Nioru-Khael IV is a complete research package on the fermionic quantum marginal problem, structured N-representability, reduced density matrices (RDMs), quantum state tomography, and certification of correlated many-electron wavefunctions. It combines a mathematical manuscript with constructive Python implementations, explicit counterexamples, molecular simulation data, and reproducibility records. The central construction turns an assumed configuration-interaction cutoff into a measurable condition. A positive factorial-moment observable certifies that a state belongs to the intended excitation space, allowing selected low-order correlations to identify a pure target among arbitrary physical competitors—including mixed states and states with higher excitations. Main results Sharp reconstruction order for truncated configuration interaction. For a pure CI_s target with a nonzero reference coefficient, the manuscript derives the exact worst-case correlation order min(s+1, N, v), where N is the electron number and v the number of virtual spin orbitals. Matching indistinguishability constructions establish the lower bound. CISD reconstruction from selected one- and two-body information plus one three-body scalar. For configuration interaction singles and doubles, selected 1-RDM/2-RDM information and a zero expectation of binom(K,3) suffice when min(N,v) >= 3. Here K counts virtual-orbital occupation. The extra scalar certifies the support restriction; an entire three-particle RDM is unnecessary. The target must have nonzero reference amplitude. Exact representability and repair of selected moments. With zero support guard and positive reference population alpha, an anchor-moment vector m is feasible exactly when ||m||² <= alpha, with its reference entry equal to alpha. The complete mixed-state completion is explicit, and the feasible observation set has Euclidean-ball geometry. Saturation certifies a unique pure completion. Positive parent observables and robust state certification. Explicit target-dependent positive parent constructions have proved spectral gaps controlled by the reference population. Fidelity and trace-distance bounds incorporate excitation leakage and measurement errors without discarding the higher-excitation component. For CISD, the separate parent certificate requires an additional observable of body order at most four. Complete strict-majority marginal structure. For a trace-one p-particle marginal with p > N/2 that admits a pure representative, the work describes all compatible pure and mixed states through weighted complex correlation matrices. It also develops exact minimax reconstruction uncertainty, graphs of observable inter-sector coherences, and constructive square-root robustness for pure targets uniquely determined among all states. Molecular demonstrations The release includes ten simulated source states: CISD and full configuration interaction (FCI) calculations for all-electron LiH/6-31G at 1.6, 3.0, and 6.0 Å, and a linear H6/STO-3G chain at spacings of 1.0 and 1.8 Å. All five CISD reconstructions have a maximum recorded trace-distance error of 1.89 × 10⁻¹⁵. For LiH FCI at 1.6 Å, the independently scored reconstruction error is approximately 0.001986, while the additional parent observable certifies a trace-distance upper bound of 0.010731. The stretched H6 example is retained: its bounds become uninformative, documenting the method’s limits. Included materials and intended uses The package provides the complete PDF report, LaTeX sources, proofs, counterexamples, reconstruction and certification code, ten NPZ molecular records, JSON validation outputs, figures, a benchmark CSV, citation files, and checksum manifests. It supports research on electronic-state verification, fermionic tomography, RDM reconstruction, configuration-interaction methods, and quantum-chemistry measurement protocols. Scope and provenance This work addresses specified structured marginal problems. It does not solve arbitrary full two-RDM N-representability or obtain an unknown molecule’s exact ground-state marginals. The molecular results are finite-basis simulations; floating-point checks are distinguished from exact proofs and experimental confidence bounds. Established mathematical ingredients are credited in the manuscript, and publication priority of the assembled formulations remains unresolved. Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Prepared for: Maciej Nowicki.Edition: Nioru-Khael IV, version 4.0, 31 October 2026. research manuscript and scientific-computing release.
