Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems
Byoungwoo Lee · Zenodo (CERN European Organization for Nuclear Research) · 2026
Overview This work develops a finite-orbit extension of reflected spectral-jet methods for degenerate minimax problems with symmetry-related active spectral branches. The main objective is to move beyond a single reflected pair of branches. A finite symmetry group acts simultaneously on normal coordinates and active branch labels, producing a finite orbit of competing spectral branches. The paper introduces group-averaged spectral jets, centered orbit deviations, primitive quadratic-isolation criteria, representation-theoretic tests for low-order branch splitting, and analytic descriptions of multi-branch coactivity geometry. The abstract framework is realized in: exact non-diagonal $C_3$- and $D_3$-symmetric Hermitian models; a fixed-scenario rank-two Hermitian compression problem; a three-location single-fault Pauli-history model; and the full eight-label independent three-bit quantum error-history process. The full-process QEC realization exhibits a particularly sharp phenomenon: truncating the history space to zero- and single-fault events can reverse the local optimization geometry of the same quantum code. Finite-orbit quadratic isolation Let $G$ be a finite group acting on a normal space $N$ and on a finite transitive branch orbit $\Omega$. For active branches $\ell_\omega$, define the centered branch deviation relative to the group-averaged jet $\mathcal J_G$: $$d_\omega = \ell_\omega - \mathcal J_G.$$ If the orbit average has uniform quadratic normal concavity, $$D_n^2\mathcal J_G \preceq -2\mu z^2 I,$$ while every centered branch deviation vanishes through second order at the group-fixed manifold and has uniformly controlled cubic remainder, then $$\max_{\omega\in\Omega} \ell_\omega(z,p,s,n) \le \mathcal J_G(z,p,s,0) - \frac{\mu}{2}z^2\vert{}n\vert{}^2$$ throughout a sufficiently small normal tube. This generalizes reflected-pair isolation to a finite active orbit. The paper also identifies a representation-theoretic sufficient condition for the disappearance of low-order centered splitting: $$\mathrm{Hom}_G(V_\Omega^0, \mathrm{Sym}^k(N)) = \{0\},$$ where $V_\Omega^0$ is the zero-sum branch-label representation. Multi-branch coactivity For analytic optimized branches $W_1,\ldots,W_m$, normalized branch-difference maps are used to define continued coactivity sets through a degenerate perturbation point. If $k+1$ branches are active and the normalized difference map has rank $k$, then the corresponding regular coactivity set is locally a codimension-$k$ real-analytic manifold. Thus: two active branches generate a codimension-one transition hypersurface; three active branches generate a codimension-two triple-coactivity locus; and higher active multiplicities generate higher-codimension coactivity strata. Near a regular coactivity point, the active envelope is locally equivalent to the standard chamber fan $$\max\{0,y_1,\ldots,y_k\}.$$ The paper also distinguishes these regular strata from singular orbit-fixed junctions, where centered branch splitting begins quadratically or cubically and the normalized difference Jacobian vanishes. Hermitian realizations An exact $C_3/D_3$-symmetric non-diagonal Hermitian prototype realizes: a proper group-fixed manifold; three distinct simple spectral branches; cubic orbit splitting; exact quadratic normal isolation; three dominance chambers; and an isolated triple-coactivity junction. A second realization starts from a fixed $4\times4$ Hermitian scenario and its $C_3$ unitary orbit. Compression to a moving rank-two graph subspace produces the exact jet $$\mathcal J_G(z, u) = 1 - \frac23 z^2\vert{} u\vert{}^2 + \frac23 z^3\vert{} u\vert{}^2 \operatorname{Re}(\omega^{-j} u) + \frac{26}{27}z^4\vert{} u\vert{}^4 + O(z^5),$$ together with an exact quadratic-isolation inequality. Three-location finite-history QEC realization The single-fault model uses the four history labels $$\{I, X_0, X_1, X_2\}$$ on three physical qubits. For the repetition code $$\mathcal C = \operatorname{span}\{\vert{}000\rangle, \vert{}111\rangle\},$$ the Knill–Laflamme scalar-compression conditions hold exactly in this restricted history sector, so the environment-record leakage vanishes. A Fourier-syndrome graph deformation produces three cyclic marginal-leakage branches with exact compressed products $$P_j(z,q) = \frac{1}{1+q} \left[ \frac{2z}{\sqrt3} \operatorname{Re}(\omega^{-j} u)\sigma_z + \frac q3\sigma_x \right], \qquad q=z^2\vert{} u\vert{}^2.