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Bias-Induced Non-Isoclinic Optimality in Finite Quantum Error Histories: Exact Separation, Thresholdless Bifurcation, and Near-Balanced All-Code Minimax

Byoungwoo Lee

Published Jul 26, 2026• 2 citations
Publisher

Abstract

## Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities \(\mu_+=\mu_-=1/5\), an explicit transverse non-isoclinic code satisfies $$L^{\mathrm{wc}}_{1/5,1/5}(V)\le \frac{99}{250}<\frac25,$$ where \(2/5\) is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy $$\eta_0=\sqrt{\frac{3\sqrt2-4}{8}},$$ and the minimax over the full rank-two code Grassmannian satisfies $$\Lambda_{\mathrm{diag}}(\delta)=\frac12-\left(\frac12-\frac{\sqrt2}{6}\right)\delta^2+o(\delta^2),\qquad \delta=1-2\mu.$$ Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with $$r=\frac12,\qquad \eta=\eta_0,\qquad e_z=o_z=0,$$ and with relative transverse-axis overlap bounded by $$|\mathbf e\cdot\mathbf o|\le\frac{\sqrt2-1}{3}.$$ Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI `10.5281/zenodo.21523950`- Balanced finite-history benchmark and complete minimizer geometry: DOI `10.5281/zenodo.21560397`

DOI
10.5281/zenodo.21598819

Related papers

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

Byoungwoo Lee

## 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`.

•Jul 27, 2026•3 citations

Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability

Byoungwoo Lee

Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: $$\inf_{V\in\mathrm{Gr}(2,4)} \sup_{\phi,\psi} \mathcal L_{\mathrm{hist}}(V) = \frac12.$$ The transverse leakage is determined exactly by $$L_{\mathrm{tr}}(\eta,t) = \max\left\{ \frac12,\, 2\eta,\, \sqrt{\frac14-\eta^2}\sqrt{1+t} \right\}.$$ All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by $$0 < \eta \le \frac14, \qquad e_z = o_z = 0, \qquad \vert{}\mathbf e \cdot \mathbf o\vert{} \le \frac{4\eta^2}{1-4\eta^2}.$$ A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With $$d_{\mathrm A}(p,q) = \left[ \sum_h \left(\sqrt{p_h}-\sqrt{q_h}\right)^2 \right]^{1/2},$$ the minimax value satisfies $$\vert{}\Lambda(p)-\Lambda(q)\vert{} \le 2d_{\mathrm A}(p,q).$$ Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.

•Jul 25, 2026•1 citation

When Is History Truncation Safe for Quantum-Code Optimization? Projective-Unitary Path Collapse and Distribution-Robust Optimizer Reversal

Byoungwoo Lee

## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form $$W_\gamma=e^{i\theta_\gamma}U_{c(\gamma)},$$ where $c(\gamma)$ is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: $$\mathcal L_{\mathrm{path}}(V)=\mathcal L_{\pi}(V).$$ For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let $\pi$ denote the induced final-parity distribution on $\mathbb F_2^3$, let $c=(1,1,1)$, and define $$\Lambda_\pi:=\sum_{t\in\mathbb F_2^3}\sqrt{\pi_t\pi_{t\oplus c}}.$$ For the three-qubit repetition code $$E=\operatorname{span}\{|000\rangle,|111\rangle\},$$ the complete complementary leakage is exactly $$\mathcal L_\pi(E)=\Lambda_\pi.$$ For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate $w$ satisfies $$\mathcal L_\pi(V(w))=\Lambda_\pi-2\Lambda_\pi|w|^2+o(|w|^2).$$ Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has $$\mathcal L_{\leq 1}(E)=0$$ and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over $\operatorname{Gr}(2,4)$, classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **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 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **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 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is $$\text{compression geometry}\longrightarrow\text{balanced global minimax}\longrightarrow\text{bias-induced selection}\longrightarrow\text{fourth-order closure}\longrightarrow\text{reflected spectral jets}\longrightarrow\text{finite-orbit coactivity}\longrightarrow\text{projective path collapse and distribution-robust reversal}.$$ ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.

