PureOne/Iyorai-IV-Quantum-Batteries-Finite-Memory-Sharp-Work

Iyorai IV: Exact Finite-Memory Capacity for Sharp Spin Work

Deterministic quantum work extraction from unequal-gap spin batteries, optimal reference curves, and certified energy-band encoding.

Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. This is the requested research pen name for an AI-generated mathematical study.

Release: Iyorai IV, 31 October 2026. This repository is a research archive containing a report, source code, exact arithmetic certificates, numerical results, and figures. The CSV and JSON files are generated scientific records, with different schemas; they are not a single machine-learning training dataset.

Start here: 13-page report · Current LaTeX source · Research notes · Reproduction guide · Citation

Abstract

How much definite work can a finite quantum memory unlock from an unknown, fully charged collection of spin dimers? Iyorai IV gives an exact answer in a specified spectral model. Each of n cells has a unique ground state, a q-fold degenerate excited level, and a positive excitation gap. The input is arbitrary within the full charged subspace and can be entangled with an external system. Extraction must conserve total energy and leave a separate load in the same pure energy eigenstate for every input.

If different occupation subsets have different total energies, a D-dimensional memory initially in a pure ground state permits a maximum sharp output equal to the sum of the largest min(n, floor(log_q D)) gaps, when its Hamiltonian and the extraction operation may be designed and its final state is unrestricted. The report supplies a converse and a coherent protocol attaining this bound. It also derives bounds for initially coherent or mixed references and nonzero failure probability.

A separate continuous-reference analysis applies established Airy-shaped wavefunctions to the approximate-return encoder. Its sufficient initial mean energy is 1.376083543343775 times residual-band-width/error, improving the compact sine construction by 12.3958% at the same derivative-based error budget. A finite-error converse matches the leading pure-reference constant as the error tends to zero. An exact rational dynamic program and rank/unrank implementation make rounded energy-band encodings classically computable with polynomial arithmetic and storage bounds.

Exact finite-memory capacity

For a charged battery with Hamiltonian

$$H_B=\sum _{i=1}^n{J}_i{P}_{T,i},J_i>0,H\_B=\backslash sum\_\{i=1\}^\{n\}J\_iP\_\{T,i\},\backslash qquad\; J\_i>0,$$

the excited subspace of each cell has dimension q; q=3 describes ideal singlet-triplet dimers. Define the ordered gaps and their partial sums by

$$J_{(1)}\ge \cdots \ge J_{(n)},V_m=\sum _{i=1}^m{J}_{(i)},V_0=0.J\_\{(1)\}\backslash ge\backslash cdots\backslash ge\; J\_\{(n)\},\backslash qquad\; V\_m=\backslash sum\_\{i=1\}^\{m\}J\_\{(i)\},\backslash quad\; V\_0=0.$$

The exact capacity is

$$\boxed{{\displaystyle w_{\max }(D)=V_{m_D},m_D=\min \{n,\lfloor {\log }_{q}D\rfloor \}\mathrm{.}}}\backslash boxed\{w\_\{\backslash max\}(D)=V\_\{m\_D\},\backslash qquad\; m\_D=\backslash min\backslash \{n,\backslash lfloor\backslash log\_qD\backslash rfloor\backslash \}.\}$$

This statement assumes:

• Distinct occupation subset sums: equal sums of gaps identify the same subset.
• An initially independent, pure ground-state memory of dimension D.
• Optimization over the memory Hamiltonian and an energy-conserving unitary.
• The same specified pure, sharp work-load output for every charged input.
• No requirement to return the memory to its initial state.

The attaining construction transfers the quantum labels of the m_D largest-gap cells into the memory, grounds those cells, and sends their stored excitation energy into the load. It preserves unknown superpositions and external correlations globally. The memory is consumed as an information resource; reset costs are outside this extraction calculation. All auxiliary memory and control systems participating in the unitary count toward the reference dimension.

For an arbitrary initially independent coherent or mixed D-dimensional reference, the upper bound becomes

$$w\le h+V_{m_D}\le M+V_{m_D},w\backslash le\; h+V\_\{m\_D\}\backslash le\; M+V\_\{m\_D\},$$

where h is the lowest initially occupied reference energy and M its initial mean energy, measured above the reference ground. The report also gives a sharper energy-sector success-probability bound and mixed-state rank refinements.

Exact arithmetic example

For gaps J_i = 1 + 2^(-i), all occupation subset sums are distinct although every gap is rational. For q=3 and n=1000, obtaining sharp work at least 600 requires D=3^600 in the optimized pure ground-memory regime. The maximum at that dimension is exactly 601 - 2^(-600). At D below 3^600, the capacity is at most 600 - 2^(-599), which is strictly less than 600.

The required dimension corresponds to 600 qutrits, or a register of at least 951 qubits. Exponential Hilbert-space dimension here means linearly growing memory-qubit count, not exponentially many physical qubits.

Approximate-return references and optimal curves

The continuous-reference encoder is a separate resource regime: input information remains encoded in the battery, the load receives exact sharp work, and the reference returns approximately while absorbing bounded residual energy shifts.

