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Every step from Adam's momentum and denominator to the spectral hitting law is formalized inside the closed state-evolution model, and it is proved that a stronger arbitrary-gradient Adam theorem is impossible, and that exact two-step linear-network loss dynamics do not identify factor spectra or heavy-tail hitting tim...
Theoretically, it is proved that exact early loss dynamics in linear networks do not determine factor spectral tails, that Adam recurrences alone do not imply spectral redistribution, and that projected singular-basis spreading implies contraction of a spectral-tail potential and hence a dimension-corrected hitting-tim...
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