L2. Identifying 3: The Cumulant Tower as Effective Action (CTEA)

Situation. L1 (https://doi.org/10.5281/zenodo.20336872) gave the chart: probability simplex, zero-sum hyperplane $H_0$, sphere pullback, Fisher–Rao metric, rotation group at the uniform. But the chart supplies only one geometric object: a metric. A working theory needs more: a notion of position, a notion of curvature, a notion of force, and a notion of how truncations of the theory fail. Where do these come from, and is there a unifying source for all of them? Task. Identify a single scalar functional on [L1](https://doi.org/10.5281/zenodo.20336872)'s chart whose successive derivatives generate the geometric objects we need, with no ad-hoc additions. Action. Use the **log-partition function** $\Phi(\eta) = \log Z(\eta)$ of the categorical exponential family. The natural parameters $\eta$ live on $H_0$ (dual to [L1](https://doi.org/10.5281/zenodo.20336872)'s centered chart); the mean parameters $\mu = \nabla \Phi$ are exactly the probabilities $p$. Take successive cumulants of $\Phi$. Result. The derivatives of $\Phi$ form a tower: 1. $\nabla\Phi = \mu = p$ — the mean parameter; the position. 2. $\nabla^2 \Phi = \mathrm{Cov}(X)$ — the covariance of the sufficient statistic, equal to the **Fisher information** at $p$; the metric, and simultaneously the inertia. 3. $\nabla^3 \Phi = T$ — the **Amari–Chentsov 3-tensor**; the cubic structure that defines information geometry as its own subject beyond Riemannian geometry. 4. $\nabla^4 \Phi$ and higher — projection-closure defects; where 3rd-order truncations fail. One functional $\Phi$, four successive geometric objects. The leap from order 2 (Fisher metric / Riemannian) to order 3 (Amari–Chentsov / dually flat) is the load-bearing distinction — the third cumulant is the first invariant Riemannian geometry cannot see. This is the "3" of the trilogy title, made concrete: the cumulant order at which information geometry separates from ordinary differential geometry. This is L2 of the trilogy *Identifying 3*. L1 supplied the chart; L2 develops its successor structure; L3 reads the resulting tensors through Pontryagin character orthogonality on the conductor lattice. Take the probabilities of \(N\) outcomes and write one number (the log of their normalization) as a function of the parameters driving them. Successive derivatives of this single scalar generate every geometric object information geometry needs: the first recovers the probabilities themselves; the second gives the Fisher information metric (how distinguishable nearby distributions are); the third gives the Amari-Chentsov 3-tensor (the cubic "force law" that distinguishes information geometry from ordinary Riemannian geometry); the fourth records where any third-order truncation stops closing. Four named geometric objects from one functional, no extras. A Gaussian's third derivative vanishes, so Riemannian geometry sees its whole geometry; the categorical distribution does not stop at order 2, and its third cumulant is the first invariant Riemannian geometry cannot see: the "3" that the trilogy title names.