Persistence--First Law--Space Exponents and Quantum Advantage over PFHS--TRoT--Blum Law--Time Semigroups

This preprint develops a persistence-first, implementation-independent complexity theory over law-time semigroups and embeds both classical and quantum computation into a common PFHS–TRoT–Blum framework. Building on persistence-first holographic systems (PFHS), fibered Bures–HK entropy–transport geometry (FBHK) and holographic observation quotients (HOQ), the paper defines PFHS–compatible law-time cost increments and a Blum-type complexity measure that track transport action, persistence change and HOQ gap along law-time trajectories. From these objects it extracts law-space exponents EPersEPersEPers that are invariant under polynomially bounded changes of size function, persistence functional, HOQ structure and Blum model, provided these are TRoT-compatibly polynomially equivalent. Within this setting, the work gives an explicit notion of “persistence-first quantum advantage”: a quantum law-time family outperforms a classical one when it realises a strictly smaller PFHS–HOQ law-space exponent for the same problem under a shared law-space background. The paper constructs a fully worked Grover search example, modelling the algorithm as a weak PFHS law-time flow on the two-dimensional Grover subspace endowed with Fubini–Study/Bures geometry and a canonical HOQ gap. Using standard query lower bounds, it shows that the quantum PFHS–HOQ complexity scales as Θ(N)Θ(√N)Θ(N) in law-time steps while a classical random search scales as Θ(N)Θ(N)Θ(N), yielding an exponent gap EPersq=1/2<1=EPersclEPersq = 1/2 < 1 = EPersᶜˡEPersq=1/2<1=EPerscl. The manuscript also sketches how thermodynamic constraints, such as Landauer-type bit-erasure costs and stochastic thermodynamics of relaxation, can be incorporated into PFHS-compatible cost. No new complexity-class separations (such as P vs NP or P vs BQP) are claimed; instead, the contribution is a law-space bridge between PFHS superintelligence theory, standard complexity models and thermodynamic resource accounting.