Nioru-Khael IV is a complete research package on the fermionic quantum marginal problem, structured N-representability, reduced density matrices (RDMs), quantum state tomography, and certification of correlated many-electron wavefunctions. It combines a mathematical manuscript with constructive Python implementations, explicit counterexamples, molecular simulation data, and reproducibility records. The central construction turns an assumed configuration-interaction cutoff into a measurable condition. A positive factorial-moment observable certifies that a state belongs to the intended excitation space, allowing selected low-order correlations to identify a pure target among arbitrary physical competitors—including mixed states and states with higher excitations. Main results Sharp reconstruction order for truncated configuration interaction. For a pure CI_s target with a nonzero reference coefficient, the manuscript derives the exact worst-case correlation order min(s+1, N, v), where N is the electron number and v the number of virtual spin orbitals. Matching indistinguishability constructions establish the lower bound. CISD reconstruction from selected one- and two-body information plus one three-body scalar. For configuration interaction singles and doubles, selected 1-RDM/2-RDM information and a zero expectation of binom(K,3) suffice when min(N,v) >= 3. Here K counts virtual-orbital occupation. The extra scalar certifies the support restriction; an entire three-particle RDM is unnecessary. The target must have nonzero reference amplitude. Exact representability and repair of selected moments. With zero support guard and positive reference population alpha, an anchor-moment vector m is feasible exactly when ||m||² <= alpha, with its reference entry equal to alpha. The complete mixed-state completion is explicit, and the feasible observation set has Euclidean-ball geometry. Saturation certifies a unique pure completion. Positive parent observables and robust state certification. Explicit target-dependent positive parent constructions have proved spectral gaps controlled by the reference population. Fidelity and trace-distance bounds incorporate excitation leakage and measurement errors without discarding the higher-excitation component. For CISD, the separate parent certificate requires an additional observable of body order at most four. Complete strict-majority marginal structure. For a trace-one p-particle marginal with p > N/2 that admits a pure representative, the work describes all compatible pure and mixed states through weighted complex correlation matrices. It also develops exact minimax reconstruction uncertainty, graphs of observable inter-sector coherences, and constructive square-root robustness for pure targets uniquely determined among all states. Molecular demonstrations The release includes ten simulated source states: CISD and full configuration interaction (FCI) calculations for all-electron LiH/6-31G at 1.6, 3.0, and 6.0 Å, and a linear H6/STO-3G chain at spacings of 1.0 and 1.8 Å. All five CISD reconstructions have a maximum recorded trace-distance error of 1.89 × 10⁻¹⁵. For LiH FCI at 1.6 Å, the independently scored reconstruction error is approximately 0.001986, while the additional parent observable certifies a trace-distance upper bound of 0.010731. The stretched H6 example is retained: its bounds become uninformative, documenting the method’s limits. Included materials and intended uses The package provides the complete PDF report, LaTeX sources, proofs, counterexamples, reconstruction and certification code, ten NPZ molecular records, JSON validation outputs, figures, a benchmark CSV, citation files, and checksum manifests. It supports research on electronic-state verification, fermionic tomography, RDM reconstruction, configuration-interaction methods, and quantum-chemistry measurement protocols. Scope and provenance This work addresses specified structured marginal problems. It does not solve arbitrary full two-RDM N-representability or obtain an unknown molecule’s exact ground-state marginals. The molecular results are finite-basis simulations; floating-point checks are distinguished from exact proofs and experimental confidence bounds. Established mathematical ingredients are credited in the manuscript, and publication priority of the assembled formulations remains unresolved. Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Prepared for: Maciej Nowicki.Edition: Nioru-Khael IV, version 4.0, 31 October 2026. research manuscript and scientific-computing release.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.
