Ǝnuoχ · Iïraŋ: The Optimal Frequency Cost of Quantum Haar Ergodicity-Exact Quasiperiodic Constructions, Smooth Bounded Approximation, and Regularity Obstructions

This research release presents analytical constructions, obstruction theorems, and reproducible numerical checks for quantum Haar ergodicity in finite-dimensional driven systems. Its central question is how many independent external base frequencies are required for a deterministic quantum evolution to reproduce Haar-distributed unitary behavior in the long-time limit. The manuscript derives the count m = d(d − 1) for every Hilbert-space dimension d ≥ 2 within its stated regular quasienergy-reducible setting. This is a conditional optimality statement about a specified class of dynamics. The construction addresses the gap between the d² − 2 sufficient construction and the d² − d conditional lower bound discussed in Pilatowsky-Cameo, Marvian, Choi, and Ho, “Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs,” Physical Review X 14, 041059 (2024), DOI: 10.1103/PhysRevX.14.041059. Established Haar-coordinate methods, including the work of Spengler, Huber, and Hiesmayr, are credited explicitly. The proposed contribution is their use in a frequency-optimal dynamical construction, together with approximation and obstruction results. PureOne/Enuokh-Iirang-Optimal-Quantum-Haar-Ergodicity · Datasets at Hugging Face The main results are: Exact construction at the conditional minimum. A recursive special-unitary flag map uses d(d − 1)/2 radial coordinates and the same number of phase coordinates. All d − 1 independent diagonal angles are supplied by quasienergy evolution rather than additional external drive frequencies. Joint rational independence gives a Haar SU(d) long-time limiting law. Its balanced unitary moments and projective unitary channels agree with those of Haar U(d); the construction does not generate an independent random global U(1) phase. The exact control has integrable, unbounded cusp peaks, an absolutely continuous propagator, and finite action on every finite interval. Finite action does not generally imply finite squared-amplitude power. Smooth bounded approximation at the same frequency count. For any prescribed 0 < ε < 1, radial truncation and smooth convolution yield globally C∞, uniformly bounded controls whose long-time limiting law lies within total variation ε of Haar. Consequently, every finite-order parallel unitary twirl has limiting half-diamond error at most ε, and every initial pure state satisfies the corresponding limiting moment bound. The guarantee is uniform over moment order after taking the infinite-time limit separately at each fixed order. It supplies no finite-time estimate uniform over all orders. A matching all-order lower bound. Under the manuscript’s regular quasienergy assumptions, using fewer than d(d − 1) external base frequencies forces the supremum of the limiting all-order half-diamond errors to equal one. The argument combines a dimension obstruction with a tomography-based relation between tensor-power moments and total variation. Together with the smooth construction, this identifies the same minimum count for uniform all-order limiting error strictly below one in the declared class. A global regularity obstruction. Exact complete unitary ergodicity with globally C¹ quasienergy micromotion requires m ≥ d(d − 1) + 1. The corresponding state-ray result requires m ≥ 2d − 1. These are necessary bounds; sufficiency at the raised counts is not proved. Their hypotheses concern the global regularity of quasienergy functions, rather than smoothness of the physical Hamiltonian alone. They therefore do not establish that every exact physical realization must use unbounded controls. Complementary temporal-window and measurement results. Part II retains a sharp frontier relating projector-population variance to second-moment deviation, a macroscopic-window obstruction to Haar second-moment ergodicity, a no-go theorem for the substitution drive 0 → 010, 1 → 111, and a finite two-copy measurement certificate. The variance concerns fluctuations of expectation values across states or times, rather than the measurement variance within one state. The two-copy certificate assumes independently prepared copies evaluated at the same uniformly selected time. The generic critical substitution 0 → 001, 1 → 11 remains unresolved. The release includes the final manuscript, LaTeX source, analytical derivations, a reusable Python generator, scientific figures, numerical ledgers, fixed-seed verification scripts, reproduction instructions, and internal proof-audit records. The core generator supports the exact cusp construction and a bounded piecewise-smooth truncation. A separate adaptive-quadrature prototype checks the globally C∞ construction; it is not an exact digital representation of the analytic smoothing kernel. Recorded checks include 9,000 SU(d) map cases in dimensions 2–10; 200,000 torus samples compared with 200,000 independent QR-Haar samples; low-order state and balanced unitary moments; 150 analytic-derivative checks; 1,500 bounded-Hamiltonian norm samples; six direct ODE intervals; eight finite-action quadratures; and 12 finite quasiperiodic moment windows. The smooth prototype adds 670 profile and derivative checks, ten error-budget configurations, 101 Hamiltonian samples, and an ODE interval crossing a cusp. These computations test ideal mathematical models. Exactness, all-order statements, and infinite-time conclusions rest on the analytical arguments rather than finite numerical sampling. Scope and research status: frequency count means independent base-frequency rank, not the number of Fourier harmonics, pure sinusoidal controls, hardware channels, or gates. No finite-bandwidth guarantee, efficient many-qubit scaling, or quantitative mixing-time advantage is established. The lower bounds do not cover arbitrary irreducible quantum dynamics. Internal agent audits are not external peer review or proof-assistant certification, and the release contains no experimental hardware validation. Worldwide priority remains unconfirmed. Release identity: Ǝnuoχ · Iïraŋ (Enuokh Iirang), FINAL Version 2.0. Research identity: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.