Soryun IV: Erasure-Optimal Quantum Memory under Noncommuting Spatial Fields and Non-Markovian Quantum Baths - Capacity Theorems and Quantum Error Correction

Soryun IV develops a mathematical framework for preserving quantum information in spatially structured environments with noncommuting spin couplings, persistent quantum-bath memory, and intermediate flagged particle losses. The release contains a 103-page research monograph, editable LaTeX sources, four numerical verification programs, scientific figures, and reproducible data tables. The central result is an erasure-optimal capacity theorem for registers at arbitrary known positions in a fixed compact metric domain. Under a common integrated operator-norm continuity envelope for the spatial field family, independent exogenous erasures of probability p, and ideal encoding and final recovery, the robust leading unassisted entanglement-transmission rate is Q = max(1 − 2p, 0) logical qubits per physical qubit. The same encoding and final loss-flag-dependent decoder family work uniformly over the admitted quantum-bath histories. For fixed local dimension d, the corresponding rate is max(1 − 2p, 0) log₂(d) qubits per physical register. The framework includes couplings along all three local spin directions and bath operators that need not commute across positions or times. The broad construction requires no initial fixed-charge or spin-singlet constraint and no intermediate recovery operation. Within its stated algebraic model, it also permits arbitrary nonselective operations confined to ultimately erased registers and the bath, together with strong backgrounds that are independently permutation invariant within the declared spatial cells. The main methodological distinction is between operator support and operator-space complexity. Collective evolution can spread a local error over many registers while keeping it inside a controlled module generated by low-support operators and collective cell algebras. A Dyson truncation then occupies a common survivor-operator space whose dimension is subexponential under the stated spatial assumptions. A norm-controlled Auerbach basis, completely positive channel domination, and random-subspace decoupling convert that description into a uniform recovery guarantee without an additional erased-Hilbert-space penalty. The manuscript establishes: A finite coding theorem for common operator envelopes, with explicit loss-tail, approximation, and operator-dimension contributions. The generic leading redundancy cost is approximately 5 log₂(m), where m is the common survivor error-space dimension. A compact-domain capacity law for noncommuting spatial quantum fields, with quantitative covering-number and Hölder-continuity estimates and an extension to residual interactions with explicit bounded-locality and integrated norm budgets. Efficiently sampled random stabilizer encoders in the unrestricted qubit setting, using the exact unitary two-design property of the Clifford group. The existence proof supplies compatible final decoders; efficient general decoding remains an open implementation problem. Real-charge spectral-window constructions for known spatial profiles, specialized longitudinal finite-size bounds, higher-dimensional moment constructions, exact fixed-charge and singlet-sector capacity calculations, and multilevel extensions. Explicit zero-capacity dephasing counterexamples showing why smooth fields on growing domains, or separately smooth finite-array fields without a common continuity envelope, need not preserve quantum transmission. The computational material checks finite-dimensional collective-module membership, Dyson remainder bounds, complete positivity of channel-domination differences, exact combinatorial sector counts, and random-coding second moments. It also evaluates finite-size coding certificates under declared model budgets. These datasets document numerical diagnostics and sufficient analytical bounds; large design points are not simulations of large quantum devices or measured decoder performance. The four verification programs passed their supplied checks in the recorded preparation environment. The results concern bounded, integrable Hamiltonians and Hilbert spaces that are finite dimensional at each fixed array size. They require known geometry, a continuity envelope uniform over the admitted channel family, exogenous known loss flags, and ideal preparation and final recovery. Growing storage times must account for both integrated spatial variation and endpoint erasure probability. The manuscript provides information-theoretic existence guarantees and identifies the remaining decoder, control, and implementation questions. Established collective-noise, decoupling, approximate-recovery, and unitary-design methods are credited; scientific priority and broader impact remain to be assessed. Creative inspiration included Suoming from Wuthering Waves.