For residual shift width Delta and derivative-based error budget epsilon, an established Airy reference gives

$$M_{\mathrm{A}\mathrm{i}\mathrm{r}\mathrm{y}}=C_A\frac{\mathrm{\Delta }}{ϵ},C_A=1.376083543343775\dots \mathrm{.}M\_\{\backslash rm\; Airy\}=C\_A\backslash frac\{\backslash Delta\}\{\backslash epsilon\},\backslash qquad\; C\_A=1.376083543343775\backslash ldots.$$

This is a 12.3958% reduction from the compact sine construction's coefficient pi/2. The corresponding unified bounds include joint trace-distance error at most epsilon, encoded-channel half-diamond error at most epsilon^2/4, and reference entropy at most h_2(epsilon^2/4) + 1.102 epsilon^2/4, in the small-error regime specified in the report, using natural logarithms. The Airy packet has an infinite energy tail with finite mean; the sine packet has compact support.

Airy optimization and its constant are credited to prior phase-estimation and battery-assisted gate research. The report's finite-error pure-reference lower bound matches the leading Airy cost as epsilon tends to zero. The translation-return dimension bound D >= epsilon^(-r) concerns r rationally independent actual shifts under explicit translation assumptions; it is not a general replacement for the work-capacity theorem.

Certified energy-band compiler

band_codec.py uses exact rational rounding, suffix dynamic-programming counts, and reversible rank/unrank maps. At fixed gap bounds, q, and rounding allowance, its bounds are O(n^3) big-integer operations, O(n)-bit counts, and O(n^4) stored bits; ranking or unranking after precomputation uses O(n) big-integer operations.

For the supplied 100-cell example, the certificate gives initial energy 98.94, output band [37.96, 40.16], and sharp work 58.78 if the reference lift is implemented. The passive-energy interval [39.26676585, 39.46676585] bounds the work penalty by 0.89323415. This is a different gap array from the rational example above. The compiler permits coincident subset sums and constructs a logical encoding; physical pulse synthesis remains unresolved.

Contents

File or directory	Purpose
EVE_Iyorai_Spin_Energy_Exact_Solution.pdf	Current 13-page Iyorai IV report
manuscript_v4.tex	Current editable manuscript
verify_final_capacity.py	Exact capacity and full-code coherent transfer checks
capacity_dimension_results.json, .csv	Exact rational and numerical capacity certificates
verify_airy.py, airy_verification.json	Reference curves, overlaps, error conventions, entropy
band_codec.py, codec_verification.json	Exact rational encoder and its verification
REPRODUCIBILITY.md, requirements-reproduce.txt	Reproduction commands and checked environment pins
verification/	Re-executed check logs and environment record for this release
prior_v3/, prior_v2/, baseline/	Preserved Iyorai III, II, and I reports and sources
Earlier scripts, JSON, CSV, and figures	Preserved earlier calculations; see research notes
CITATION.cff, citation.bib, ABSTRACT.txt	Citation metadata and plain-text abstract
PROVENANCE.json, SHA256SUMS.txt	Source preservation record and exact file hashes

The older root manuscript.tex, manuscript_v2.tex, and manuscript_v3.tex are historical sources. Build manuscript_v4.tex for the current report. Historical documents retain their original claims and attribution; the current README and IV report give the current interpretation.

Reproduce the current checks

Use a fresh Python environment and run these commands from the repository root:

python -m pip install -r requirements-reproduce.txt
python verify_final_capacity.py
python verify_airy.py
python band_codec.py

The codec uses only the Python standard library. Numerical verification uses NumPy, SciPy, and Matplotlib; no GPU, hosted API, external dataset, or network access is needed after dependency installation. Each check writes its own results and figures. See REPRODUCIBILITY.md for manuscript compilation and optional earlier checks.

The current release checks passed: 88 capacity cases, five reference-spectrum cases, 20 success bounds, complete transfer of the 81-state charged input code, Airy quadrature comparisons, all 65,536 eight-cell output states, all 6,561 eight-cell source round trips, sampled 32- and 100-cell cases, and edge cases. The coherent-transfer error was approximately 2.22e-16; transition energy differences were exactly zero in integer arithmetic. Integer capacities and rational certificates are exact. Quadratures, matrix exponentials, spectral errors, and plot coordinates use floating-point arithmetic.

Scope, prior work, and research status

The candidate specialized contribution is the ordered-gap capacity staircase together with its arbitrary-reference energy-sector success law and the related resource-accounting results. Scientific priority is not established. Support-based deterministic extraction, pure assistants as entropy sinks, dimension costs in other thermodynamic models, Airy reference optimization, and enumerative/reversible compression have prior literature. The report contains the full comparison and primary references, including:

• Salvia and Giovannetti, Zero-fluctuation quantum work extraction.
• Berry and coauthors, Optimal Heisenberg-style bounds for the average performance of arbitrary phase estimates.
• Castellano and coauthors, Exact requirements for battery-assisted qubit gates.
• Boes and coauthors, By-passing fluctuation theorems with a catalyst.
• Chaki and coauthors, Role of energy-invariant assistants in energy extraction from quantum batteries.
• Cover, Enumerative source encoding.
• Chuang and Modha, Reversible arithmetic coding for quantum data compression.

The finite-memory theorem depends on exact spectral assumptions. Output-energy tolerance, uncertain gaps, triplet splitting, extra conserved symmetries, local molecular control, memory reset, and complete device overhead require further analysis. The continuous reference is infinite dimensional. These are mathematical and computational results for an explicit model, not an experimental battery or a new energy source.

Informal research assessment retained from the source: T6/9. It is not a peer-review designation. No independent experimental validation is claimed.

Citation and reuse

Use CITATION.cff or citation.bib. The byline is a research pen name and the study is AI-generated. No DOI or unverified institutional affiliation is assigned. No license declaration was supplied with the source archive, so this preparation adds no license grant.