•Jul 30, 2026•0 citations

Finite Quantum Histories: Holonomy Spectra, Minimal Clocks, and Exact Clock-Change Covariance

Maxim V. Churilov

We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy $M=U_{L-1}\cdots U_0$. Its spectrum is $λ_{a,k}=1-\cos((2πk-θ_a)/L)$, where $e^{iθ_a}\in\mathrm{spec}(M)$. Thus the exact history sector is isomorphic to $\mathrm{Fix}(M)$, while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step $U$, the minimal number of clock events is the projective order of $U$. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by $U(r)\times\mathbb{Z}_L$ on a rank-$r$ history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.

quant-pharXiv:2608.05748•Aug 7, 2026•0 citations

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

Byoungwoo Lee

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 $$\mathcal J_G(z,\nu) = 1 - \frac23 z^2\vert{}\nu\vert{}^2 + \frac23 z^3\vert{}\nu\vert{}^2 \operatorname{Re}(\omega^{-j}\nu) + \frac{26}{27}z^4\vert{}\nu\vert{}^4 + O(z^5),$$ 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 $$P_j(z,q) = \frac{1}{1+q} \left[ \frac{2z}{\sqrt3} \operatorname{Re}(\omega^{-j}\nu)\sigma_z + \frac q3\sigma_x \right], \qquad q=z^2\vert{}\nu\vert{}^2.$$ 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 $$\mathcal L_8(V) = \max_{\vert{}u\vert{}=1} \left\vert{} \sum_{s\ne0} (u\cdot\mathbf k_s) D_p P_s D_p \right\vert{}_1,$$ 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.

•Jul 28, 2026•5 citations

Global Minimax Readout of a Qubit Direction

Abbas Taherpour, Amirhossein Taherpour, Tamer Khattab

We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length $η$. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For $N$ copies, inverse-Fisher $A$ loss has the exact value $2/[Nf(η)]$, where $ f(η)= \frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely $A$- and $D$-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have $A$-loss coefficient $4/η^2$, whereas collective measurements attain $2(1+η)/η^2$. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.

quant-pharXiv:2608.16840•Aug 18, 2026•0 citations

Overview This theorem-focused paper studies one logical qubit encoded into two physical qubits and exposed to two independent temporally ordered dephasing events, one collective and one differential, whose binary history labels are coherently stored by an environment. The objective is to minimize the worst-case logical information leaked into the environment’s history register over all rank-two quantum codes. A probability-adapted Bernoulli basis reduces the complementary output difference for arbitrary independent event probabilities to a traceless three-dimensional Hermitian arrow matrix. Phase-reflection and event-exchange symmetries then show that the fixed-code worst-phase leakage and its all-code minimax depend on the event probabilities only through their Bernoulli variances. ## Main results At equal event probabilities (\mu_+=\mu_-=1/5), an explicit transverse non-isoclinic code satisfies L1/5,1/5wc(V)≤99250<25,L^{\mathrm{wc}}_{1/5,1/5}(V)\le \frac{99}{250}<\frac25,L1/5,1/5wc​(V)≤25099​<52​, where (2/5) is the exact optimum over the complete parity-isoclinic manifold. The strict upper bound is certified using exact rational tensor-Bernstein arithmetic. Distributional stability extends this separation to an explicit two-parameter rectangle whose open interior is a region of forced non-isoclinic optimality. On the symmetric independent line, the paper proves a thresholdless bifurcation: every nondeterministic nonbalanced distribution destroys parity-isoclinic global optimality. Near the balanced point, probability bias selects the universal finite anisotropy η0=32−48,\eta_0=\sqrt{\frac{3\sqrt2-4}{8}},η0​=832​−4​​, and the minimax over the full rank-two code Grassmannian satisfies Λdiag(δ)=12−(12−26)δ2+o(δ2),δ=1−2μ.\Lambda_{\mathrm{diag}}(\delta)=\frac12-\left(\frac12-\frac{\sqrt2}{6}\right)\delta^2+o(\delta^2),\qquad \delta=1-2\mu.Λdiag​(δ)=21​−(21​−62​​)δ2+o(δ2),δ=1−2μ. Every asymptotically minimizing code accumulates on the transverse balanced-minimizer stratum with r=12,η=η0,ez=oz=0,r=\frac12,\qquad \eta=\eta_0,\qquad e_z=o_z=0,r=21​,η=η0​,ez​=oz​=0, and with relative transverse-axis overlap bounded by ∣e⋅o∣≤2−13.|\mathbf e\cdot\mathbf o|\le\frac{\sqrt2-1}{3}.∣e⋅o∣≤32​−1​. Thus probability bias breaks the balanced continuum of globally minimizing codes and selects a genuinely non-isoclinic geometric phase with a universal leading anisotropy over the full code Grassmannian. ## Scope The paper does not claim an exact finite-bias formula for the complete minimax, a determination of the order at which the remaining transverse-axis overlap is selected, or a complete off-diagonal phase boundary for unequal event variances. These are identified as follow-up problems. ## Reproducibility package This record contains: - the manuscript PDF;- the complete LaTeX source;- a standalone reproducibility bundle;- seven deterministic exact-arithmetic theorem verifiers;- machine-readable JSON verification reports;- a SHA-256 manifest and verification utility. No floating-point comparison enters a verifier’s PASS decision. The scripts certify the exact rational or symbolic identities used in the proofs. Compactness, localization, perturbative-envelope, and Grassmannian liminf arguments remain analytic parts of the manuscript and are not replaced by computation. ## Version Version v0.3r2, July 2026. ## Related public records - Directional code-geometry foundation: DOI 10.5281/zenodo.21523950- Balanced finite-history benchmark and complete minimizer geometry: DOI 10.5281/zenodo.21560397