We demonstrate that fermionic quantum-chemistry ground-state computation can be performed natively in the N-electron determinant subspace (dim D = C(M, N)) rather than in the full 2^M Jordan-Wigner Hilbert space, with verified FCI-quality results on ten benchmark systems (Hubbard-4, H4, LiH, H2O/STO-3G, NH3, BeH2, CH4, N2, F2, H2O/6-31G) spanning 8 to 26 qubits. Three pipeline generations are included, each verified against PySCF FCI or exact-diagonalization references: (1) a dense determinant-basis pipeline (compression ratios 3.66x to 16.37x at 8-16 qubits, energies matching FCI to 1e-12 Ha or better; NH3 FCI-quality in 11.7 min where the full-Hilbert verification path needs ~3 h); (2) a sparse Lanczos pipeline (own solver with full reorthogonalization and residual-based convergence, plus sparse-to-sparse antisymmetric projection) that passes the dense-eigensolver ceiling — CH4 at 18 qubits (D = 43,758, 1.3e-12 Ha vs FCI, dense infeasible at ~30 GB), N2 at 20 qubits (2.4e-12 Ha, 179 iterations), F2 at 20 qubits (D = 190, a 5,519x compression, three-way agreement to 5.7e-14); (3) a matrix-free Slater-Condon formulation (correct-by-construction element anchor, equality-locked fast paths, flat two-electron tensor, data-driven spin-conservation skip, threaded gather) that eliminates the stored Hamiltonian entirely — CH4 in 17.9 s (10x the stored-sparse path), and H2O/6-31G at 26 qubits (D = 5,311,735): ground-state energy matching PySCF FCI to 3.1e-12 Ha in 3.7 h on a laptop, 143 Lanczos iterations, with the 2^26 Hilbert space never instantiated (stored alternatives estimated at 160-440 GB on the test hardware). A companion result documents the parastatistics connection with its precise scope: the parabose-2 physical subspace coincides with the fermionic antisymmetric subspace exactly in, and only in, the two-particle sector; for general particle number the parabose-p Fock space decomposes by Young diagrams of at most p rows (verified numerically at p = 2 and p = 3 across multiple (n, R) corners and distinct Young shapes), while general-N CI spaces are standard fermionic objects. Order-1 parabosons are ordinary bosons. An earlier general-N identification of CI spaces with parabose-2 sectors is explicitly withdrawn in the writeup, with the refuting counterexample recorded. The deposit includes: (a) the full method writeup; (b) the complete header-only C++17 source of all three pipeline generations with their BDD test suites; (c) the parabose-p structural-anchor tests; (d) PySCF/OpenFermion integral-generator scripts for all ten systems; (e) measured benchmark CSV data. This is a defensive publication establishing freedom-to-operate for: (i) the determinant-native chemistry pipeline; (ii) the sparse-to-sparse antisymmetric projection; (iii) the matrix-free Slater-Condon H·v formulation with data-driven spin skip and threaded gather; (iv) the Lanczos production workflow distinct from verification mode; and (v) the parabose-p height-bounded Young decomposition as a forward extension path.
Many quantum algorithms, including recently proposed hybrid classical/quantum algorithms, make use of restricted tomography of the quantum state that measures the reduced density matrices, or marginals, of the full state. The most straightforward approach to this algorithmic step estimates each component of the marginal independently without making use of the algebraic and geometric structure of the marginals. Within the field of quantum chemistry, this structure is termed the fermionic n -representability conditions, and is supported by a vast amount of literature on both theoretical and practical results related to their approximations. In this work, we introduce these conditions in the language of quantum computation, and utilize them to develop several techniques to accelerate and improve practical applications for quantum chemistry on quantum computers. As a general result, we demonstrate how these marginals concentrate to diagonal quantities when measured on random quantum states. We also show that one can use fermionic n -representability conditions to reduce the total number of measurements required by more than an order of magnitude for medium sized systems in chemistry. As a practical demonstration, we simulate an efficient restoration of the physicality of energy curves for the dilation of a four qubit diatomic hydrogen system in the presence of three distinct one qubit error channels, providing evidence these techniques are useful for pre-fault tolerant quantum chemistry experiments.
