Ŋyōsha ∿ Lhūmira VI: Exact Drift-Invariant Spectral Tomography through Known-Mean Restoration, Two-Spectrum Sparse Coupling Recovery, and Closed-Cycle Measurements

This work develops an exact acquisition framework for recovering coupling products and closed-cycle information from the spectra of finite-dimensional complex networks under calibrated diagonal controls and independently varying uniform complex drift. Its central insight is that removing an unknown common spectral shift need not discard the known mean of the applied control. Restoring that known component after spectral centering preserves drift cancellation while recovering information lost by balanced measurements. The resulting fourth-moment construction identifies round-trip coupling products, W_ij = Z_ij Z_ji, without the additive vertex ambiguity of the preceding balanced disjoint protocol. Opposite random sign controls then produce an exact quadratic observation from two complete spectra, with one mask-independent intercept estimated jointly with the products. The construction also supports sparse change detection around a dense baseline and an extension to unequal, fixed, calibrated actuator gains. Its algebra applies to arbitrary finite complex matrices without requiring Hermiticity, normality, diagonalizability, or an infinitesimal-control approximation. A self-contained small-ball argument establishes uniform, stable recovery for complex-valued sparse and approximately sparse product vectors. For k nonzero or changed products among E = N(N−1)/2 candidate pairs, the sufficient record count scales as O((k+1) log(E+1)), including the unknown intercept coordinate. The theorem provides conservative explicit constants and noise stability; these sufficient bounds establish the sampling order rather than predicting practical reconstruction thresholds. The same restoration calculus extends to closed-cycle invariants through finite-amplitude parity contrasts of higher spectral moments. A triangle construction uses eight complete spectra and moments through degree six to detect directed loops whose pair products all vanish. Pair and triangle observables also yield a discriminant for unequal opposing loop products. For complex-symmetric matrices with a nonzero anchor star, the manuscript gives a constructive reconstruction up to vertex signs and an unknown global scalar. Remaining ambiguities from diagonal similarity, global transposition, and invisible acyclic directed couplings are stated explicitly. The numerical release includes 23 successful verification groups, 212 exact rational polynomial assertions, 260 sparse-record linear-programming solves, and 100 fitted-eigenvalue error-propagation cases. In the main synthetic study, a 48-site system has 1,128 candidate products and four signed nonzero changes. Without receiving the true sparsity, the solver achieved noiseless exact recovery in 40/40 trials at each of 48, 64, and 96 records. At 48 records, corresponding to 96 complete-spectrum equivalents per snapshot, independent bounded record noise in [−0.02, 0.02] yielded correct support in 20/20 trials and a median relative product error of 1.852%. These are finite-trial observations, not a universal success probability. The separate eigenvalue-error study supplies the true support and therefore tests error propagation rather than support discovery. The deposit contains the complete manuscript and proofs in PDF, editable Word, Markdown, and LaTeX formats; implementation and reproduction scripts; trial-level datasets and summary metrics; exact certificates; six original figures; an illustrative full reconstruction dataset; an experimental calibration and falsification protocol; references; provenance; and integrity records. The results concern the stated finite-matrix model and synthetic computations. Physical application requires a justified effective network model, complete complex-spectrum or equivalent moment information, calibrated diagonal actuation, and a fixed non-scalar background and control amplitude during each two-spectrum snapshot. Unknown nonuniform drift and control-dependent coupling changes are outside the cancellation guarantee. No detector-level experiment, fabricated device, or measured photon-budget or acquisition-time advantage is reported. The proposed contribution is the exact information-preserving acquisition construction and its explicit recovery guarantees. Spectral Hamiltonian tomography, quadratic sensing, small-ball methods, and cycle invariants are established foundations cited in the manuscript. The framework offers a mathematically specified route toward controlled network characterization and sparse change detection in coupled-resonator, nanophotonic, and other linear-network models. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki