Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems

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 C₃- and D₃-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 Ω. For active branches _ω, define the centered branch deviation relative to the group-averaged jet JG: d_ω = _ω - JG. If the orbit average has uniform quadratic normal concavity, Dₙ² JG -2μ z² I, while every centered branch deviation vanishes through second order at the group-fixed manifold and has uniformly controlled cubic remainder, then maxω∈Ω _ω(z,p,s,n) ≤ JG(z,p,s,0) - μ/2z²n² 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: HomG(V_Ω⁰, Symᵏ(N)) = \0\, where V_Ω⁰ is the zero-sum branch-label representation. Multi-branch coactivity For analytic optimized branches W₁,…,Wₘ, 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,y₁,…,yₖ\. 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 C₃/D₃-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 4×4 Hermitian scenario and its C₃ unitary orbit. Compression to a moving rank-two graph subspace produces the exact jet JG(z,ν) = 1 - 23 z²ν² + 23 z³ν² Re(ω⁻ʲν) + 26/27z⁴ν⁴ + O(z⁵), together with an exact quadratic-isolation inequality. Three-location finite-history QEC realization The single-fault model uses the four history labels , X₀, X₁, X₂\ on three physical qubits. For the repetition code C = span\000, 111\, 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 Pⱼ(z,q) = 1/1+q [ 2z/√3 Re(ω⁻ʲν)σz + q3σₓ ], q=z²ν². 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 C₃ 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 ∈ \0,1\³ Wₕ = X₀h₀X₁h₁X₂h₂. For an arbitrary rank-two code, the complete complementary-register leakage admits the exact Bloch reduction L₈(V) = max_u=1 ∑s≠0 (u· kₛ) Dₚ Pₛ Dₚ ₁, where kₛ is the Bloch vector of the traceless part of the compressed history product V^ Wₛ V. At the repetition code, the full process has an exact logical-complement obstruction: L₈( C) = 8 ∏ⱼ₌₀² √εⱼ(1-εⱼ). For equal independent bit-flip probabilities (εⱼ=ε), let τ = [ε(1-ε)]3/2. Then L₈( C) = 8τ. For the Fourier-syndrome graph family, the nontrivial history products split into: a weight-one C₃ orbit; a weight-two C₃ orbit; and the triple-event logical-complement singleton. In the intrinsic graph coordinate w, the complete leakage satisfies L₈(w) = 8τ - 16τw² + o(w²) (w→0). 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, Lsingle( C) = 0, and every nontrivial sufficiently small graph deformation produces positive leakage. In the full eight-label process, L₈( C) = 8τ, and the same deformation decreases leakage quadratically. Thus the paper proves an exact sector-truncation reversal: local protection in the truncated process 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 Qeff(α,β,n) = H(a₀,t) + β + 12 Qα,α + cn², β≥0. 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.