This preprint develops a projected Fock-space channel framework for finite-rank tensor correctors. The selected two-particle fluctuation channel is realized inside the condensate-orthogonal bosonic Fock space through a symmetric Gram metric with exchange terms. The manuscript proves a correspondence isometry, formulates the projected Bogoliubov comparison as a Duhamel leakage identity, and separates the projected-channel error into resolved leakage, three-body leakage, and finite-dimensional model discrepancy. The paper also provides sufficient spectral-tail conditions for resolved leakage, a regular smoothed Fourier benchmark on the three-dimensional torus, a bounded three-body leakage functional linked to microscopic obstruction diagnostics, and a metric-conditioning transfer estimate for the symmetric two-particle Gram matrix. The results are finite-dimensional and conditional: no full many-body derivation, trace-norm propagation of chaos, automatic N^{-1/2} rate, or cutoff-free Coulomb closure is claimed. This version v0.8 incorporates audit-correction and submission-readiness changes, including metric-consistent symmetric lifting, complex test-density convention, form-domain Duhamel clarification, explicit Fourier lattice-tail constants, and expanded conditioning estimates.
This preprint develops a projected Fock-space channel framework for finite-rank tensor correctors. The selected two-particle fluctuation channel is realized inside the condensate-orthogonal bosonic Fock space through a symmetric Gram metric with exchange terms. The manuscript proves a correspondence isometry, formulates the projected Bogoliubov comparison as a Duhamel leakage identity, and separates the projected-channel error into resolved leakage, three-body leakage, and finite-dimensional model discrepancy. The paper also provides sufficient spectral-tail conditions for resolved leakage, a regular smoothed Fourier benchmark on the three-dimensional torus, a bounded three-body leakage functional linked to microscopic obstruction diagnostics, and a metric-conditioning transfer estimate for the symmetric two-particle Gram matrix. The results are finite-dimensional and conditional: no full many-body derivation, trace-norm propagation of chaos, automatic N^{-1/2} rate, or cutoff-free Coulomb closure is claimed. This version v0.8 incorporates audit-correction and submission-readiness changes, including metric-consistent symmetric lifting, complex test-density convention, form-domain Duhamel clarification, explicit Fourier lattice-tail constants, and expanded conditioning estimates.
This preprint develops a certified rank-selective finite-rank tensor correction mechanism for a three-dimensional cutoff-regularized BBGKY benchmark with density-selected phase response. The construction uses an observable-dual topology matched to selected response/readout channels and a finite response menu with outcomes 3, 8, or fail. The main constructive branch is a rank-three dipole closure theorem under explicit dipole-admissibility and scale-safety gates; the rank-eight quadrupole branch is retained as a conditional diagnostic extension. The manuscript includes manufactured diagnostics and a reproducibility scaffold for the finite-dimensional response matrices and conservative exponent witness.
This preprint develops a microscopic pair-correlation framework for cutoff-regularized Coulomb-type BBGKY defect channels. A smooth radial cutoff separates the transition shell, an active mesoscopic annulus, and the macroscopic projected-response region. On the active annulus the paper uses an amplitude formulation F_epsilon = u_epsilon^2, where the radial amplitude belongs to a uniform shell-admissible class allowing correlation-hole behavior without imposing a cutoff-independent positive lower bound. The second-hierarchy defect current is decomposed, in a tested distributional sense, into macroscopic projected, shell-pair, pair-residual, three-body, and stress components. The scalar-pair-generated defect class is closed in the calibrated dual norm, while three-body residues outside this class are detected by annihilating test functionals. A model angular example illustrates how non-radial shell components can have positive distance from scalar radial pair corrections. The manuscript formulates basis-dependent leakage conditions preventing shell-scale residuals from contaminating the macroscopic Gram/Riesz certificate and provides sufficient conditions for convergence of the selected defect identity in a fixed tested distributional channel. No cutoff-free Coulomb closure, trace-norm propagation of chaos, full many-body derivation, or complete Bogoliubov fluctuation theorem is claimed.
Preprint. This manuscript has not undergone external peer review. We characterize exact projection channels on all tensor powers of a finite-dimensional system. Fix d ≥ 2. Suppose the channels are unital, completely positive, trace preserving and idempotent, covariant under collective SU(d) transformations and tensor permutations, consistent with identity embeddings, and completely depolarizing on one site. Then the entire family is either complete depolarization or collective Haar twirling. The proof combines standard conditional-expectation theory, the two-site Schur–Weyl commutant and an elementary consequence of infinite exchangeability. It requires no general classification of invariant subalgebras. For qubits, any nontrivial family necessarily preserves a logical qubit on three spins, excluding a nontrivial purely classical retained algebra under these assumptions. We illustrate the obstruction with a total-spin measurement channel and give sharp three-spin disturbance bounds. The entanglement-breaking bound is an application of the established trine state-reconstruction benchmark. An explicit example separates small normalized-Choi consistency defects from uniform operational closeness, clarifying a limitation of approximate extensions.
Abstract We obtain an operator-norm bound on the deviation of the band-projected density algebra of an isolated rank-one Chern band from the full projected-density algebra of the lowest Landau level, including the Landau-level form factor. The bound depends only on the band projector through two geometric control parameters, the nonuniformity of the Berry curvature and the deviation of the Fubini–Study metric from the isotropic Landau-level value, and is independent of the bandwidth; bandwidth and interband-mixing effects enter separately, at the level of an explicit Schrieffer–Wolff projection error controlled by W / Δ . Under a conditional gap hypothesis, quasi-adiabatic continuation extends the single-particle comparison to the many-body setting. We introduce a finite-volume algebraic proximity measure E Λ , L and evaluate it by exact diagonalization in the anisotropic Harper–Hofstadter and kagome lattice models. In both model families, E Λ , L is the sharpest single-particle indicator of the finite-volume many-body gap (Pearson | r | > 0.93 ); the projector-geometric combination δ F + δ g , motivated by the right-hand side of the main theorem, provides a cheaper estimator, and the curvature-only proxy 2 π σ B is weaker than either.
The reduced k-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of k-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of k-body operators for all k.
## 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), where 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). 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 denote the induced final-parity distribution on F23, let c=(1,1,1), and define Λπ:=∑t∈F23πtπt⊕c. For the three-qubit repetition code E=span{∣000⟩,∣111⟩}, the complete complementary leakage is exactly Lπ(E)=Λπ. For every fixed full-support final-parity law, the intrinsic Fourier-syndrome graph coordinate w satisfies 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 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), 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. ## 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.