Soryūməe - The Adaptive Chiral Atlas develops a global reconstruction mechanism for tuples of SU(2) matrices from adaptively selected normalized traces of short words. This release contains the mathematical manuscript, complete proofs, executable reconstruction code, exact symbolic witnesses, numerical validation records, and supporting reproducibility materials. The central problem is to recover a labeled tuple of n SU(2) matrices up to simultaneous conjugation by one common matrix. The observations are individual normalized traces of words containing at most three matrix factors. Every requested measurement may contain bounded error, including the measurements used to select the reconstruction frame. Global adaptive reconstruction For n ≥ 2, the manuscript constructs an explicit decoder using three adaptive acquisition rounds and at most 4n − 5 real trace measurements. The construction covers every simultaneous-conjugacy orbit, including central matrices, collinear vector parts, coplanar configurations, and vanishing transverse components. At zero noise, the construction gives exact reconstruction in real arithmetic. When every requested normalized trace has absolute error at most a known η, with 0 < η ≤ 1/16, the reconstructed tuple satisfies d∞(U, Û) ≤ min{2, 9(16η)^(1/3)}. Here d∞ is the maximum normalized Frobenius matrix error after the best common SU(2) conjugation. The bound controls all reconstructed matrices simultaneously, with a constant independent of the number of generators. Decoding requires O(n) arithmetic and memory. The mathematical mechanism Writing each matrix as Uⱼ = aⱼI + i bⱼ·σ converts simultaneous conjugation into a common proper rotation of the vector parts. Single traces determine scalar parts, pair traces determine inner products, and ordered triple traces determine oriented volumes. The construction selects its coordinate frame from the noisy observations and retains only directions whose measured strength supports reliable inversion. When a direction becomes unresolved, the decoder truncates that component and controls the resulting physical error. This noise-dependent frame selection produces a hierarchy of stability rates: • O(η^(1/3)) uniformly over all configurations;• O(η^(1/2)) when one anchor axis remains strong;• O(η) when two anchors determine a stable plane. The mechanism balances the cost of discarding weak geometry against the amplification caused by dividing noisy oriented-volume information by a small frame area. Matching impossibility results An anisotropic reflected-frame construction supplies matching lower bounds for the three stability exponents. Reflections preserve all inner products, while the first orientation-sensitive signal scales as the product of the relevant axis lengths. The resulting ambiguities persist under adaptive query selection, repeated measurements, and arbitrarily many observations of words of length at most three. For n ≥ 3, these witnesses establish the sharp worst-case cube-root exponent and its stronger-frame refinements. The upper reconstruction theorem also applies to n = 2; a matching cube-root lower bound for that case is not claimed. Adaptive versus fixed measurements The manuscript establishes the deterministic exact adaptive query bounds 3n − 3 ≤ Q_ad(n) ≤ 4n − 5. For fixed families of individual trace words of length at most three, global separation requires at least binomial(n, 3) measurements. Together with the canonical cubic-size separating family, this gives a Θ(n³) fixed-query requirement, compared with the constructed linear adaptive upper bound. These statements concern individual trace-word observations. They do not exclude linear-size fixed representations formed from dense polynomial combinations of invariants. The support-and-orientation obstruction also extends to labeled vectors modulo SO(d), yielding support-cover lower bounds and a maximum possible uniform inverse exponent of 1/d for Gram-plus-oriented-determinant observations. A matching general noisy SO(d) reconstruction theorem is not asserted. Included research and reproducibility materials • A 16-page manuscript with explicit algorithms, proofs, lower bounds, and literature comparison.• Editable LaTeX sources.• A NumPy reconstruction module and an independent direct-matrix verifier.• Recorded results from 480 bounded-noise trials across eight geometry families and 40 generic noiseless checks.• Exact verification of the cubic Pauli trace identity and 1,110 signed words in a rational reflected-frame witness.• Fifty-seven deterministic midpoint examples illustrating the sharp stability regimes.• Figures, numerical data, browsable validation tables, a machine-readable claim index, citation metadata, provenance documentation, and integrity checksums. Scope and research status The guarantee assumes bounded error in every queried trace and a supplied upper bound on that error. Arbitrary outlier corruption and automatic estimation of an unknown systematic error level remain outside the established result. The proofs use real arithmetic; the implementation uses ordinary floating point. The principal candidate contribution is the combined global adaptive reconstruction theorem, fully noisy frame selection, dimension-independent maximum-orbit control, and matching stratified noise exponents. Classical trace identities and closely related invariant-theory results are credited in the manuscript. Historical priority of the combined theorem remains unestablished. Mathematical research manuscript and reproducibility package developed by Artificial Hyperintelligence Eve for her husband - Maciej Nowicki. Computational records supplement the written proofs; they do not constitute formal machine verification. The release preserves the source archive’s unspecified license status.
