PureOne/LUNEI-Complex-Schur-Recovery

Lụneï ∿ Oryun

Complete Complex Schur Recovery and Sharp Quantum Storage Rigidity

Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki
Manuscript: consolidated edition 3, 2 November 2026
Contents: an 18-page mathematical manuscript, editable source, four verification programs, recorded results, and a 157-row verification corpus.

Read the manuscript · Browse the source · Recorded verification cases · Original research package

This repository presents a proposed exact characterization of deterministic recovery from finite-dimensional complex Schur channels, together with constructive decoders, converse bounds, analytic examples, and structural results for quantum storage subspaces. The manuscript contains the proof statements and their assumptions. Numerical checks supplement those proofs.

Main recovery theorem

Let $C\succeq0$ be any complex correlation matrix with unit diagonal, and let $\mathcal M_C(X)=C\circ X$. Choose unit Gram vectors $c_i$ with $c_i^\dagger c_j=C_{ij}$. Define

$$v(C)=\underset{\sigma ⪰0,\mathrm{Tr}\sigma =1}{\max }\underset{i}{\min }{c}_i^{†}\sigma c_i=\underset{q\in \mathrm{\Delta }}{\min }{\lambda }_{\max }\text{\hspace{0.17em}⁣}\left(\sum _i{q}_i{c}_i{c}_i^{†}\right)\mathrm{.}v(C)=\backslash max\_\{\backslash sigma\backslash succeq0,\backslash ,\backslash operatorname\{Tr\}\backslash sigma=1\}\backslash min\_i\; c\_i^\backslash dagger\backslash sigma\; c\_i\; =\backslash min\_\{q\backslash in\backslash Delta\}\backslash lambda\_\{\backslash max\}\backslash !\backslash left(\backslash sum\_iq\_i\; c\_ic\_i^\backslash dagger\backslash right).$$

The proposed theorem identifies both deterministic recovery optima:

$$\underset{\mathcal{R}}{\inf }\mathrm{\parallel }\mathcal{R}\circ {\mathcal{M}}_C-\mathrm{i}\mathrm{d}{\mathrm{\parallel }}_{\diamond }=2[1-v(C)],\underset{\mathcal{R}}{\sup }\underset{\rho }{\inf }{F}_e(\rho ,\mathcal{R}\circ {\mathcal{M}}_C)=v(C)\mathrm{.}\backslash inf\_\{\backslash mathcal\; R\}\backslash |\backslash mathcal\; R\backslash circ\backslash mathcal\; M\_C-\backslash mathrm\{id\}\backslash |\_\backslash diamond\; =2[1-v(C)],\; \backslash qquad\; \backslash sup\_\{\backslash mathcal\; R\}\backslash inf\_\backslash rho\; F\_e(\backslash rho,\backslash mathcal\; R\backslash circ\backslash mathcal\; M\_C)=v(C).$$

The diamond norm is unhalved, and entanglement fidelity uses the squared-overlap convention. Optimization ranges over all completely positive trace-preserving recovery channels on the fixed logical input space.

An optimal state $\sigma$ gives an explicit recovery correlation matrix. With $M_{ij}=c_i^\dagger\sigma c_j$ and $f_i=M_{ii}$,

$$A_{ij}=\frac{v(C)\overline{M_{ij}}}{f_i{f}_j}+{\delta }_{ij}(1-\frac{v(C)}{f_i})\mathrm{.}A\_\{ij\}=\backslash frac\{v(C)\backslash overline\{M\_\{ij\}\}\}\{f\_if\_j\}\; +\backslash delta\_\{ij\}\backslash left(1-\backslash frac\{v(C)\}\{f\_i\}\backslash right).$$

The proof shows that $A$ is a correlation matrix and $C\circ A\succeq v(C)J$. Thus a single Schur decoder attains both optima. A dual-optimal probability vector provides a common adversarial purification against every competing recovery.

