P vs NP as Observer-Dependent Computational Hardness: A Reinterpretation through Dynamic Gate Numbers

P vs NP as Observer-Dependent Computational Hardness:
A Reinterpretation through Dynamic Gate Numbers

This paper offers a reinterpretation of the P vs NP problem through the framework of Dynamic Gate Numbers, treating computational hardness as a relational and state-dependent quantity rather than an intrinsic, observer-independent property of problem instances.

The work does not claim to resolve P vs NP in its classical formulation, nor does it challenge established results in complexity theory such as NP-completeness or time hierarchy theorems. Instead, it examines why the P vs NP question has remained persistent by relaxing three implicit assumptions of classical complexity theory: fixed computational models, observer-independent hardness, and memoryless computation.

Within the Dynamic Gate Numbers framework, computational cost is modeled as a trajectory-valued quantity that depends on an agent’s internal state and the set of admissible transitions (gates) available to that agent. Hardness emerges from the interaction between problem structure and gate-constrained agents, rather than from problem instances alone.

The paper establishes three main results:

1. Impossibility of universal hardness measures — when agents learn and adapt, no observer-independent scalar function can faithfully represent computational difficulty.

2. Gate-dependent complexity — the same problem instance may exhibit P-like behavior for one agent and NP-like behavior for another, without any modification to the problem itself.

3. Persistence of NP-hardness — apparent NP-hardness persists when an agent’s gate capacity is structurally bounded, regardless of learning, indicating that hardness endurance is a capacity phenomenon rather than an intrinsic property of problems.

Classical complexity theory is recovered as a special case of this framework, corresponding to fixed gates and state-independent cost functions. The reinterpretation preserves all standard definitions of P and NP while situating them within a broader class of adaptive computational systems.

This work is theoretical and interpretive. It introduces no new algorithms, simulations, or empirical claims. Its contribution lies in clarifying the structural conditions under which computational hardness appears stable, relative, or persistent, and in providing a mathematical language for discussing learning and adaptation within complexity theory.