Wrapped Phase, Protected Qubits

Wrapped Phase, Protected Qubits: Minimal Phase-Channel Model for Syndrome-Guided Branch Tracking in Quantum Error Correction This paper introduces Harmonic Consistency and Oversampled Phase-Lock (HCOPL) as a signal-processing-native interpretation layer for phase-dominant quantum error correction. It is derived from Scale-Time Theory 10.0 and Engineering Note EN-001, Relative-Ratio Bookkeeping for Kinematic Aliasing, and develops the idea that certain phase-channel errors may be treated as wrapped-phase branch ambiguities under finite sampling. HCOPL does not claim to replace standard Quantum Error Correction, and it is not presented as a hardware-ready fault-tolerant protocol. Instead, it proposes a minimal, falsifiable phase-channel model in which the relative phase coordinate of an encoded logical relation is redundantly represented across harmonic channels. Errors are detected as violations of harmonic consistency, and correction is framed as classical branch tracking of syndrome-compatible wrapped-phase information. A central distinction in this version is the separation between normalized phase in cycles, ν, and physical radian phase, θ, with θ_log = 2πν_log. The paper defines a minimal harmonic-channel model, introduces a branch-invariant pairwise congruence residual, and presents a candidate Method-F-like weighted least-squares solver for recovering the logical phase branch under prior continuity constraints. The oversampling ratio (R_A) is defined as a control-frame parameter relative to the effective bandwidth of the targeted phase-error process, not the physical qubit carrier. This keeps the proposal grounded as a baseband/control-frame simulation target rather than an unrealistically high-frequency hardware requirement. The paper also gives a cautious conceptual bridge to Continuous-Variable QEC and Gottesman-Kitaev-Preskill codes. The correspondence is limited to shared mathematical themes such as modulo structure, displacement-like errors, branch selection, and thresholded recovery. HCOPL is not claimed to be a GKP code and does not inherit GKP thresholds, bosonic encoding properties, or fault-tolerance guarantees. The immediate validation target is a classical phase-channel simulation. The proposed roadmap is to encode a tracked logical phase into harmonic channels, inject controlled drift and phase slips, compute congruence and decoder residuals, run the candidate branch-tracking solver, and quantify whether oversampling depth (R_A) and harmonic depth (M) reduce logical phase-channel error rates. Later stages may compare the resulting model against baseline stabilizer-code simulations under matched phase-drift noise. This release is intended as a cautious research hypothesis and simulation proposal. Its value lies in defining a model that can be implemented, challenged, and falsified.