Notation Reference — Operator World Models¶
A standalone glossary for the Operator World Models series. Every symbol used across the series, grouped by role, with a "read as" column. Keep it open in a second tab while reading.
This series writes the encoder as \(E\), matching the Action Operators foundation. The same encoder is written \(\varphi_\psi\) in the GRL literature — the same object, \(E \leftrightarrow \varphi_\psi\).
Representation — the JEPA side¶
| Symbol | Read as | Meaning |
|---|---|---|
| \(x\) | "x" | a raw observation — an image, a window of behavioral sensor signal, a protein structure |
| \(s\) | "s" | the underlying system state in observation/state space \(\mathcal{S}\) — the physically real thing an action transforms (a person's full physiological state, a protein's atomic coordinates) |
| \(E\) | "E" | the encoder: a neural network mapping an observation to a latent. Same object as \(\varphi_\psi\) in the GRL literature |
| \(z = E(x)\) | "z equals E of x" | the latent embedding — a vector capturing meaning, not surface form. Lives in latent space \(\mathcal{Z}\) |
| \(z_t\) | "z at t" | the latent at time \(t\), \(z_t = E_\xi(x_{\le t})\) — the encoded history up to \(t\) |
| \(E_\xi\) | "E-xi" | the online encoder, trainable weights \(\xi\) (Greek xi) |
| \(E_{\bar\xi}\) | "E-xi-bar" | the target encoder: a slow exponential-moving-average (EMA) copy of \(E_\xi\), used to produce prediction targets. Stop-gradient — no backprop flows into it |
| \(\bar\xi \leftarrow \tau\bar\xi + (1-\tau)\xi\) | — | the EMA update of the target weights; \(\tau\) close to 1 makes the target drift slowly |
| \(\tau\) | tau | the EMA rate (e.g. \(0.999\)): the fraction of the old target weights kept each update |
| \(\mathrm{sg}\) | "stop-grad" | stop-gradient: treat the argument as a constant during backprop |
Operators — state space and latent space¶
| Symbol | Read as | Meaning |
|---|---|---|
| \(\hat O\) | "O-hat" | an action operator: a function \(\hat O:\mathcal{S}\to\mathcal{S}\) that turns a state into its successor. The hat marks "operator (a function), not a number" — the physics convention (as in \(\hat H\)), not the statistics "estimate" hat of \(\hat\theta\) |
| \(\hat O_\theta\) | "O-hat-theta" | the specific state operator chosen by parameters \(\theta\) — the physically meaningful transformation acting on the real state \(s\). Often inaccessible: you only observe \(s\) and \(s'\), never the operator itself |
| \(f_\theta\) | "f-theta" | the latent operator: how \(\hat O_\theta\) acts on the latent \(z=E(s)\) instead of on \(s\). This is the object you actually compute with |
| \(\theta\) | theta | operator parameters — the knobs that configure the operator: a complete description of how it transforms the state (for a continuous \(\theta\) the operator is synthesized from \(\theta\), not looked up from a menu) |
| \(\Theta\) | capital Theta | the space of all operator parameters |
Building the latent operator¶
| Symbol | Read as | Meaning |
|---|---|---|
| \(B_i\) | "B-i" | the generator basis — a set of \(m\) matrices (each \(D\times D\), where \(D=\dim\mathcal{Z}\)) that span the allowed operator directions. The basis choice sets what kind of operator is possible |
| \(\alpha\) | alpha | the coefficient vector over the basis; \(\alpha = \theta\). Component \(\alpha_i\) says "how much of generator \(B_i\)" |
| \(M_\theta = \sum_i \alpha_i B_i\) | "M-theta" | the flow generator — a Lie-algebra element; the "infinitesimal" form of the operator, living in a flat vector space |
| \(A_\theta = \exp(M_\theta)\) | "A-theta" | the operator matrix — the group element obtained by exponentiating the generator. \(\exp\) is the matrix exponential |
| \(b_\theta\) | "b-theta" | an optional affine bias term in latent space |
| \(f_\theta(z) = A_\theta z + b_\theta\) | — | the latent operator in its concrete linear form, \(\exp(M_\theta) z + b_\theta\) |
Conditioning and policy¶
| Symbol | Read as | Meaning |
|---|---|---|
| \(c_t\) | "c at t" | the context / intervention covariates at time \(t\) — the known causes of state change (hours slept, medication taken, a stressor; a mutation, a ligand) |
| \(\pi_\psi\) | "pi-psi" | the context policy: emits the operator coefficients from the latent and context, \(\theta \sim \pi_\psi(z_t, c_t)\), with weights \(\psi\) |
| \(\Delta(\Theta)\) | "simplex over Theta" | the set of probability distributions over \(\Theta\). A stochastic policy emits one of these; a deterministic policy emits a single \(\theta\) |
| \(g_\phi\) | "g-phi" | JEPA's predictor with weights \(\phi\) — the query-conditioned operator the action operator generalizes |
| \(q_{\Delta t}\) | "query at delta-t" | the predictor's query: in vanilla temporal JEPA it carries only the time offset \(\Delta t\); the action operator replaces it with \(\theta(c_t)\) |
Energy, spectrum, loss¶
| Symbol | Read as | Meaning |
|---|---|---|
| \(E(\hat O)\ge 0\) | "energy of O-hat" | the energy functional: how large a transformation the operator is. Here \(E(\hat O)=\lVert M_\theta\rVert_F^2\) — a least-action / parsimony penalty that keeps the operator near identity unless the data demands otherwise. (Distinct from the encoder \(E\); context disambiguates) |
| \(\lambda\) | lambda | the weight on the energy penalty |
| \(\mathrm{Re}(\lambda_i)\) | — | the real part of an eigenvalue of the generator \(M_\theta\). A positive real part in any mode flags locally growing (destabilizing) dynamics |
| \(\mathcal{L}\) | "script L" | the training loss — latent-space squared error between the predicted and target latents |
| \(\lVert v\rVert^2\) | — | squared Euclidean (\(\ell_2\)) norm |
| \(\lVert M\rVert_F^2\) | — | squared Frobenius norm — the sum of squared matrix entries |
One overload to keep straight¶
The symbol \(E\) does double duty: the encoder \(E\) (and its variants \(E_\xi\), \(E_{\bar\xi}\)) and the energy functional \(E(\hat O)\). They never appear in the same role, and the argument disambiguates — \(E(x)\) encodes an observation, \(E(\hat O)\) scores an operator.
See every symbol at work. A table defines a symbol; it does not teach one. For all of this notation threaded through a single concrete story — CGM, insulin, carbs, exercise, and medical codes for one person managing diabetes — see the worked example, where each symbol is earned by something real before it is named.
Series home: Operator World Models. The foundation: Augmenting JEPA with Action Operators.