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 minimax protection against finite two-event quantum error histories by rank-two quantum codes in the symmetric near-balanced regime. The central phenomenon is a singular form of degenerate geometric selection. Within every compact subset of the balanced overlap interior, the fully optimized worst-case environment-record leakage remains exactly independent of the transverse-axis overlap. Consequently, the interior degeneracy is not lifted by the second-order term, the fourth-order term, or any finite Taylor coefficient within the continued analytic phase. Overlap selection occurs instead through a singular active-set transition near the boundary of the balanced overlap cap. The paper establishes: exact fixed-interior overlap persistence; a uniform singular overlap-cap boundary-layer law; a locally unique analytic fourth-order transition curve for the full phase/input envelope; the bias-induced displacement of the optimal code anisotropy; the optimized transverse minimax expansion; and fourth-order minimax closure over the full Grassmannian of rank-two codes. The transverse code set used in the global argument is defined intrinsically as the Grassmannian closure of the regular parity cosine–sine locus with balanced parity loading and vanishing longitudinal parity-axis components. This formulation removes endpoint gauge ambiguity in the cosine–sine coordinates. Main result Let $$c_*=\frac12-\frac{\sqrt2}{6}, \qquad c_4=\frac{479\sqrt2-492}{3024}.$$ 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 provides the matching recovery sequence. A reflected one-axis lower jet, balanced-normal coercivity, compact localization, fiberwise transverse retraction, and a full-space liminf argument show that nontransverse rank-two codes cannot improve the minimax value through fourth order. Exact interior persistence Let $$t=\vert{}e\cdot o\vert{}$$ denote the transverse-axis overlap and let $$t_*=\frac{\sqrt2-1}{3}$$ be the balanced overlap cap. For every compact set $$K\Subset[0,t_*),$$ the fully optimized leakage of the selected-anisotropy transverse family agrees exactly with the continued one-axis analytic branch for all sufficiently small bias, uniformly for $t\in K$. Thus the overlap is not selected by any finite Taylor coefficient within the fixed interior analytic phase. The selection mechanism is instead generated by the activation of a second adversarial orbit near the cap boundary. Singular overlap selection In the second-order boundary layer $$t=t_*-\rho\delta^2,$$ the leakage coefficient is governed by $$C_{\mathrm{BL}}(\rho) = \min\left\{ c_*, \frac14+\kappa_0^2\rho \right\},$$ where $$\kappa_0^2=\frac{3(2-\sqrt2)}8.$$ More precisely, for every finite $R>0$, the boundary-layer expansion is uniform for $$\rho\in[0,R].$$ The cap equality classification and compact spectral isolation remain uniform throughout the corresponding $O(\delta^2)$ boundary layer and the finer $O(\delta^4)$ crossing window. Therefore no third phase/input branch enters the local full envelope. The leading retreat threshold is $$\rho_*=\frac{2-\sqrt2}{9}.$$ At fourth order, the locally unique transition curve at fixed selected anisotropy is $$t_c^{(\eta_0)}(\delta) = \frac{\sqrt2-1}{3} -\frac{2-\sqrt2}{9}\delta^2 +\frac{181-149\sqrt2}{1134}\delta^4 +O(\delta^6).$$ Below this curve the continued one-axis orbit controls the full local envelope; above it the mixed cap orbit controls the envelope. Moving anisotropy and optimized transverse value The optimal anisotropy moves according to $$ \eta_\delta = \eta_0 + \eta_0 \frac{376\sqrt2-379}{2352}\delta^2 + O(\delta^4), $$ where $$\eta_0 = \sqrt{\frac{3\sqrt2-4}{8}}.$$ Optimizing the transverse family over both anisotropy and overlap yields the same fourth-order coefficient $c_4$ as the full-Grassmannian minimax theorem. Full-space geometry For $O(\delta^4)$-near minimizers, the full-space lower bound gives the mixed-scale localization $$\eta-\eta_0=O(\vert{}\delta\vert{}),$$ while the three balanced hard-normal coordinates satisfy $$ r-\frac12=O(\delta^4), \qquad e_z=O(\delta^4), \qquad o_z=O(\delta^4). $$ 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. The transverse subset entering this argument is defined intrinsically as the Grassmannian closure of the regular transverse parity cosine–sine locus. The quantities $r$ and $a$ (or $\eta$) are determined by the trace and spectral gap of the compressed even-parity projection, so the global formulation does not depend on a nonunique endpoint choice of parity axes. Relation to the preceding theorem 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. The preceding work supplies: the common finite-history leakage model; the leading near-balanced minimax value; the selected anisotropy $\eta_0$; and the leading accumulation geometry of asymptotically minimizing codes. The present paper proves new results beyond that leading-order theory: exact fixed-interior overlap persistence; the cap active-set classification; the uniform boundary-layer envelope; fourth-order transition curves; the moving-anisotropy correction; the optimized fourth-order transverse coefficient; and the fourth-order full-Grassmannian liminf and recovery closure. No principal fourth-order theorem of the present work is contained in the preceding paper. Scope The results are perturbative and apply to 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 standalone research bundle contains: the manuscript PDF and LaTeX source; theorem-facing proof notes; five deterministic SymPy verification scripts; machine-readable JSON verification reports; a tested Python/SymPy environment specification; a portable SHA-256 manifest verifier; a complete SHA-256 manifest; and package-status and change-log files. The symbolic verifiers check exact algebraic identities, characteristic-polynomial expansions, radical constants, Hessian signs, transition coefficients, moving-anisotropy coefficients, and the algebraic inputs to the full-space effective functional. Compactness, analytic continuation, uniform branch isolation, branch compatibility, fiberwise retraction, and liminf/recovery arguments are supplied analytically in the manuscript and proof notes rather than claimed as fully machine-verified. All proof-critical scripts reject optimized Python mode, and no floating-point comparison enters a PASS decision.