Millennium Problem II: P≠NP via PAL-Constrained Event Cascades
This paper tackles the Clay Millennium Problem II (P vs NP) using Pattern Field Theory (PFT), the Allen Orbital Lattice (AOL), and coherence constrained computation. By imposing the Pattern Alignment Lock (PAL) on event cascades, the paper defines a new family of machines and demonstrates that coherence is a genuine computational resource that cannot be removed by standard simulation.
Coherence-Constrained Computation Theory (CCCT)
The work introduces Coherence-Constrained Computation Theory (CCCT) and the Allen Orbital Lattice Machine (AOL-Machine). Active states live on the AOL and must satisfy a PAL coherence bound:
\[ \forall u,v \in S_t,\quad \cos\big(\theta_t(u) - \theta_t(v)\big) \ge 1 - \frac{1}{p_u p_v}, \]
where \(S_t\) is the active state set at time \(t\), \(\theta_t(\cdot)\) is the phase on the lattice, and \(p_u, p_v\) are the prime indices attached to those sites. Computations are carried out as event cascades along PAL permitted paths.
P, NP, and PAL coherence
Within this framework, the paper defines deterministic and nondeterministic coherence classes:
- PAOL – deterministic PAL coherent computations on the AOL-Machine.
- NPAOL – nondeterministic PAL coherent computations.
The central results are:
- PAOL ⊂ NPAOL – deterministic and nondeterministic coherence classes are strictly separated.
- PAOL ⊂ P – not all P time Turing machines admit PAL coherent simulations.
- Cobham's invariance principle fails under PAL – coherence cannot be simulated away by standard machine encodings.
Together, these establish coherence as a third axis of complexity, orthogonal to time and space. The P vs NP question is reframed in a setting where coherence capacity, not only runtime, becomes decisive.
PAL-constrained event cascades
Computations are represented as PAL constrained event cascades on the Allen Orbital Lattice. Each cascade:
- loads an input as a finite excitation pattern on the AOL,
- propagates along PAL permitted transitions,
- is subject to coherence capacity limits over the active set, and
- terminates in accepted or rejected equilibrium configurations.
Nondeterministic machines can exploit a larger portion of the PAL permitted configuration space than deterministic ones, creating a coherence gap that cannot be bridged by polynomial time simulation. This coherence gap is the structural core of the P≠NP separation in the CCCT/AOL setting.
Position in the Millennium series
This paper is the second in the Pattern Field Theory Millennium series. The first paper addresses the Riemann Hypothesis through the Allen Orbital Lattice equilibrium condition. This second instalment extends the same structural toolkit to computational complexity and P vs NP, with PAL coherence providing the bridge between number theoretic structure and algorithmic behaviour.
Read the full paper
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Citation
James Johan Sebastian Allen, Millennium Problem II: P≠NP via PAL-Constrained Event Cascades, Pattern Field Theory Millennium Series, November 2025.