Axiomatic Emergence Z; The Physical Origin of Lower Bounds of Computational Complexity

This paper starts from two fundamental principles—Information Conservation(A1) and Finite-Step Computability (A2)—to argue the mechanism for the naturalemergence of inherent lower bounds of computational complexity in physically realizable computational processes. Research shows that core concepts of computationalcomplexity (such as polynomial-time upper bounds, circuit depth lower bounds) arenot mere mathematical abstractions but are quantitative laws that must be followedby physical systems when performing information transformations under fundamental constraints such as energy, time, and information conservation. Specifically, thequantum speed limit together with Axiom A2 jointly sets a polynomial-time upperbound for any finite-energy physical system; while the Information Conservationaxiom (A1) combined with the principle of minimal physical realization complexity(A3) ensures that the logical difficulty of a computational task is inevitably reflected in the topological complexity of the spatiotemporal structure of its physicalrealization. This sets a lower bound for the circuit depth determined by topological invariants, and in quantum many-body systems, it directly manifests asvolume-law growth of entanglement entropy, whose coefficient is determined by theunderlying hardware architecture. To test this theoretical framework, this paperdesigns two cross-scale experimental paths: at the microscopic level, using superconducting quantum processors to precisely measure the volume-law coefficient ofentanglement in random quantum circuits; at the macroscopic cosmological level,inferring the “computational complexity” growth information of the early-universeinflationary process by detecting the phase coherence of the primordial gravitationalwave background. These schemes aim to place foundational issues of computationalcomplexity within the domain of empirically testable physical science.