## 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 δ\deltaδ, the full-Grassmannian minimax leakage satisfies Λdiag(δ)=12−c∗δ2−c4δ4+o(δ4).\Lambda_{\mathrm{diag}}(\delta)=\frac12-c_*\delta^2-c_4\delta^4+o(\delta^4).Λdiag​(δ)=21​−c∗​δ2−c4​δ4+o(δ4). Equivalently, in the variance coordinate A↑1/4A\uparrow 1/4A↑1/4, Λind(A,A)=12−c∗(1−4A)−c4(1−4A)2+o ⁣((1−4A)2).\Lambda_{\mathrm{ind}}(A,A)=\frac12-c_*(1-4A)-c_4(1-4A)^2+o\!\left((1-4A)^2\right).Λind​(A,A)=21​−c∗​(1−4A)−c4​(1−4A)2+o((1−4A)2). 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∗=2−13.t_*=\frac{\sqrt2-1}{3}.t∗​=32​−1​. In the second-order boundary layer t=t∗−ρδ2+o(δ2),t=t_*-\rho\delta^2+o(\delta^2),t=t∗​−ρδ2+o(δ2), the leakage coefficient is governed by CBL(ρ)=min⁡{c∗,14+κ02ρ},C_{\mathrm{BL}}(\rho)=\min\left\{c_*,\frac14+\kappa_0^2\rho\right\},CBL​(ρ)=min{c∗​,41​+κ02​ρ}, where κ02=3(2−2)8.\kappa_0^2=\frac{3(2-\sqrt2)}8.κ02​=83(2−2​)​. At fourth order, the locally unique transition curve at fixed selected anisotropy is tc(η0)(δ)=2−13−2−29δ2+181−14921134δ4+O(δ6).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).tc(η0​)​(δ)=32​−1​−92−2​​δ2+1134181−1492​​δ4+O(δ6). The optimal anisotropy moves according to ηδ=η0+η03762−3792352δ2+O(δ4),\eta_\delta=\eta_0+\eta_0\frac{376\sqrt2-379}{2352}\delta^2+O(\delta^4),ηδ​=η0​+η0​23523762​−379​δ2+O(δ4), with η0=32−48.\eta_0=\sqrt{\frac{3\sqrt2-4}{8}}.η0​=832​−4​​. ## Full-space geometry For O(δ4)O(\delta^4)O(δ4)-near minimizers, the full-space lower bound gives the mixed-scale localization η−η0=O(∣δ∣),\eta-\eta_0=O(|\delta|),η−η0​=O(∣δ∣), 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 Qfull(α,β)=−c4+β+8(22+3)27α2,β≥0.\mathcal Q_{\mathrm{full}}(\alpha,\beta)=-c_4+\beta+\frac{8(2\sqrt2+3)}{27}\alpha^2,\qquad\beta\ge0.Qfull​(α,β)=−c4​+β+278(22​+3)​α2,β≥0. Its minimum is −c4-c_4−c4​, 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`.
Overview This preprint develops a finite-history framework for quantifying how coherently stored environment records reveal logical quantum information. Instead of considering only which error occurred, the analysis treats the full complementary environment channel associated with a labeled family of finite quantum-error histories. For two selected history labels, the leakage is expressed exactly through the numerical-range diameter of the compressed relative operator. This yields global finite-history compression bounds, a zero-leakage characterization equivalent to Knill–Laflamme scalarity, and lower and upper bounds for recovery by a single common channel. Main results For a balanced independent two-step dephasing model on one logical qubit encoded into two physical qubits, the paper solves the worst-case optimization over the full rank-two code Grassmannian: inf⁡V∈Gr(2,4)sup⁡ϕ,ψLhist(V)=12.\inf_{V\in\mathrm{Gr}(2,4)} \sup_{\phi,\psi} \mathcal L_{\mathrm{hist}}(V) = \frac12.infV∈Gr(2,4)​supϕ,ψ​Lhist​(V)=21​. The transverse leakage is determined exactly by Ltr(η,t)=max⁡{12, 2η, 14−η21+t}.L_{\mathrm{tr}}(\eta,t) = \max\left\{ \frac12,\, 2\eta,\, \sqrt{\frac14-\eta^2}\sqrt{1+t} \right\}.Ltr​(η,t)=max{21​,2η,41​−η2​1+t​}. All global minimizers are classified. Besides the parity-isoclinic orthogonal-axis family, the minimizer set contains a genuine non-isoclinic region characterized by 0<η≤14,ez=oz=0,∣e⋅o∣≤4η21−4η2.0 < \eta \le \frac14, \qquad e_z = o_z = 0, \qquad \vert{}\mathbf e \cdot \mathbf o\vert{} \le \frac{4\eta^2}{1-4\eta^2}.0<η≤41​,ez​=oz​=0,∣e⋅o∣≤1−4η24η2​. A universal quartic inequality needed for the all-code theorem is reduced analytically to an exact polynomial-positivity problem and closed using rational Bernstein-basis verification. Distributional stability On a fixed labeled history-operator family, the complementary channel, worst-parameter leakage, and all-code minimax value are Lipschitz-stable under changes of the history probability distribution. With dA(p,q)=[∑h(ph−qh)2]1/2,d_{\mathrm A}(p,q) = \left[ \sum_h \left(\sqrt{p_h}-\sqrt{q_h}\right)^2 \right]^{1/2},dA​(p,q)=[∑h​(ph​​−qh​​)2]1/2, the minimax value satisfies ∣Λ(p)−Λ(q)∣≤2dA(p,q).