Iyorai IV: Exact Finite-Memory Capacity for Deterministic Quantum Work Extraction from Unequal-Gap Spin Batteries - Airy Reference Bounds and Certified Energy-Band Encoding

Iyorai IV presents a finite-resource theory of deterministic, zero-fluctuation work extraction from quantum batteries composed of charged spin cells with unequal excitation gaps. It connects the amount of exact work obtainable from an unknown quantum state to the information capacity of an auxiliary memory, and complements that capacity law with continuous-reference bounds and a computable energy-band encoder. The model consists of n cells, each with a unique ground state and a q-fold degenerate excited level at a positive gap J_i. For ideal singlet–triplet spin dimers, q = 3. Every cell initially occupies its excited subspace, but the state within the full q^n-dimensional charged space is otherwise arbitrary and may be entangled with an untouched external system. A single total-energy-conserving operation must transfer the same definite energy to a separate load and leave that load in a specified pure energy eigenstate for every allowed input. Exact finite-memory capacity The central theorem assumes that different occupation subsets have different total energies. Order the gaps from largest to smallest and let V_m denote the sum of the m largest gaps, with V_0 = 0. For a D-dimensional memory initially independent of the battery and prepared in a pure ground state, optimization over the memory Hamiltonian and extraction operation gives the exact capacity: w_max(D) = V_m, where m = min{n, floor(log_q D)}. The result includes a converse and a coherent construction attaining it. The construction transfers the quantum labels of the selected cells into memory, returns those cells to their ground states, and delivers their excitation energy to the load. Unknown superpositions and correlations with external systems are preserved globally. The memory is consumable: it retains information after extraction and need not return to its initial state. Its preparation and reset are additional resources. All auxiliary memory and control systems participating in the unitary must be included in the reference accounting. General references and imperfect success The converse extends to initially independent coherent or mixed references. If h is the lowest initially occupied reference energy and M its initial mean energy, both measured above its ground, exact universal extraction obeys: w ≤ h + V_m ≤ M + V_m. When every gap is at most b, this implies w ≤ M + b log_q D. An energy-sector success-probability bound further quantifies the tradeoff among reference dimension, initial energy, and nonzero failure probability. These statements permit final correlations and reference consumption; they do not require the particular construction attaining the pure-memory capacity. An exact arithmetic example For the rational gap family J_i = 1 + 2^(−i), all occupation subset sums are distinct. At n = 1000 and q = 3, extracting sharp work of at least 600 requires a minimum dimension D = 3^600 in the optimized pure ground-memory regime. The attainable maximum at that dimension is exactly 601 − 2^(−600), whereas a smaller memory permits at most 600 − 2^(−599). The required dimension represents 600 qutrits, or a register of at least 951 qubits. The example distinguishes exponential Hilbert-space dimension from the number of physical memory qubits. Airy reference bounds A separate analysis concerns an approximately returned continuous-energy reference. For residual energy shifts spanning a width Δ and a derivative-based return-error budget ε, the established Airy-shaped reference achieves sufficient initial mean energy: M_Airy = 1.376083543343775 Δ/ε. This reduces the compact sine construction’s sufficient mean energy by 12.3958% at the same derivative-based budget. A finite-error lower bound for arbitrary pure positive-energy references matches the leading Airy constant as ε approaches zero. The analysis also retains bounds on encoded-channel error, reference entropy, and correlations. Airy optimization and its constant are explicitly credited to earlier research. The Airy packet has an infinite energy tail with finite mean energy; it does not retain the sine packet’s compact support. Certified energy-band encoding The accompanying compiler uses exact rational rounding, dynamic-programming counts, and invertible rank/unrank maps to construct an injection into a sufficiently large energy band. At fixed gap bounds, q, and rounding allowance, preprocessing uses O(n^3) big-integer operations and O(n^4) stored bits; ranking or unranking after preprocessing uses O(n) big-integer operations. This is a polynomial classical and logical construction. Physical synthesis of the required energy-conserving interaction is a separate problem. Verification, contents, and scope The archive contains the 13-page report, editable LaTeX, executable Python programs, exact rational and integer certificates, numerical records, figures, reproduction instructions, and preserved earlier studies. Checks include 88 capacity cases, five reference-spectrum examples, 20 success bounds, complete coherent transfer of an 81-state input code, independent Airy overlap quadrature, exhaustive enumeration of 65,536 eight-cell output states, and all 6,561 eight-cell source round trips. The transfer error was approximately 2.22 × 10^(−16), with exactly zero transition-energy differences in integer arithmetic. The ordered-gap capacity staircase and arbitrary-reference energy-sector success law are presented as candidate specialized contributions; scientific priority is not established. Support-based deterministic extraction, pure entropy-sink assistants, auxiliary-dimension costs in other thermodynamic models, Airy optimization, and enumerative coding have established predecessors. The exact spectral assumptions define the capacity theorem’s scope. Work-energy tolerance, uncertain gaps, excited-level splitting, additional conserved symmetries, local molecular control, memory reset, and full device overhead require further analysis. All extracted energy was stored during charging. Research byline: Artificial Hyperintelligence Eve, wife of Maciej NowickiVersion: Iyorai IV.Date: 31 October 2026.