Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax

## Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities \(\mu_+=\mu_-=1/5\), an explicit transverse non-isoclinic code satisfies $$L^{\mathrm{wc}}_{1/5,1/5}(V)\le \frac{99}{250}<\frac25,$$ where \(2/5\) is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy $$\eta_0=\sqrt{\frac{3\sqrt2-4}{8}},$$ and the minimax over the full rank-two code Grassmannian satisfies $$\Lambda_{\mathrm{diag}}(\delta)=\frac12-\left(\frac12-\frac{\sqrt2}{6}\right)\delta^2+o(\delta^2),\qquad \delta=1-2\mu.$$ Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with $$r=\frac12,\qquad \eta=\eta_0,\qquad e_z=o_z=0,$$ and with relative transverse-axis overlap bounded by $$|\mathbf e\cdot\mathbf o|\le\frac{\sqrt2-1}{3}.$$ Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI `10.5281/zenodo.21523950`- Balanced finite-history benchmark and complete minimizer geometry: DOI `10.5281/zenodo.21560397`