\vert{}\Lambda(p)-\Lambda(q)\vert{} \le 2d_{\mathrm A}(p,q).∣Λ(p)−Λ(q)∣≤2dA​(p,q). Consequently, key quantitative implications of the balanced complete solution—including the minimax value, principal near-minimizer coordinate constraints, and a universal operational recovery lower bound—remain stable under nearby biased or correlated perturbations.
## Overview This paper studies when a reduction of a finite quantum error-history model preserves the local direction of quantum-code optimization, and when it reverses that direction. For finite coactive spectral branches, the paper first proves a second-order local truncation-safety certificate. A strict optimizer of a truncated model persists when the full-versus-truncated effective-jet discrepancy is smaller than the coercivity margin of the truncated objective. Opposite leading jets provide a certificate of local optimizer reversal. The paper then identifies an exact boundary between genuine temporal structure and removable path redundancy. For a finite projective unitary error group, any open-loop path operator has the form Wγ=eiθγUc(γ),W_\gamma=e^{i\theta_\gamma}U_{c(\gamma)},Wγ​=eiθγ​Uc(γ)​, where c(γ)c(\gamma)c(γ) is the final group product. Consequently, an arbitrary temporally correlated path law is isometrically equivalent, for complete complementary leakage and for every code, to the single-shot distribution of the final projective product: Lpath(V)=Lπ(V).\mathcal L_{\mathrm{path}}(V)=\mathcal L_{\pi}(V).Lpath​(V)=Lπ​(V). For Pauli trajectories, this remains true even when ordered errors do not commute, because different orderings can differ only by projective phases after contraction to the final channel. Thus open-loop final-channel leakage cannot identify temporal organization beyond the final projective Pauli-product distribution. Intermediate measurements, controls, syndrome extraction, or adaptive recovery are required for genuinely intervention-resolved multi-time code geometry. ## Distribution-robust optimizer reversal For an arbitrary three-location bit-flip trajectory law, let π\piπ denote the induced final-parity distribution on F23\mathbb F_2^3F23​, let c=(1,1,1)c=(1,1,1)c=(1,1,1), and define Λπ:=∑t∈F23πtπt⊕c.\Lambda_\pi:=\sum_{t\in\mathbb F_2^3}\sqrt{\pi_t\pi_{t\oplus c}}.Λπ​:=∑t∈F23​​πt​πt⊕c​​. For the three-qubit repetition code E=span⁡{∣000⟩,∣111⟩},E=\operatorname{span}\{|000\rangle,|111\rangle\},E=span{∣000⟩,∣111⟩}, the complete complementary leakage is exactly Lπ(E)=Λπ.\mathcal L_\pi(E)=\Lambda_\pi.Lπ​(E)=Λπ​. For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate www satisfies Lπ(V(w))=Λπ−2Λπ∣w∣2+o(∣w∣2).