Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions

## Overview This paper develops a constructive selection framework for finite-dimensional degenerate spectral minimax problems with reflected adversarial branches. The central proof architecture separates four geometric mechanisms: - nonsmooth hard-normal coercivity;- Morse-type selection in soft code coordinates;- exact persistence along flat tangential directions;- and singular activation of coactive spectral branches at a boundary. For a compact minimax problem $$\mathcal L_z(x)=\sup_{y\in Y}\lambda_{\max}D_z(x,y),\qquad\Lambda(z)=\inf_{x\in X}\mathcal L_z(x),$$ the framework derives branchwise analytic even jets from physically admissible reflected spectral witnesses, assembles them through fiber-compatible active branches, and obtains a global effective selection theorem. ## Main variational theorem Assume that, near the balanced minimizer set, the objective admits the uniform lower jet $$\mathcal L_z(x)\ge\Lambda_0+m(x)-F(a(x))z+H(a(x),t(x))z^2-C\bigl(m(x)z+z^3\bigr),$$ where: - $m(x)\ge0$ is a hard-normal excess;- $a$ is a soft selector coordinate;- $t$ is a residual tangential coordinate;- $F$ has a unique nondegenerate maximizer $a_0$. With a fourth-order recovery upper bound, the minimax value satisfies $$\Lambda(z)=\Lambda_0-F(a_0)z+h_*z^2+o(z^2),$$ where $$h_*=\min_t H(a_0,t).$$ Every $O(z^2)$-near minimizer obeys the mixed localization laws $$m(x_z)=O(z^2),$$ and $$a(x_z)-a_0=O(\sqrt z).$$ The corresponding effective lower functional is $$\mathcal Q_{\mathrm{eff}}(t,\alpha,\beta)=H(a_0,t)+\beta+\frac12\langle Q\alpha,\alpha\rangle,\qquad\beta\ge0,$$ with $$Q=-\nabla^2F(a_0)>0.$$ ## Reflected spectral-jet realization The paper distinguishes two analytic mechanisms. ### Regular reflected optimizer jets When the balanced optimizer Hessian is nondegenerate, the analytic implicit-function theorem produces locally optimized signed spectral branches. Physical reflection pairs their odd coefficients, yielding an analytic even jet in $z$. ### Normalized reflected lower witnesses When the raw optimizer Hessian vanishes at $z=0$, the lower-order jet is removed and the optimizer is continued through a normalized blown-up functional. The resulting reflected average is treated as an analytic lower witness. The paper explicitly distinguishes this reflected average from the actual maximum of the two reflected branches. Identification with the actual pair max-envelope requires an additional equalization-and-isolation gate. ## Quadratic pair isolation from reflected tangency A principal new result derives pair-envelope isolation from primitive local conditions. Let $$\mathcal J=\frac{\ell_++\ell_-}{2},\qquad\Delta=\ell_+-\ell_-,$$ and $$\mathcal M=\max\{\ell_+,\ell_-\}=\mathcal J+\frac12|\Delta|.$$ Suppose that, in normal coordinates $n$ around a reflection-fixed manifold, $$D_n^2\mathcal J\preceq-2\mu z^2 I,$$ the reflected average is even in $n$, the branch difference is odd in $n$, $$D_n\Delta(z,p,s,0)=0,$$ and $$\|D_n^3\Delta\|_{\mathrm{op}}\leLz^2.$$ Then, on a sufficiently small normal tube, $$\mathcal M(z,p,s,n)\le\mathcal J(z,p,s,0)-\frac{\mu}{2}z^2\|n\|^2.$$ Thus quadratic pair isolation follows from: $$\text{normal concavity}+\text{reflected tangency}+\text{uniform cubic control}.$$ This replaces a result-level pair-isolation hypothesis by locally verifiable differential conditions. ## Flat tangential persistence and boundary activation The framework also treats a residual tangential coordinate that remains exactly flat throughout the regular interior. Under exact tangential decoupling and uniform active-set isolation, $$\mathcal L_z(s(a,t))=W_{\mathrm{one}}(z,a)$$ on every compact subset of the regular interior. Consequently, no finite Taylor coefficient within the same analytic phase selects the tangential coordinate. At a coactive boundary, a second optimized branch may enter. In the scaled chart $$t=t_*(a)-\rho z+\sigma z^2,$$ the retreat coefficient and fourth-order crossing coefficient are $$\rho_*(a)=\frac{F(a)-g(a)}{\kappa_b(a)},$$ and $$\sigma_*(a)=\frac{h_1(a)-h_m(a,\rho_*(a))}{\kappa_b(a)}.$$ This yields a locally unique analytic transition curve $$t_c(z,a)=t_*(a)-\rho_*(a)z+\sigma_*(a)z^2+O(z^3).$$ ## Hermitian realization examples Two explicit non-diagonal $2\times2$ Hermitian models are included. The first realizes a pointwise-equalized reflected pair and a normalized optimizer with a degenerate raw Hessian but a nondegenerate normalized Hessian. The second has a proper reflection-fixed set and reflected branches that split cubically away from it. An exact positive-semidefinite certificate proves quadratic isolation of the actual pair max-envelope. ## Quantum error-history application The abstract framework is related to the rank-two, two-event finite quantum error-history minimax problem through a formal model-verification proposition. The application identifies: - the Grassmannian hard-normal excess;- the soft anisotropy selector;- reflected one-axis spectral jets;- exact overlap persistence;- optimized transverse recovery;- and the coactive mixed boundary branch. The primitive reflected-tangency theorem is presented as a sufficient abstract route for future models. The previously established two-event quantum error-history result continues to rely on its model-specific determinant and active-set analysis. ## Scope The framework concerns finite-dimensional compact spectral minimax problems with a finite analytic active cover, isolated spectral branches, reflected admissible witnesses, and suitable hard-normal coercivity. It does not claim that: - every nonsmooth eigenvalue minimax problem admits this structure;- reflection symmetry alone implies pair-envelope identification;- every branch crossing has quadratic isolation;- all tangential degeneracies are resolved at a boundary;- or the theory extends automatically to infinite-dimensional operator problems. ## Reproducibility The accompanying bundle contains: - the manuscript PDF and LaTeX source;- a theorem-facing note for quadratic pair isolation;- exact symbolic verifiers for both non-diagonal Hermitian examples;- machine-readable JSON reports;- a portable SHA-256 manifest verifier;- a complete SHA-256 manifest;- and package audit information. The symbolic scripts verify exact matrix-reflection identities, characteristic expansions, normalized quadratic coefficients, cubic branch splitting, positive-semidefinite certificates, and transition data. Compactness, analytic continuation, implicit-function arguments, and variational liminf proofs are supplied in the manuscript.