# Spectral Structure in Finite Free Information Inequalities and p-Stam Phase Transitions

arXiv: https://arxiv.org/abs/2604.11922 · HTML: https://mlsanity.com/paper/2604.11922/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 24 references that are in the map.
- First posted on arXiv 13 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [There are natural scores: Full comment on Shafer, "Testing by betting: A strategy for statistical and scientific communication"](https://mlsanity.com/paper/2102.05569/) (2021)
- [Can Transformers Do Enumerative Geometry?](https://mlsanity.com/paper/2408.14915/) (2024)

## Abstract (arXiv)

Using FlowBoost, a closed-loop deep generative optimization framework for extremal structure discovery, we investigate ℓp-generalizations of the finite free Stam inequality for real-rooted polynomials under finite free additive convolution boxplusn. At p=2, FlowBoost finds the Hermite pair as the unique equality case and reveals the spectral structure of the linearized convolution map at this extremal point. As a result, we conjecture that the singular values of the doubly stochastic coupling matrix En on the mean-zero subspace are 2-k/2:k=1,ldots,n-1, independent of n. Conditional on this conjecture, we obtain a sharp local stability constant and the finite free CLT convergence rate, both uniform in n. We introduce a one-parameter family of p-Stam inequalities using ℓp-Fisher information and prove that the Hermite pair itself violates the inequality for every p>2, with the sign of the deficit governed by the ℓp-contraction ratio of En. Systematic computation via FlowBoost supports the conjecture that p^*!=2 is the sharp critical exponent. For p<2, the extremal configurations undergo a bifurcation, meaning that they become non-matching pairs with bimodal root structure, converging back to the Hermite diagonal only as p→ 2^-. Our findings demonstrate that FlowBoost, can be an effective tool of mathematical discovery in infinite-dimensional extremal problems.