\mathcal L_\pi(V(w))=\Lambda_\pi-2\Lambda_\pi|w|^2+o(|w|^2).Lπ​(V(w))=Λπ​−2Λπ​∣w∣2+o(∣w∣2). Hence the repetition code is a strict local maximum throughout this two-real-dimensional graph chart for every full-support final-parity distribution, independently of temporal correlations, stationarity, roundwise independence, identical error rates, or cyclic spatial symmetry. By contrast, a conditioned zero-and-one-fault truncation with positive coarse single-location weights has L≤1(E)=0\mathcal L_{\leq 1}(E)=0L≤1​(E)=0 and strictly positive leakage at every nonzero sufficiently nearby graph code. The same reference code is therefore a strict local minimum for the truncated model and a strict local maximum for the complete process. The central conclusion is: > Exact aggregation of paths that implement the same projective physical operator preserves the entire code-optimization landscape, whereas deletion of higher-fault history fibers can reverse the local optimizer geometry. ## Scope The results concern random-unitary, open-loop finite-history models and complete complementary-channel leakage. The optimizer reversal is proved in a specified Fourier-syndrome graph chart around the three-qubit repetition code. The paper does not claim: - a full-Grassmannian classification for the three-qubit model;- a decoder or recovery-circuit construction;- a logical-error threshold;- a circuit-level hardware noise model;- collapse of an intervention-resolved process tensor;- equivalence in the presence of intermediate syndrome measurements, controls, or adaptive recovery. These intervention-resolved settings form the natural next boundary of the theory. ## Relation to the preceding research program This paper is a concluding bridge theorem paper connected to a six-paper exact finite-history QEC and spectral-minimax research line. ### 1. Compression-geometric foundation **Sharp Complementarity, Complete Attainable Region, and Metric Rigidity of Rank-Two Quantum Codes under Collective and Differential Dephasing** Version v0.5r2b DOI: 10.5281/zenodo.21523950 This paper determined the complete attainable compression-defect region for rank-two codes under collective and differential two-qubit dephasing, classified the equality geometry, and established sharp metric rigidity. ### 2. Balanced finite-history minimax **Environment-Record Leakage for Finite Quantum Error Histories: Balanced Global Minimax, Complete Minimizer Geometry, and Distributional Stability** Version v0.5r2 DOI: 10.5281/zenodo.21560397 This paper introduced the complete environment-record leakage objective for finite error histories, solved the balanced two-step minimax problem globally over Gr⁡(2,4)\operatorname{Gr}(2,4)Gr(2,4), classified all balanced minimizers, and established distributional stability. ### 3. Bias-induced geometric selection **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 This paper proved exact non-isoclinic separation, an open non-isoclinic parameter region, thresholdless loss of isoclinic optimality under arbitrarily small bias, and the leading full-Grassmannian near-balanced minimax selector. ### 4. Fourth-order overlap selection **Flat-Interior Persistence and Singular Overlap Selection in Near-Balanced Quantum Error Histories: Fourth-Order Transition Curves and Full-Grassmannian Minimax Closure** Version v0.1r3 DOI: 10.5281/zenodo.21622029 This sequel resolved the higher-order overlap degeneracy, established flat-interior persistence, derived singular boundary-layer transition curves, and closed the near-balanced full-Grassmannian minimax expansion through fourth order. ### 5. Abstract reflected spectral-jet framework **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 This paper extracted a reusable abstract framework for hard-normal isolation, soft tangential persistence, reflected active branches, and coactive boundary transitions in degenerate spectral minimax problems. ### 6. Finite-orbit generalization and first QEC truncation reversal **Beyond Reflected Pairs: