# Integrating Out, Twice:The Open-System Case That Neural-Network Ensemble Theory Is Missing

arXiv: https://arxiv.org/abs/2606.09950 · HTML: https://mlsanity.com/paper/2606.09950/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 35 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)

## Abstract (arXiv)

Averaging a neural network over its random parameters and marginalizing a Gaussian sector are the same operation, the Schur complement of the eliminated block, and when that block is closed it returns a covariance and its inverse. That is all a network ensemble produces, the closed case. The open case is missing, and nuclear reaction theory has it worked out. Projecting a scattering problem onto a chosen set of channels, with the rest carrying probability irreversibly to a continuum, leaves a non-Hermitian effective generator that conserves and itemizes exactly what it loses: the nuclear optical model and its generalized optical theorem. I set the two cases side by side using only the moments of a distribution, the algebra of Gaussians, and block inversion, no field theory, and give the closed-case dictionary in full: the neural tangent kernel is the Fisher sensitivity kernel, the infinite-width Gaussian limit is the Gaussian-process emulator, and the lazy-to-feature transition is the validity boundary of a reduced-basis emulator. I then test the open export on a truncated attention map, a token-level transfer operator, and a sparse expert router, and report a mostly negative result. The conserved flux ledger ports wherever openness is genuinely present, but its distinctive content is absent, an artifact of the chosen partition, or pinned near a floor by the training objective, and the operationally useful uncertainty turns out to be epistemic, living in the closed half of the correspondence, not the open one. The negative has a structural reason this note makes precise: the open case needs an eliminated sector with a continuous spectrum and wave-like, not relaxational, dynamics, which mainstream learning's finite or dissipative objects do not supply. This is a note, not a result; its main finding is that negative one, and its value is the map that locates it.