Lụneï ∿ Oryun: Exact Minimax Recovery for Complex Schur Channels, Semidefinite Packing, and Sharp Quantum Storage Rigidity

Lụneï ∿ Oryun presents an exact characterization of optimal deterministic recovery for every finite-dimensional complex Schur channel, together with constructive decoders, matching converse bounds, and structural results for quantum erasure-correcting subspaces. The release includes an 18-page mathematical manuscript, editable LaTeX source, four numerical verification programs, recorded computational results, and reproducibility materials. The central result identifies the smallest achievable diamond-norm recovery error and the largest achievable worst-case entanglement fidelity through a single semidefinite optimization. One explicitly constructed recovery channel attains both optima. A common adversarial input proves optimality against every deterministic completely positive trace-preserving recovery, including recoveries outside the Schur and unitary classes. PureOne/LUNEI-Complex-Schur-Recovery · Datasets at Hugging Face The work also develops sharp code-repair bounds, an analytic separation between general recovery and mixed-unitary correction, and an obstruction to completing approximately maximum-dimension codes. Exact optimal recovery for arbitrary complex Schur channels Let (C) be a complex positive semidefinite matrix with unit diagonal. Its associated Schur channel acts by entrywise multiplication: [ M_C(X)=C∘ X.] Choose unit Gram vectors (c_i∈ C^r), where (r=rankC), satisfying (c_i^ c_j=Cᵢⱼ). Define minq∈Δλₘₐₓ!(∑_iq_i c_ic_i^),] where (Δ) is the probability simplex. The manuscript proves the operational identities [infR| R∘ M_C-id|_=2[1-v(C)]] and [Rinf__e!(ρ, R∘ M_C)=v(C).] Both optimizations range over all deterministic completely positive trace-preserving recovery channels on the fixed logical space. The diamond norm is unhalved, and entanglement fidelity uses the squared-overlap convention. These identities characterize optimal approximate recovery; perfect recovery occurs when (v(C)=1). The optimization uses an (r× r) density matrix, or an equivalent spectral minimization over classical probabilities. This replaces an unrestricted search over recovery channels with a smaller problem on the Gram span. Constructive decoder and universal optimality witness An optimal density matrix (σ) determines a recovery correlation matrix explicitly. Writing [Mᵢⱼ=c_i^σ c_j, f_i=Mᵢᵢ,] the construction is {v(C){Mᵢⱼ}}{f_if_j}+δᵢⱼ(1-v(C)/f_i).] The resulting Schur recovery ( M_A) attains both operational optima. Its recovered channel satisfies [C∘ A v(C)J,] where (J) is the all-ones matrix. This yields a convex decomposition containing an identity-channel component of weight (v(C)). The decomposition describes the deterministic channel; it does not assert an experimentally heralded successful branch. A dual-optimal probability distribution supplies a single reference-entangled input witnessing the converse bound against every competing recovery. Feasible primal and dual solutions also provide quantitative bounds on the optimal value and achievable recovery error. Connection to established semidefinite packing theory For the rank-one projectors (P_i=c_ic_i^), the reciprocal (1/v(C)) equals the established classical–quantum semidefinite packing number. The manuscript credits the packing framework studied by Duan and Winter rather than claiming the parameter as a new construction. The proposed contribution is its precise deterministic recovery interpretation: 2(1-1/ A(P)),ₒₚₜ=1/ A(P),] together with the explicit common optimal decoder and universal adversarial witness. This interpretation concerns ordinary deterministic recovery and does not require a no-signalling assistance resource. Further operational and structural results Tensor-product recovery. The value obeys [ v(C⊗ E)=v(C)v(E). ] Product optimal decoders therefore attain the optimum against arbitrary collective recovery on the fixed tensor-product logical input space. This statement does not optimize additional redundant encodings across channel uses. Strict advantage over mixed-unitary correction. An explicitly solved tetrahedral tight-frame family exhibits a recovery channel that strictly outperforms every mixed-unitary recovery. For the four-dimensional family (C_t=(1-t)I+tG), the exact values are [ v(C_t)=1+t/4, Dₒₚₜ=3-t/2. ] The strict separation holds for (t>0). A convex quadratic subclass. For real entrywise nonnegative correlation matrices, [ v(C)=minq∈ΔqTCq, ] and identity recovery is optimal. The general complex theorem covers phases for which this simpler conclusion fails. 1- { √(1+a)(1+b) + √(1-a)(1-b) }{2}. ] A sharp consequence is [ Dₒₚₜ≤1-√1-ε^2 ] when the compression has operator norm at most (ε). Complete classification at maximum dimension. For complement-invariant binary supports of size (2K≥4) and minimum Hamming distance at least two, every dimension-(K) code correcting any one known-coordinate erasure is classified by phase-weighted complementary pairs. Sharp constructive repair. Every rank-(K) trial projector on such a support admits an explicitly constructed closest exact code. Its squared Frobenius distance is bounded by the centered local compression defect divided by the support’s minimum Hamming distance. The uniform coefficient is globally sharp. Operator-norm repair. In the full central excitation sector, a complementary stability theorem controls the distance to an exact maximum-dimension code using local operator-norm defects. Its coefficient depends on physical blocklength rather than logical dimension. Failure of uniform near-capacity completion. A family of exactly erasure-correcting codes approaches the maximum dimension ratio while remaining far from every maximum-dimension code within the specified support. This establishes a limitation on extending the rank-half rigidity theorem to smaller ranks. Computational verification and release contents The accompanying programs record 157 case-level checks: 47 rigidity cases, 50 real minimax-recovery cases, 17 structural-extension cases, and 43 complex-recovery cases. Coverage includes explicit decoders, independent semidefinite formulations, primal–dual comparisons, tensor products, tight-frame examples, sharpness witnesses, and noncompletion constructions. The release contains the manuscript and source, verification scripts, full JSON reports, a consolidated verification table, dependency specifications, citation metadata, and integrity checksums. The prepared repository also provides a runner that preserves the recorded results while writing regenerated reports separately. These computations supplement the analytic proofs. They are finite floating-point checks rather than formal verification or interval-arithmetic certificates. Solver statuses and numerical residuals are retained in the reports. Research status and scope The proposed novelty concerns the precise operational equivalence, constructive optimal recovery, mixed-unitary separation, and stated structural theorems. Established packing parameters, packing multiplicativity, complement-pair constructions, and recovery methods are credited in the manuscript. Worldwide priority remains unconfirmed. The principal recovery theorem applies to finite-dimensional Schur channels on a fixed logical space. The erasure and rigidity results require their stated support and rank assumptions. The release makes no claim of solving unrestricted quantum recovery or demonstrating a physical fault-tolerant device. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Version: Consolidated edition 3, 2 November 2026.