Finite-Orbit Spectral Jets, Quadratic Isolation, and Multi-Branch Coactivity in Degenerate Minimax Problems** Version v0.4r1 DOI: 10.5281/zenodo.21644364 This paper generalized the reflected-pair theory to finite transitive orbits, developed multi-branch coactivity and representation-theoretic jet obstructions, and exhibited an exact three-location QEC realization in which restoring omitted multi-fault histories reverses the local optimizer geometry. ### 7. Present bridge theorem The present paper determines which open-loop temporal path information is exactly removable and which history reductions remain unsafe. It extends the earlier single-process reversal from independent or cyclic models to arbitrary correlated projective-unitary paths and arbitrary full-support final-parity laws. The resulting research progression is compression geometry⟶balanced global minimax⟶bias-induced selection⟶fourth-order closure⟶reflected spectral jets⟶finite-orbit coactivity⟶projective path collapse and distribution-robust reversal.\text{compression geometry}\longrightarrow\text{balanced global minimax}\longrightarrow\text{bias-induced selection}\longrightarrow\text{fourth-order closure}\longrightarrow\text{reflected spectral jets}\longrightarrow\text{finite-orbit coactivity}\longrightarrow\text{projective path collapse and distribution-robust reversal}.compression geometry⟶balanced global minimax⟶bias-induced selection⟶fourth-order closure⟶reflected spectral jets⟶finite-orbit coactivity⟶projective path collapse and distribution-robust reversal. ## Reproducibility package The accompanying bundle contains: - the manuscript PDF;- the LaTeX source;- an exact symbolic verification script;- a machine-readable verification report;- a README describing the proof-artifact contract;- a SHA-256 manifest;- version and package-audit records. The verifier performs 72 exact symbolic checks, including projective Pauli phase identities, phase-covariant path compression, arbitrary-parity polar factors, translation orthogonality, Fourier-syndrome graph compressions, and coefficient bookkeeping for the distribution-robust local expansion. The verifier deliberately rejects optimized Python mode. PASS decisions do not rely on floating-point tolerances. Compactness, maximizing-direction localization, trace-norm differentiability, and uniform asymptotic assembly remain analytic parts of the manuscript proof rather than delegated computational claims.
We solve cyclic finite-dimensional quantum histories for arbitrary time-dependent unitary steps, without assuming that one step has finite order. The propagation Hamiltonian is a unitary connection Laplacian on a cycle; its complete gauge invariant is the monodromy M=UL−1⋯U0M=U_{L-1}\cdots U_0M=UL−1​⋯U0​. Its spectrum is λa,k=1−cos⁡((2πk−θa)/L)λ_{a,k}=1-\cos((2πk-θ_a)/L)λa,k​=1−cos((2πk−θa​)/L), where eiθa∈spec(M)e^{iθ_a}\in\mathrm{spec}(M)eiθa​∈spec(M). Thus the exact history sector is isomorphic to Fix(M)\mathrm{Fix}(M)Fix(M), while frustration, the gap above a nonempty zero-energy sector, the determinant, and the finite-temperature trace are obtained in closed form. Ordinary spectral data recover the multiset of monodromy phase cosines but not phase orientation; low energy certifies proximity to an exact relational history. We then define the predictive quotient of a sharp finite clock relative to an accessible operator system as the unique coarsest event alphabet preserving all conditional statistics on a history sector. A finite-error theorem shows that threshold clustering recovers this quotient when the