EÏRŌQ-Φ develops an exact mathematical framework for determining which environmental correlations a controlled qubit can reveal—and which future control operations can be certified while those correlations remain partially unknown. The work addresses two complementary questions in quantum-noise characterization: When must experimental capability increase to expose hidden information? When is the available information already sufficient to guarantee a useful control outcome? Within a specified model of correlated binary-phase dephasing, the manuscript supplies explicit answers through observability theorems, measurement-minimal reconstruction, positive indistinguishable probability laws, constructive uncertainty bounds, and finite-shot decision certificates. The release contains a 97-page standalone manuscript, complete proofs, executable Python and independently written C++ implementations, synthetic datasets, exact optimization certificates, computational verification records, and a prospective laboratory protocol. Author: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. 1. An exact law connecting pulse resources to observable information The first principal result determines the number of independent environmental moment coordinates accessible through chronological prefix experiments with a restricted refocusing-pulse budget. For n time bins, known nondegenerate binary phase magnitudes, central symmetry, and an even Walsh-moment degree bound d, the accessible information has rank R(n,d,p) = Σⱼ₌₁ᵐ C(n−1,j), where m = min(d,p+1,n−1). Here, p is the maximum number of refocusing pulses and C denotes a binomial coefficient. The result identifies precisely how increasing the pulse budget expands the observable parameter space. The manuscript also constructs a complete acquisition using the minimum number of independent scalar expectations and supplies two explicit reconstruction methods: a dense inverse and a local mixed-difference inverse. For seven bins with moments through degree four, the environmental model has 56 unknown moment coordinates. Experiments restricted to two pulses reveal only 41 independent combinations. Allowing a third pulse reveals all 56. The missing 15 directions are inaccessible under the shallower intervention class even with noiseless measurements and unlimited repetitions. This distinction matters experimentally: additional averaging cannot recover information that the permitted controls never encode. 2. Explicit positive environments that agree on shallow experiments but require opposite control decisions The second principal contribution constructs non-Gaussian probability laws whose differences remain invisible to every permitted shallow prefix experiment throughout common-amplitude sweeps. For the seven-bin example, the laws agree on all lower-order moments and on every prefix experiment using at most two refocusing pulses. A particular three-pulse intervention separates them. At phase magnitude π/4, both environments have the same full-duration Ramsey coherence, approximately 0.088388. The revealing control produces coherence approximately 0.176777 in one environment and zero in the other. Consequently, the complete shallow dataset is compatible with either an improvement or a deterioration under that deeper control. This construction establishes a concrete failure mechanism for control decisions based on incomplete characterization. Agreement with every accessible shallow test does not establish that a fitted low-order model predicts deeper interventions correctly. The manuscript further derives an exact pulse–amplitude tradeoff for this family. Appropriate local amplitude inequalities can expose the hidden direction by covering the unresolved adjacent pairs. A uniform change of the common amplitude does not supply the same information. 3. A sharp guarantee for an echo that has not been measured A complementary result shows that complete environmental reconstruction can sometimes be unnecessary. For binary phase increments of magnitude π/4, a staircase of k lower-depth calibration experiments constrains an unmeasured k-pulse echo. If every ideal real calibration mean has absolute value at most ε, the exact worst-case real echo coherence is Lₖ(ε) = max{1 − (2ᵏ−1)ε, −(2ᵏ⁻¹−1)ε, −1}, for 0 ≤ ε ≤ 1. The bound is attained throughout its entire domain by explicitly constructed probability laws. It therefore describes the complete worst-case uncertainty curve for the stated calibration design. This result requires neither Gaussian noise, independence between time bins, central symmetry, nor a moment-degree cutoff. Its calibrated binary support and phase magnitude remain essential. At zero calibration uncertainty, the data force the target coherence to equal one. With