$$ The centered splitting is quadratic rather than cubic. The resulting upper envelope has six dominance sectors and a singular triple-coactivity junction. This gives a concrete QEC realization of the fact that finite $C_3$ symmetry alone does not eliminate quadratic orbit splitting. Full eight-label three-bit process The complete independent three-bit history process is indexed by $$h \in \{0,1\}^3 \quad \implies \quad W_h = X_0^{h_0}X_1^{h_1}X_2^{h_2}.$$ For an arbitrary rank-two code, the complete complementary-register leakage admits the exact Bloch reduction $$\mathcal L_8(V) = \max_{\vert{}u\vert{}=1} \left\vert{} \sum_{s e0} (u\cdot\mathbf k_s) D_p P_s D_p \right\vert{}_1,$$ where $\mathbf k_s$ is the Bloch vector of the traceless part of the compressed history product $V^\dagger W_s V$. At the repetition code, the full process has an exact logical-complement obstruction: $$\mathcal L_8(\mathcal C) = 8 \prod_{j=0}^{2} \sqrt{\varepsilon_j(1-\varepsilon_j)}.$$ For equal independent bit-flip probabilities ($\varepsilon_j=\varepsilon$), let $$\tau = [\varepsilon(1-\varepsilon)]^{3/2}.$$ Then $$\mathcal L_8(\mathcal C) = 8\tau.$$ For the Fourier-syndrome graph family, the nontrivial history products split into: a weight-one $C_3$ orbit; a weight-two $C_3$ orbit; and the triple-event logical-complement singleton. In the intrinsic graph coordinate $w$, the complete leakage satisfies $$\mathcal L_8(w) = 8\tau - 16\tau\vert{}w\vert{}^2 + o(\vert{}w\vert{}^2) \qquad (w\to0).$$ Consequently, the repetition code is a strict local maximum of the complete eight-label leakage throughout the two-real-dimensional Fourier-syndrome graph chart. Sector-truncation reversal The same code and the same graph deformation have opposite local effects in the truncated and complete history models. In the four-label single-fault sector, $$\mathcal L_{\mathrm{single}}(\mathcal C) = 0,$$ and every nontrivial sufficiently small graph deformation produces positive leakage. In the full eight-label process, $$\mathcal L_8(\mathcal C) = 8\tau,$$ and the same deformation decreases leakage quadratically. Thus the paper proves an exact sector-truncation reversal: $$\boxed{ \text{local protection in the truncated process} \quad\longleftrightarrow\quad \text{local leakage maximality in the full process} }$$ The restoration of double- and triple-event histories does not merely change the leakage quantitatively. It changes the local minimax geometry itself. Variational consequence The finite-orbit isolation mechanism enters hard-normal minimax selection through the effective lower functional $$\mathcal Q_{\mathrm{eff}}(\alpha,\beta,n) = H(a_0,t) + \beta + \frac12 \langle Q\alpha,\alpha\rangle + c\vert{}n\vert{}^2, \qquad \beta\ge0.$$ This yields compactness, localization, and accumulation constraints for exact and near minimizers. Scope The paper does not claim: a finite-parameter closed formula for the complete eight-label leakage away from the local graph regime; a full-Grassmannian minimax theorem for the three-bit process; a complete classification of all minimizing or zero-leakage codes; a fourth-order classification of the interaction between the weight-one and weight-two history orbits; or a treatment of semisimple active spectral clusters. The active spectral branches in the abstract framework are assumed to be locally simple. Matrix-valued spectral jets associated with multiple eigenvalue clusters remain a separate extension. Reproducibility The accompanying standalone research bundle contains: the manuscript PDF and LaTeX source; theorem-facing proof notes for the single-fault and full eight-label QEC models; a deterministic SymPy verification script; a machine-readable JSON verification report; a tested Python/SymPy environment specification; a portable SHA-256 manifest verifier; a complete SHA-256 manifest; and package-status, audit, and change-log files. The verifier checks exact covariance identities, orbit averages, centered jet cancellations, characteristic polynomials, Hermitian compression formulas, Pauli-product compressions, leakage coefficients, logical-complement constants, and intrinsic graph-coordinate expansions. Compactness, analytic continuation, uniform branch localization, trace-norm asymptotics, variational liminf arguments, and recovery constructions 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. Related work This paper extends the reflected-pair framework developed in: Reflected Spectral-Jet Realization for Degenerate Minimax Problems: Hard-Normal Selection, Flat Tangential Persistence, and Coactive Boundary Transitions, Version v0.2r1, DOI: 10.5281/zenodo.21636022. Its QEC applications are also related to the finite-history minimax line developed in: 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.