minimum diamond separation of inequivalent event channels exceeds four times the estimation error, and proves an optimal record-count bound. With full matrix access and homogeneous step UUU, the minimal number of clock events is the projective order of UUU. We distinguish the normalizer of the clock algebra from transformations preserving the coherent history code and classify oriented exact sharp clock changes by U(r)×ZLU(r)\times\mathbb{Z}_LU(r)×ZL​ on a rank-rrr history sector; without orientation the cyclic factor becomes dihedral. Reversible changes of full-information clock fibers are necessarily unitary, so irreversible coarse-graining is not exact clock covariance. Minimal realizations of a complete history Gram kernel are uniquely unitarily equivalent, with a finite-data Procrustes bound. Independent finite-matrix code verifies the main results.
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 C3C_3C3​- and D3D_3D3​-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 GGG be a finite group acting on a normal space NNN and on a finite transitive branch orbit Ω\OmegaΩ. For active branches ℓω\ell_\omegaℓω​, define the centered branch deviation relative to the group-averaged jet JG\mathcal J_GJG​: dω=ℓω−JG.d_\omega = \ell_\omega - \mathcal J_G.dω​=ℓω​−JG​. If the orbit average has uniform quadratic normal concavity, Dn2JG⪯−2μz2I,D_n^2\mathcal J_G \preceq -2\mu z^2 I,Dn2​JG​⪯−2μz2I, while every centered branch deviation vanishes through second order at the group-fixed manifold and has uniformly controlled cubic remainder, then max⁡ω∈Ωℓω(z,p,s,n)≤JG(z,p,s,0)−μ2z2∣n∣2\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{}^2maxω∈Ω​ℓω​(z,p,s,n)≤JG​(z,p,s,0)−2μ​z2∣n∣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: HomG(VΩ0,Symk(N))={0},\mathrm{Hom}_G(V_\Omega^0, \mathrm{Sym}^k(N)) = \{0\},HomG​(VΩ0​,Symk(N))={0}, where VΩ0V_\Omega^0VΩ0​ is the zero-sum branch-label representation. Multi-branch coactivity For analytic optimized branches W1,…,WmW_1,\ldots,W_mW1​,…,Wm​, normalized branch-difference maps are used to define continued coactivity sets through a degenerate perturbation point. If k+1k+1k+1 branches are active and the normalized difference map has rank kkk, then the corresponding regular coactivity set is locally a codimension-kkk 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}.\max\{0,y_1,\ldots,y_k\}.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/D3C_3/D_3C3​/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 4×44\times44×4 Hermitian scenario and its C3C_3C3​ unitary orbit. Compression to a moving rank-two graph subspace produces the exact jet JG(z,ν)=1−23z2∣ν∣2+23z3∣ν∣2Re⁡(ω−jν)+2627z4∣ν∣4+O(z5),\mathcal J_G(z,\nu) = 1 - \frac23 z^2\vert{}\nu\vert{}^2 + \frac23 z^3\vert{}\nu\vert{}^2 \operatorname{Re}(\omega^{-j}\nu) + \frac{26}{27}z^4\vert{}\nu\vert{}^4 + O(z^5),JG​(z,ν)=1−32​z2∣ν∣2+32​z3∣ν∣2Re(ω−jν)+2726​z4∣ν∣4+O(z5), together with an exact quadratic-isolation inequality. Three-location finite-history QEC realization The single-fault model uses the four history labels {I,X0,X1,X2}\{I, X_0, X_1, X_2\}{I,X0​,X1​,X2​} on three physical qubits. For the repetition code C=span⁡{∣000⟩,∣111⟩},\mathcal C = \operatorname{span}\{\vert{}000\rangle, \vert{}111\rangle\},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)=11+q[2z3Re⁡(ω−jν)σz+q3σx],q=z2∣ν∣2.P_j(z,q) = \frac{1}{1+q} \left[ \frac{2z}{\sqrt3} \operatorname{Re}(\omega^{-j}\nu)\sigma_z + \frac q3\sigma_x \right], \qquad q=z^2\vert{}\nu\vert{}^2.Pj​(z,q)=1+q1​[3​2z​Re(ω−jν)σz​+3q​σx​],q=z2∣ν∣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 C3C_3C3​ 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.h \in \{0,1\}^3 \quad \implies \quad W_h = X_0^{h_0}X_1^{h_1}X_2^{h_2}.h∈{0,1}3⟹Wh​=X0h0​​X1h1​​X2h2​​. 