The Structural Gap to ModelingToolkit

design open-problem

ModelingToolkit as an Acausal Relation asks what MTK and Lenticulum have in common. This note asks the harder question in the other direction: what does MTK contain, structurally, that Lenticulum does not?

Five things, ranked by how deep they go. Two of them are additions; three are not. The design that follows from them is Time as a Base.

Sources: original to this vault (design and analysis; no single paper).

Theory (CT-ML wiki): Hypergraph Category · Frobenius Monoid · Decorated Cospan · Bayesian Inversion · Graphical Linear Algebra · Parametric Lens · Variational Free Energy · Lens · Cospan

0. The ranking

	gap	addable?
1	models are terms, not closures	no — a different design
2	a distinguished independent variable, and a derivation on it	no — structure on the variable category, not a factor
3	connectors carry two interacting structures, not one	yes, and it corrects Acausal Composition is a Hypergraph Category
4	subsystems are first-class and instantiable	yes; the formalisation is already in the vault
5	events — a system that rewrites its own equations	no

Items 1 and 2 are the ones that matter. 3 is a correction. 4 is unfinished work. 5 is a different subject.

1. The models are terms, not closures

An MTK equation is a symbolic expression in a term algebra. A Lenticulum factor is a Julia object with methods — opaque to the framework.

Mycelium knows which channels a factor touches; that is the incidence graph, and ModelingToolkit as an Acausal Relation §4 shows it is the same bipartite graph MTK’s structural analysis runs on. What Mycelium cannot know is how.

So MTK can ask questions Lenticulum structurally cannot:

• is this equation linear in $x$?
• are these two equations aliases of one another?
• can I solve this one for $x$ symbolically and substitute it into the others?

Which is exactly what mtkcompile is: inspection followed by rewriting. Alias elimination, tearing and Pantelides index reduction all require looking inside a node.

The gap surfaces precisely in Composition is Elimination. That note argues elimination is the composition operation of the framework, and the only version Lenticulum can implement is the numerical one — marginalisation, or a solve. MTK does it symbolically, exactly, at compile time.

This one is not addable

Making factors symbolic means making them terms in a free algebra rather than objects carrying methods. That is a different architecture, not a feature — and it would cost the thing the current design buys, namely that a factor’s internals can be an arbitrary neural network (Lux as a Parametric Lens). You cannot do equational reasoning on a U-Net.

The realistic conclusion is a division of labour, not a merge. See Time as a Base §8.

2. Time is external and shared; Lenticulum’s is internal and sealed

Lenticulum does have time. NeuralODEFactor has a tspan; VPSDE has $t$. The difference is where it lives:

MTK’s time is external and shared — every equation in the system lives over one base. Lenticulum’s time is internal and private — sealed inside a single factor and integrated out before that factor talks to the graph.

NeuralODEFactor exposes $(z_0,z_1)$ and nothing else. You cannot attach a factor to $z(0.5)$. And that is a little damning, because it is the same move DEQ as a Relation criticises DeepEquilibriumNetwork for: sealing the solve inside and handing back the endpoints. The factor un-seals the direction and keeps the time sealed.

The algebraic content

MTK’s variables live in a differential ring: D = Differential(t) is a derivation, and equations relate variables and their derivatives. Lenticulum’s variables live in a plain space — Channel{name,S}(space) carries a space and nothing else, and there is no derivation anywhere in LenticulumCore or Mycelium.

And D cannot be added as a factor. It relates a variable to itself at infinitesimally nearby times, which a bipartite graph cannot express. You would introduce $\dot{x}$ as a second variable with $\dot{x}=D(x)$ — but that relation is not an arbitrary factor. It is a fixed, universal, non-learnable relation that the solver has to know about. It is structure on the category of variables, not content in the graph.

A factor graph’s honest answer to time is discretisation into a chain, which is The Linear Gaussian Chain — i.e. Kalman filtering. That is a legitimate answer and a different one, not a worse one. Time as a Base is about what the continuous answer would look like.

3. A connector carries two interacting structures, not one

This corrects Acausal Composition is a Hypergraph Category §2.

