Gaussian–T³ Spectral Inference: Statistical Identifiability and Stable Representation

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We study a concrete statistical class in which latent or predictive structure is represented on the three-torus and observation acts through a Gaussian heat channel. The heat operator diagonalizes in the Fourier basis with nonvanishing multipliers exp(−τ|m|2). This elementary spectral fact yields several exact consequences: injectivity and completeness for toroidal Gaussian proxy models; explicit necessary and sufficient range conditions for proximal outcome and treatment bridge functions; a precise separation between population identifiability and severely ill-posed function recovery; logarithmic minimax L2 recovery rates over Sobolev balls; and a stable-representation theorem characterizing when a continuous minimal predictive statistic can genuinely live in T3. We also audit attempts to use the same architecture to remove the computational–statistical gap in generic sparse PCA. Fixed Fourier embeddings, unfiltered synchronization, edge thresh- olding, finite-dimensional Hubbard–Stratonovich compression, torus-charge Wick projectors, and determinant/principal-minor compression are each shown to encounter explicit obstructions. In particular, a rank obstruction prevents exact compression of a generic Wishart quadratic form to three Gaussian auxiliary coordinates; torus orthogonality cannot remove self-pairing noise; the highest-order principal-minor score reduces exactly to a leverage score and recovers the familiar k2-type sample barrier; and finite-sample null and spiked Gaussian laws are mutually absolutely continuous, excluding exact separation by an almost-sure topological sector invariant. The third audit adds a held-out validation theorem showing that even a polynomial-size list containing one constant-overlap candidate would immediately give a polynomial-time … [source excerpt; see full PDF]

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