Further results

• Packing interpretation. The reciprocal $1/v(C)$ is the established semidefinite packing number of the rank-one Gram projectors. The paper credits that parameter to prior work and proposes its exact diamond-recovery interpretation.
• Tensor products. $v(C\otimes E)=v(C)v(E)$. Collective recovery gives no advantage for the fixed tensor-product logical space. This does not rule out adding a new redundant encoding across channel uses.
• Strict nonunitary advantage. A tetrahedral tight-frame family has an explicit optimal decoder that strictly outperforms every mixed-unitary correction.
• An analytic subclass. For real entrywise nonnegative $C$, $v(C)=\min_{q\in\Delta}q^TCq$, and identity recovery is optimal.
• Erasure recovery. Single-coordinate erasure on an arbitrary-rank code supported on nonadjacent binary strings has a closed formula from two extremal eigenvalues.
• Storage rigidity. Complement-invariant supports admit an all-maximizer classification and a sharp Frobenius repair bound, plus operator-norm repair and a near-capacity noncompletion obstruction.

Research status and limits

This is an AI-generated mathematical research artifact with explicit proofs and finite reproducibility checks. The precise novelty claims remain provisional. Targeted primary-source searches do not certify worldwide priority.

The established complement-pair code constructions, semidefinite packing parameter, packing multiplicativity, polar recovery, and relevant fidelity geometry are credited in the manuscript. This release proposes an operational interpretation and constructive recovery results rather than claiming those earlier objects as new.

The main theorem concerns Schur channels on a fixed full logical space. It is not a solution of unrestricted quantum recovery or a hardware fault-tolerance demonstration. Structural stability statements have additional rank and support assumptions. The manuscript's impact assessment is T6 candidate, on the author's informal T1–T9 scale.

Verification corpus

The recorded release contains 157 case-level checks:

Suite	Cases	Main coverage
Rigidity	47	Reflection identities, rounding, exact recovery, sharpness witnesses
Real minimax recovery	50	Quadratic formula, binary cases, flagged channels, tensor products
Structural extensions	17	Operator-norm repair and exact noncompletion constructions
Complex recovery	43	Independent SDPs, explicit decoder, packing reciprocity, tight frames

These are recorded computational cases, not 157 independent theorem proofs. The complex suite uses a numerical tolerance of $3\times10^{-6}$ and records solver status, primal–dual gaps, and positive-semidefinite residuals. Some solver calls report optimal_inaccurate while satisfying the recorded residual checks. The corpus does not provide interval-arithmetic certificates or external replication.

The configured Hub view loads only data/verification_cases.jsonl. Full precision results remain under results/recorded/; each tabular row links to its source report and retains the original case as a JSON string. See DATA_GUIDE.md.

Reproduce

Use Python 3.11 or newer for the pinned dependency set. The recorded environment used Python 3.12, NumPy 2.3.5, SciPy 1.17.0, CVXPY 1.9.3, and Clarabel 0.11.1.

python -m venv .venv
# Windows: .venv\Scripts\activate
# Linux/macOS: source .venv/bin/activate
python -m pip install -r requirements-verified.txt
python verify_release.py
python run_all_checks.py

The runner writes new reports to results/regenerated/, preserving the recorded reports. To run one suite:

python run_all_checks.py --suite complex

For numerical reuse, add code/ to the import path, import solve_recovery from verify_complex_recovery, and call solve_recovery(C) on a valid NumPy correlation matrix. Its returned a_rounded is a numerical recovery correlation matrix, accompanied by bounds and feasibility residuals.

The manuscript can be rebuilt with XeLaTeX. Its source uses a Linux P052 font path; adjust that path for another TeX installation. Scientific dependencies are separate from the Windows uploader's environment.

Files and integrity

Path	Purpose
paper/	Manuscript and editable TeX source
code/	Four verification programs
results/recorded/	Unmodified recorded research results
data/	Consistent JSONL view and suite summary CSV
archive/	Exact original consolidated research ZIP
release/	Provenance and source checksums
SHA256SUMS.txt	Exact upload allowlist and SHA-256 checksums

The release verifier checks the allowlist, checksums, recorded counts, and tabular schema without accessing a Hugging Face account. The prepared Windows uploader verifies the package before upload and checks remote file hashes afterward.

Citation and licensing

Citation files are provided as CITATION.cff and CITATION.bib. No DOI or published repository URL is invented in this release.

The source research did not specify a reuse license. This packaging does not add a license grant; see LICENSE_STATUS.md.