That note argues a Mycelium variable of degree $d$ is a Frobenius spider — all legs carry equal values, and the spider theorem says a $d$-way junction has no internal structure. That is right for MTK’s across variables: voltage, temperature, position, pressure. Equal at the junction.

But @connector also declares through variables — current, force, heat flow, mass flow — and those do not copy. They sum to zero.

So a connector is not one spider. It is a copying structure and an adding structure on the same object, which in the vocabulary that note already cites (Bonchi–Sobociński–Zanasi, Interacting Hopf Algebras) is exactly an interacting Hopf algebra — strictly richer than the single special commutative Frobenius algebra.

	across / effort	through / flow
examples	voltage, temperature, position	current, heat flow, force
junction law	all equal	sum to zero
algebra	the copy comonoid — the spider	the add monoid, with the antipode for sign
in Mycelium	a shared variable node	not available

And this is precisely why Kirchhoff’s law had to be written as an explicit LinearConstraintFactor in the divider test of ModelingToolkit as an Acausal Relation §7: the sum-to-zero structure is not available in the variable node, so it has to be added as a separate factor.

The deeper consequence: in MTK the interconnection semantics are attached to the port type, not chosen at the call site. That is what makes acausal composition work uniformly across electrical, mechanical, thermal and hydraulic domains — you connect two pins and the right equations appear, because the pins know what they are. LenticulumCore.Channel carries a dimension. No domain, no units, no through/across flag.

This one is addable, and cheaply

A through/across flag on Channel, plus generating the balance equation at connect!-time rather than making the user write it. The algebra is richer than what Acausal Composition is a Hypergraph Category describes, but it is well understood and the graphical-linear-algebra literature has a complete axiomatisation of it.

4. Subsystems are first-class

MTK has @component, hierarchical namespacing (circuit.resistor1.p.v), and instantiation — one resistor model, twenty instances, flattened by mtkcompile.

Lenticulum has AbstractLenticulumContainerFactor{factors}, which nests parameters, and flat graphs with one symbol namespace. There is no “collapse this subgraph into a factor” operation.

The formalisation is already in the vault and is not the problem: Acausal Composition is a Hypergraph Category §6 gives it as decorated cospans — an open system is a subgraph with a boundary, and a boundary is a set of channels, so an open subgraph is a factor. Composition is Elimination says what collapsing one means. §8 of ModelingToolkit as an Acausal Relation sketches it as MTKFactor.

Nothing implements it. This is unfinished work rather than a structural obstacle — and Time as a Base §4 notes that it is the same missing capability as the temporal extension, because a belief over a trajectory is itself a chain-structured subgraph.

5. Events

MTK has continuous and discrete callbacks: a system that changes its own equations at event times. A ball bounces; a switch closes; a controller saturates.

Lenticulum has no notion of a graph that rewires itself. That is genuinely deep — it is a dependent structure, in which the graph is a function of the state, and none of the machinery in this project (fixed FactorGraph, fixed schedules, fixed counting numbers, a Bethe free energy whose Euler characteristic is computed once) survives a graph that changes shape mid-inference.

A smaller one, with a connection

MTK’s initialization system solves a nonlinear system for consistent initial conditions — necessary for any DAE of index $\ge 1$, because not every assignment of values satisfies the hidden constraints.

That is precisely the problem whose absence appeared as the bootstrap deadlock in constraint.md §4.2: a loopy acausal graph where every message needs every other channel to be informative, so the first sweep produces nothing and so does every sweep after it. MTK has an answer; Lenticulum works around it with weak regularising priors.

6. The other direction, for honesty

MTK has no probability, no beliefs, no learned factors, and no notion of an inexact inversion whose cost is measured. Its equations are hard and known, and its parameters are calibrated rather than learned.

That is the whole of Lenticulum’s claim over it, and it is a real one. But it is narrower than “Lenticulum is MTK plus noise” — items 1–5 above are the price, and items 1 and 2 are not payable.

Related: ModelingToolkit as an Acausal Relation, Time as a Base, Acausal Composition is a Hypergraph Category, Composition is Elimination, Differential Algebra and DAE Factors, The Linear Gaussian Chain, Lux as a Parametric Lens