The Decoherence Exponent: Stable Phase Noise and Constraints on Objective State Reduction
arXiv:2608.18335
2026
Architecture
1 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper identifies a sharp mathematical family of positive-definite similarity kernels: scale-covariant symmetric Lévy phase noise necessarily yields K_t(q,q') = exp(-Dt|q-q'|^alpha) with 0 < alpha <= 2, while alpha > 2 fails positive definiteness already on three equally spaced points. This gives neural architectures a learnable fractional-distance bias with a hard stability constraint rather than an unconstrained distance exponent. The most direct transfer is a stable-charge attention or kernel-mixing branch in which each token receives a learned scalar charge and interactions are modulated by this kernel. The theorem does not guarantee better task accuracy by itself, so experiments should test optimization stability, extrapolation to longer sequences, and whether learned fractional exponents outperform fixed Gaussian attention biases.
Ideas from this paper
Unverified
2026
Attach each token or graph node a learned scalar charge q_i and add a fractional stable kernel K_ij = exp(-tau D |q_i-q_j|^alpha) to the interaction mechanism. Constrain 0 < alpha <= 2, the exact range in which the kernel is positive semidefinite for arbitrary finite real charge sets, and optionally make tau layer-dependent to obtain multiscale interactions. This provides a principled alternative to unconstrained learned distance biases and can be used either as an attention-logit bias or as a…
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