Morphological Physics already produces closed-form velocity fields for pressure-driven flow through arbitrary duct cross-sections. Heat transfer under forced convection follows directly: the thermal energy equation reduces to a Poisson problem on the same cross-section, with the velocity-derived source as the inhomogeneous term. The eigenfunctions used to represent the flow serve equally well for the temperature field. No new machinery is required — the same boundary-driven expansion carries through.

Nothing remains at the mathematical level. The construction is complete and the results match established benchmarks. What remains is engineering: connecting the analytical layer to an interactive interface where a user can adjust cross-section geometry, boundary conditions, and Reynolds and Prandtl numbers, and see the resulting temperature field and Nusselt number update.

The framework computes analytical results directly from the shape of a boundary rather than from a discretized volume. Plasma confinement performance depends strongly on the geometry of the confinement surface — the shape of the last closed flux surface, the magnetic well contour, the field-line topology at the boundary. A method whose primary input is boundary and well morphology, like molecular wave-function modeling, has a natural entry point into plasma confinement questions.

Plasma is not a passive medium adopting the geometry of its container. Instabilities, drift, and self-organized structure change the effective confinement shape on timescales the analytical framework does not currently model. Coupling boundary geometry to plasma-internal dynamics is the open research question.

Morphological Physics already provides intrinsic tools for modeling densities, computing the mean of a density, sampling from a density model, and comparing both densities and distributions in multi-dimensional space. All four capabilities extend directly to statistical problems on shapes: a shape is a boundary, a boundary is a member of a shape space, and populations of shapes are distributions on that space.

Statistical shape analysis is straightforward to state as an applied problem but requires some of the most demanding mathematical theory of the open directions on this page. The high dimensionality of shape boundaries is the primary obstacle. Comparison of distributions of unit-normalized shapes is already available using the existing toolkit; sampling from shape densities is not. The work to solve distribution comparison in R^n gave Morphological Physics a new mathematical foundation for the entire theory. Extending that foundation to shape spaces requires implementing the R^n machinery for shape data, which has not yet been done.

Retail consumer data lives naturally as sparse consumer-product matrices — thousands of consumers, tens of thousands of products, most cells empty. Morphological Physics treats these matrices as observed samples from an underlying multidimensional density. Once the density is represented intrinsically, downstream questions — expected lifetime value, propensity to purchase a category, similarity between consumers or between products — become analytical operations on the density rather than fitted estimates. The construction also produces correlation coefficients between variables directly as a byproduct of the density fit, without a separate estimation step. The interaction terms that couple variables in these consumer models are the same mathematical object as the coupling matrices Morphological Physics uses in molecular electronic structure — the framework carries a single formalism across domains that appear unrelated on the surface. The density models are also Bayesian by default; the underlying density is a probability model, not a fitted estimator, so all outputs carry their probabilistic interpretation naturally, and the framework fits cleanly onto graphical-model representations of consumer behavior. Current work uses directed graph structures; whether the framework's underlying formalism requires direction, or whether undirected representations serve equally well, is itself an open question.

The applied problem is straightforward to state, but consumer data is where the framework meets several practical obstacles at once. Sparsity is extreme — a typical consumer purchases fewer than one percent of the products available, so the density model must handle regions where data is thin without overfitting where it is thick. Purchase histories are non-stationary — consumer behavior drifts with season, life stage, and market conditions, so a density fit at one point in time may not describe behavior six months later. Available datasets are also limited in size compared to what modern consumer analytics operations run on internally; whether more data would resolve the current characterization limits or expose deeper structural questions remains open.

Attention operators in transformers act on normalized token representations — vectors that live, up to scaling, on a hypersphere. Morphological Physics provides analytical models for distributions on the hypersphere, and more broadly has repeatedly replaced numerical methods with analytical ones across disciplines — DFT for molecular electronic structure, FEM for fluid transport, KDE for probability density estimation. Extending the hyperspherical models to describe trained attention gives an analytical construction of what an attention operator should look like, produced without training. Attention is another instance of a class of problems normally treated numerically that the framework has indicated can be treated analytically.

Attention has been the workhorse of modern machine learning for nearly a decade and the mathematical literature around it is enormous. Establishing that an analytical construction agrees with a trained operator in a way that is not artifactually circular — the analytical construction cannot use the trained operator itself to fix its geometry — is the current experimental challenge. The natural first test failed. A weaker and more interesting result survives: the principal modes of the analytical construction match the principal modes of the trained operator, while a naive baseline does not. The decisive experiment builds the geometry from data alone and repeats the comparison. A separate open question, developed at length in the institute's internal work, is whether training a model on the analytical construction from the start would be more productive than trying to recover the construction from a model trained conventionally.