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)DpPsDp∣1,\mathcal L_8(V) = \max_{\vert{}u\vert{}=1} \left\vert{} \sum_{s\ne0} (u\cdot\mathbf k_s) D_p P_s D_p \right\vert{}_1,L8​(V)=max∣u∣=1​​∑s=0​(u⋅ks​)Dp​Ps​Dp​​1​, where ks\mathbf k_sks​ is the Bloch vector of the traceless part of the compressed history product V†WsVV^\dagger W_s VV†Ws​V. At the repetition code, the full process has an exact logical-complement obstruction: L8(C)=8∏j=02εj(1−εj).\mathcal L_8(\mathcal C) = 8 \prod_{j=0}^{2} \sqrt{\varepsilon_j(1-\varepsilon_j)}.L8​(C)=8∏j=02​εj​(1−εj​)​. For equal independent bit-flip probabilities (εj=ε\varepsilon_j=\varepsilonεj​=ε), let τ=[ε(1−ε)]3/2.\tau = [\varepsilon(1-\varepsilon)]^{3/2}.τ=[ε(1−ε)]3/2. Then L8(C)=8τ.\mathcal L_8(\mathcal C) = 8\tau.L8​(C)=8τ. For the Fourier-syndrome graph family, the nontrivial history products split into: a weight-one C3C_3C3​ orbit; a weight-two C3C_3C3​ orbit; and the triple-event logical-complement singleton. In the intrinsic graph coordinate www, the complete leakage satisfies L8(w)=8τ−16τ∣w∣2+o(∣w∣2)(w→0).\mathcal L_8(w) = 8\tau - 16\tau\vert{}w\vert{}^2 + o(\vert{}w\vert{}^2) \qquad (w\to0).L8​(w)=8τ−16τ∣w∣2+o(∣w∣2)(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,\mathcal L_{\mathrm{single}}(\mathcal C) = 0,Lsingle​(C)=0, and every nontrivial sufficiently small graph deformation produces positive leakage. In the full eight-label process, L8(C)=8τ,\mathcal L_8(\mathcal C) = 8\tau,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\boxed{ \text{local protection in the truncated process} \quad\longleftrightarrow\quad \text{local leakage maximality in the full process} }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⟩+c∣n∣2,β≥0.\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.Qeff​(α,β,n)=H(a0​,t)+β+21​⟨Qα,α⟩+c∣n∣2,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.
We determine the exact worst-direction Fisher-information cost of using a parameter-independent readout to estimate an unknown qubit direction at known Bloch-vector length ηηη. Every fixed-local architecture, including recorded classical randomization, heterogeneous single-copy measurements, and arbitrary outcome spaces, reduces exactly to a zero-barycenter probability measure on the Bloch ball. For every full-rank qubit and every trace-balanced spectral Fisher loss, the resulting minimax problem is rigid: the unique optimal aggregate design is the spin-coherent Haar positive-operator-valued measure (POVM). For NNN copies, inverse-Fisher AAA loss has the exact value 2/[Nf(η)]2/[Nf(η)]2/[Nf(η)], where $ f(η)= \frac{2η-(1-η^2)\log[(1+η)/(1-η)]}{4η}$. This uniqueness has an immediate finite-readout consequence. No finite-support measurement attains the unrestricted mixed-state optimum, while at the smallest globally regular support the tetrahedral symmetric informationally complete (SIC) measurement is uniquely AAA- and DDD-minimax, with exact worst-direction values. Relaxing the fixed-readout constraint separates the asymptotic resources: one-way local operations and classical communication (LOCC), unrestricted LOCC, and separable measurements have AAA-loss coefficient 4/η24/η^24/η2, whereas collective measurements attain 2(1+η)/η22(1+η)/η^22(1+η)/η2. We further classify the rigidity conditions for unequal contrasts and show that Haar uniqueness survives at the nonregular pure-state endpoint.