Covariant quantum error correction in a three-layer quantum brain model: computational analysis of layer-specific coherence dynamics

Abstract: Proposals for quantum coherence in neural computation lack quantitative frameworks for evaluating when -- and whether -- coherence provides computational benefits at biologically calibrated parameters. Here we construct such a framework by integrating a three-layer model parameterized by \textit{ab initio} spin Hamiltonian calculations of monoamine oxidase~A (MAO-A) with approximate covariant quantum error correction (CQEC) based on energy-conserving recursive swap tests. The three layers -- ${}^{31}$P nuclear spin memory ($d = 4$, $T_2 = 3.2$~ms), electron spin quantum-classical interface ($d = 8$, $T_2^e = 1.1$~ns), and classical radical-pair electrochemistry -- are evaluated on error correction benchmarks and a symmetric binary decision task. We find a layer-specific dichotomy: Layer~1 operates in the naturally coherence-preserving regime ($γ\mathrm{eff} \approx 10^{-6}$) while Layer~2 is decoherence-dominated ($γ\mathrm{eff} \approx 4.5$, $F \approx 0.51$ versus random baseline $F = 0.125$). In the decision task, CQEC maintains L$\leftrightarrow$R tunneling coherence (up to 168-fold at $γ= 0.5$), extending the time window during which a symmetric double-well system can oscillate between degenerate states before decoherence-induced symmetry breaking. Crucially, a matched classical stochastic model with equivalent noise structure reproduces the symmetry-breaking phenomenon but not the oscillatory dynamics, establishing coherent tunneling as a genuinely quantum signature. We explicitly identify what this toy model cannot address: state preparation at 310~K, spatial entanglement distribution, metabolic costs of error correction, and the 62-fold gap between nuclear spin $T_2$ (3.2~ms) and behaviorally relevant timescales ($\sim$200~ms). These limitations define the quantitative targets that any serious quantum brain proposal must meet.