⁴ is a scale constant. This potential is zero if and only if the current assignment satisfies the formula, has polynomial range, and is computable in polynomial time with O(log n)-bit arithmetic.

(ii) A harmonic neighbourhood descent operator. Rather than flipping a single variable, the algorithm computes a gradient-directed multi-variable correction within the coupled neighbourhood of the first unsatisfied clause. This multi-flip has visibility into the cross-clause coupling structure and can coordinate corrections that a single-flip operator cannot.

(iii) A recursive cascade correction with adaptive decay. The neighbourhood descent may perturb clauses beyond its immediate scope. To manage this, the algorithm deploys a self-replicating cascade of progressively smaller corrections. At each cascade level, the actual perturbation arriving at boundary clauses is measured and a calibrated correction is applied. The cascade terminates in O(log n) levels, and the total correction work is polynomial. The net global potential decrease is a provable constant fraction of the initial local improvement.

Together, these components form a composite descent operator that guarantees strict global potential decrease of at least 1/(2B) at every step. Since the potential has range at most m + 1, the algorithm terminates after at most O(m · n⁴) accepted descents, each requiring polynomial work. The total runtime is polynomial in the size of the input formula.

When the formula is satisfiable, the algorithm terminates with a satisfying assignment. When the formula is unsatisfiable, a terminal plateau triggers extraction of a contradictory core — a minimal unsatisfiable subset of clauses with a polynomial-time verifiable refutation.