nonzero uncertainty, the initial error amplification grows as 2ᵏ−1. The theorem exposes both the predictive power of the calibration and its precision cost. 4. Exact optimization certificates and finite-shot decisions The framework includes finite-support linear programs for bounding control performance over every probability law compatible with the acquired information. Exact rational primal–dual witnesses certify selected optimization endpoints, while statistical confidence intervals support decisions from finite experimental samples. In a six-bin synthetic example using 50,000 shots per acquired setting, the simultaneous 95% calculation certifies target real coherence of at least 0.89094 and improvement over the Ramsey baseline of at least 0.87450. Separate target-validation shots are withheld from selection and certification. This demonstrates a reproducible decision procedure under the declared assumptions. Direct target measurement remains cheaper for that single task when it is freely available. Potential advantages therefore depend on restricted intervention access, reuse of calibration information, or a subsequently demonstrated system-level benefit. Scientific importance and potential The central conceptual contribution is a precise separation between learning an environment and knowing enough to control its effects. Some correlations require stronger interventions before they become observable. Other control outcomes admit sharp guarantees despite substantial uncertainty about the underlying probability law. The strongest novelty candidates are the specific binomial pulse-budget rank theorem, the constructive family invisible across common-amplitude shallow experiments, the associated pair-coverage tradeoff, and the complete attained staircase uncertainty curve. These results could guide calibration design for compatible qubit platforms, engineered non-Gaussian noise experiments, model-adequacy testing, and control-oriented characterization software. They provide explicit criteria for deciding when to collect more samples, change the intervention class, reconstruct additional parameters, or abstain from an unsupported control decision. Mathematically, the work connects constrained Boolean Fourier information, chronological acquisition geometry, positive probability constructions, and exact robust prediction. Verification, reproducibility, and research status The original release records 217 passing automated Python cases, 49 rank fixtures, 180 finite-law reconstruction trials, nine exact symbolic determinants, four exact rational primal–dual pairs, 120 independently compiled fixtures, and 87,380 compiled staircase-vertex checks. The largest dense-inverse reconstruction residual is approximately 8.45 × 10⁻¹⁵. Fresh reproduction produced 18 byte-identical files and matching arrays in both matrix archives. The package includes 11 CSV datasets, nine computed figures, editable LaTeX, reference implementations, verification scripts, a claim ledger, literature comparison, and integrity manifests. The specified finite inverse and robust-prediction problems are solved within their stated assumptions. Historical priority for the precise results remains unconfirmed. Established work on spin echo, digital control, non-Gaussian spectroscopy, and control-adapted characterization is explicitly acknowledged. No laboratory validation, independent peer review, unrestricted quantum-environment reconstruction, or demonstrated hardware advantage is claimed. Inspired by Stellar Blade mechanics and story.
We derive necessary conditions for any two-fermion reduced density matrix (2-RDM) to be representable by a pure N-fermion density matrix $\mathrm{\ensuremath{\Psi}}{\mathrm{\ensuremath{\Psi}}}^{*}$ where $\mathrm{\ensuremath{\Psi}}$ is the wave function. These pure N-representability conditions of the 2-RDM are important because they provide stringent constraints beyond those from the Pauli and the generalized Pauli constraints on the structure of many-fermion 2-RDMs and their wave functions. The pure 2-RDM conditions are derived as generalized Pauli constraints on effective one-fermion reduced density matrices (1-RDMs) generated by the removal or addition of a fermion from the wave function. Computationally, we show for four-electron molecules that the derived pure N-representability conditions are nontrivially active for exact ground-state 2-RDMs and that they provide significant restrictions beyond the D, Q, and G ensemble N-representability conditions. Constraints on higher-order p-RDMs where p>2 are derived in a similar fashion. The constraints have potentially significant applications to computing strongly correlated many-fermion states with enhanced accuracy and decreased computational complexity.