Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure

## Overview This work develops a fourth-order perturbative theory for the minimax protection of finite two-event quantum error histories by rank-two quantum codes in the near-balanced regime. The central phenomenon is an unusual form of degenerate selection. Within every compact subset of the balanced overlap interior, the fully optimized worst-case leakage remains exactly independent of the overlap parameter. Consequently, the interior degeneracy is not lifted by the second-order term, the fourth-order term, or any finite Taylor coefficient within the same analytic phase. Overlap selection instead occurs through a singular boundary layer near the balanced overlap cap. The paper establishes: - exact fixed-interior overlap persistence;- the singular overlap-cap boundary-layer law;- a locally unique analytic fourth-order transition curve;- the bias-induced displacement of the optimal code anisotropy;- the optimized transverse minimax expansion;- and the fourth-order minimax closure over the full Grassmannian of rank-two codes. ## Main result Let $$c_*=\frac12-\frac{\sqrt2}{6},\qquadc_4=\frac{479\sqrt2-492}{3024}.$$ For the symmetric near-balanced bias parameter $\delta$, the full-Grassmannian minimax leakage satisfies $$\Lambda_{\mathrm{diag}}(\delta)=\frac12-c_*\delta^2-c_4\delta^4+o(\delta^4).$$ Equivalently, in the variance coordinate $A\uparrow 1/4$, $$\Lambda_{\mathrm{ind}}(A,A)=\frac12-c_*(1-4A)-c_4(1-4A)^2+o\!\left((1-4A)^2\right).$$ The optimized transverse family supplies the matching recovery sequence, while a reflected one-axis lower jet, balanced-normal coercivity, compact localization, and a full-space liminf argument show that nontransverse rank-two codes cannot improve the minimax value through fourth order. ## Singular overlap selection The balanced overlap cap is $$t_*=\frac{\sqrt2-1}{3}.$$ In the second-order boundary layer $$t=t_*-\rho\delta^2+o(\delta^2),$$ the leakage coefficient is governed by $$C_{\mathrm{BL}}(\rho)=\min\left\{c_*,\frac14+\kappa_0^2\rho\right\},$$ where $$\kappa_0^2=\frac{3(2-\sqrt2)}8.$$ At fourth order, the locally unique transition curve at fixed selected anisotropy is $$t_c^{(\eta_0)}(\delta)=\frac{\sqrt2-1}{3}-\frac{2-\sqrt2}{9}\delta^2+\frac{181-149\sqrt2}{1134}\delta^4+O(\delta^6).$$ The optimal anisotropy moves according to $$\eta_\delta=\eta_0+\eta_0\frac{376\sqrt2-379}{2352}\delta^2+O(\delta^4),$$ with $$\eta_0=\sqrt{\frac{3\sqrt2-4}{8}}.$$ ## Full-space geometry For $O(\delta^4)$-near minimizers, the full-space lower bound gives the mixed-scale localization $$\eta-\eta_0=O(|\delta|),$$ while the three balanced hard-normal coordinates obey $$r-\frac12=O(\delta^4),\qquade_z=O(\delta^4),\qquado_z=O(\delta^4).$$ The corresponding fourth-order effective lower functional is $$\mathcal Q_{\mathrm{full}}(\alpha,\beta)=-c_4+\beta+\frac{8(2\sqrt2+3)}{27}\alpha^2,\qquad\beta\ge0.$$ Its minimum is $-c_4$, attained by the optimized transverse recovery family. ## Scope The results are perturbative and apply in the symmetric near-balanced regime. The paper does not claim: - an exact finite-bias formula for the full minimax value;- a complete finite-bias classification of all minimizing codes;- sixth-order full-Grassmannian geometry;- or the full unequal-variance phase diagram. The leakage minimax is operationally related to recovery obstruction through complementary-channel duality, but the present theorem does not identify it with a particular recovery-fidelity optimizer. ## Reproducibility The accompanying research bundle contains: - the manuscript PDF and LaTeX source;- theorem-facing proof notes;- deterministic SymPy verification scripts;- machine-readable JSON verification reports;- a tested Python/SymPy environment specification;- a portable SHA-256 manifest verifier;- and a complete SHA-256 manifest. The symbolic verifiers check exact algebraic identities, characteristic-polynomial expansions, radical constants, Hessian signs, transition coefficients, and the algebraic inputs to the full-space effective functional. Compactness, analytic continuation, branch compatibility, and liminf/recovery arguments are supplied in the manuscript and proof notes rather than claimed as fully machine-verified. This paper is an independent fourth-order sequel to: **Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax**, Version v0.3r2, DOI: `10.5281/zenodo.21598820`.