Quantum Error Correction Using the Chrono-Elastic Wave

This paper introduces Chrono-Elastic Refocusing (CER): a measurement-free, holonomic quantum error correction protocol derived from the Chrono-Elastic Wave (CEW) operator algebra of Pure Temporal Geometry (PTG). The CER protocol applies the six-step sequence Ĥ, P̂ (δ), Ŝ (γ), Ĥ, Ŝ (γ), P̂ (δ) to a quantum state in continuous-variable (bosonic) phase space. Five structural results are established. (1) The net composite of the six-step sequence is the identity on every initial state: F∘G = 𝟙₂, where F = P̂ (δ) ∘Ŝ (γ) and G = Ĥ∘Ŝ (γ) ∘P̂ (δ) ∘Ĥ = F⁻¹. (2) The period composite has Wirtinger pair (1, 0): it is exactly holomorphic, despite every individual step being non-holomorphic. (3) The torsion coupling κg = γδ is strictly positive and is the unique period-invariant of the protocol. (4) Symmetric SINK-type errors (identical amplitude-damping perturbations applied at steps 3 and 5 of the sequence) are cancelled exactly. (5) Symmetric PUMP-type errors (identical phase-damping perturbations applied at steps 2 and 6) are cancelled exactly. Results (4) and (5) follow from the same algebraic mechanism: the horizon operator Ĥ at step 4 reverses the sign of the phase quadrature, so that equal environmental perturbations in the two halves of the sequence cancel. The protocol requires no syndrome measurement, no ancilla qubits, and no post-selection. It addresses both amplitude damping (T₁) and phase damping (T₂) simultaneously — a capability not shared by standard dynamical decoupling sequences such as CPMG or XY-n, which address phase errors only. The three operators are implementable as native Gaussian operations on superconducting, trapped-ion, and photonic hardware.