Ŋeriū: Positive Gaussian Recovery and the Limits of Atomic Regularization - Covariance-Weight Classification, Balanced Gaussian-Mixture Counterexamples, and Exact Common-Covariance Recovery

This work establishes a classification of fixed covariance weights for positive Gaussian mixture recovery and develops a constructive method for showing when atomic regularization fails to recover a true mixing measure. It pairs a universal impossibility result over unrestricted Gaussian dictionaries with an exact recovery theorem for targets having an unknown common covariance. The final consolidated release includes the manuscript, detailed analytic proofs, reproducibility code, exact certificates, and supporting figure data. The model allows arbitrary real means and positive-definite covariance matrices. A nonnegative mixing measure represents a signal as an integral of Gaussian probability densities, with cost given by the integral of a fixed positive covariance weight W(C). The central theorem characterizes universal exact minimality: for any positive, finite-valued Borel covariance weight, every finite positive Gaussian target minimizes this cost among all positive exact representations if and only if W is constant. Constant weights measure physical mass, so distinct positive representations tie. Every nonconstant covariance weight has a strictly cheaper positive competitor for some target with at most three true components; a three-component probability target with equal physical weights can be chosen. No continuity or differentiability of the covariance-only weight is required. The constructive mechanism is positive Gaussian compensation. Symmetric mean shifts and a covariance perturbation produce a moment-matched residual of order h². A broad Gaussian mean law dominates this residual pointwise, allowing an additional true component to absorb it and yield a nonnegative exact alternative. The correction can be made cheaper than the leading covariance-weight saving. A related heat-residual argument extends the obstruction to smooth mean-dependent and covariance-dependent weights under the stated differentiability and polynomial-growth assumptions. An explicit balanced counterexample has means −1/16, 1/16, and 0; variances 1, 1, and 2; and physical mixture weights 1/3 each. It has a nonnegative exact alternative with the same density and probability mass, but lower normalized L² Gaussian atomic cost. The certified saving exceeds w(1)/8192, where w(C) is the L² norm of a Gaussian probability density. A support-transfer lemma shows that the same three-point support fails exact normalized positive-TV selection for every strictly positive amplitude vector. Related counterexamples fit arbitrarily narrow nontrivial covariance bands. Three components suffice; their minimality among all heterogeneous witnesses is not claimed. The exact recovery complement gives a necessary-and-sufficient weight criterion. Unique positive selection of every single Gaussian is equivalent to strict Loewner decrease: W(D)>W(C) whenever D≤C and D≠C. The same condition is equivalent to unique selection of every compact-mean positive target with an unknown common covariance, even when competing representations may use arbitrary covariances. No minimum mean separation is required, and the optimizer is not given the common covariance. The normalized Gaussian weight satisfies this criterion. For determinant-power coefficient penalties on unit-normalized Gaussian features, the analysis identifies the sharp exponent threshold γ=1/4 between common-covariance selection, physical-mass ties, and singleton failure. For the normalized unit-feature positive-TV and BLASSO formulations, the release proves global optimizer existence and, under the stated vanishing-noise and regularization conditions, qualitative consistency on the common-covariance success class. At the fixed balanced heterogeneous counterexample, however, fitted signals can converge to the true full L² signal while the estimated parameter measures remain inconsistent. Finite positive approximants preserve the cheaper competing cost with arbitrarily small signal residual. This separates accurate density fitting from recovery of the generating mixture. Further results address continuous-target instability without a positive Hölder recovery rate, dual-source nonattainment, sharp Gaussian feature geometry with a sharp covariant Hessian norm bound √7, and quantitative separated signed recovery. The measurement analysis distinguishes full L² observations from finite sensing: under positive absolute adversarial noise, any uniformly finite deterministic adaptive budget of bounded linear probes has minimax L² reconstruction error one over unit Gaussian translates with unbounded location. Compact parameter domains admit finite measurement realizations, although their construction dictionary can be exponentially large. An arithmetic appendix preserves the original encoding lineage and associated integer readout results. The scope is essential: exact competing representations may have continuously distributed, unbounded means. The impossibility results do not contradict identifiability among finite Gaussian mixtures and do not rule out fixed finite-model-order algorithms, adaptive weights, or other regularizers. Continuous-target rate barriers and fixed finite-target estimator inconsistency have different hypotheses. No universal finite-data Gaussian-mixture learning algorithm or iid-sample rate is claimed. This deposit documents final release 1.0, identified as NERIU-FINAL-1.0-2026-11-02. General continuum results rest on analytic proofs; executable checks audit identities and certificates and distinguish exact arithmetic from floating-point diagnostics. Novelty priority remains unestablished.