In an effort to make the abstract and technical geometric concepts more accessible, Pickhardt has provided an example through an iPython notebook available on his Github repository, where he further develops this theory and research paper. It's important to note that the current draft of the paper hosted there is preliminary and contains inaccuracies, with an updated, more precise version anticipated to be released soon.

The underlying premise of Pickhardt's work stems from previously identified limitations within models used to estimate liquidity in payment channels and compute success probabilities for payments. These models inaccurately assumed independent liquidity distributions across channels and relied on uniform distribution to model uncertainty. To address these issues, the introduction of bimodal models was proposed, which better reflect the observed distributions arising from the specific geometries of payment channel networks. This realization led to the abandonment of the assumption that liquidity distributions are independent, highlighting a significant shift in understanding the network's dynamics.

Pickhardt’s contribution through the notebook showcases a method to compute the payment success rate between two peers on the Lightning Network without requiring information about the actual liquidity distribution. This approach assumes all feasible wealth distributions within the network's topology are equally likely, allowing for the calculation of payment feasibility based solely on whether the wealth distribution after a payment remains viable. This methodology represents a departure from traditional assumptions about liquidity and wealth distribution, aiming closer to the realities of skewed power-law distributed wealth within the network.

Illustrative results from the notebook indicate how the likelihood of successfully sending a payment varies with the amount and the involved nodes, contrasting these findings against the minimum cost flow (MCF) probabilities. The examples demonstrate that even though MCF might suggest a certain probability of success, the actual feasibility of a payment depends on the underlying wealth distribution and network topology, often resulting in different success probabilities.

In conclusion, Pickhardt’s work offers a novel perspective on estimating payment success probabilities within the Lightning Network, underscoring the importance of considering feasible states and wealth distributions over simplistic liquidity estimation models. This approach highlights the inherent limitations of payment channels, suggesting that 100% success rates are unattainable due to the dynamic and constrained nature of feasible wealth distributions within the network. The ongoing development of this mathematical theory promises to provide deeper insights into the operational dynamics of the Lightning Network, potentially guiding future improvements in network design and pathfinding algorithms.