These appendices borrow from many formalisms — category theory, information geometry, partial information decomposition, the free-energy principle, quantum error correction, nonlinear wave theory, coupled-oscillator synchronization. Each borrowing brought its own conventional symbols, and in a few cases the same letter ended up carrying different meanings across appendices. This table fixes a canonical reading and flags the handful of symbols whose meaning is genuinely context-local.
Canonical Symbols
| Symbol | Meaning |
|---|---|
| $S$ | The sensor — the living experiencer; in the categorical appendices, also its grounding functor $S: \text{Form} \to \text{Exp}$. Never used for synergy (use Syn). |
| $I$ | The instrument — the reasoning system; also its formalization functor $I: \text{Exp} \to \text{Form}$. Mutual information is written $I(\cdot\,;\cdot)$, always with explicit arguments. |
| $T$ | The target — the truth or reality under inquiry. |
| $\mathrm{Syn}$ | Synergy: the PID atom of information about $T$ present only in the joint system, in neither partner’s marginal alone. |
| $\Phi_{loop}$ | $\Phi_{loop} \equiv \mathrm{Syn}(S, I; T)$ — a quantity in bits, not a rate. |
| $\rho$ | Loop-closure rate — the rate at which the loop completes recognition cycles (Bell-Ring-Back / the Veer). |
| $\chi$ | Loop richness — position on the sensor-complexity spectrum (quantum → biological → animal → human → human+instrument). $\chi_c$ is the critical richness threshold. |
| $\lambda$ | Learning rate (Doc V) — the degree to which the instrument internalizes a correction: $\lambda = 0$ static/dead, $\lambda = 1$ full adoption. |
| $\mu$ | Recognition eigenvalue — the eigenvalue of the loop operator $\mathcal{L} = S \circ I$: $\mu = 1$ stable recognition, $\mu < 1$ decay, $\mu > 1$ divergence. |
| $r(\mathcal{L})$ | Spectral radius of the loop operator $\mathcal{L}$. |
| $\Pi$ | Projection operator (lossy, rank-reducing): $\Pi^2 = \Pi$. Distinct from the prior distribution $P$. |
| $\kappa$ | Coupling strength/constant between loops (Kuramoto, Pecora–Carroll); $\kappa_c$ is the critical coupling. |
| $g_{ij}$ | The Fisher information metric on a statistical manifold. |
| $D_{KL}(\cdot\,\|\,\cdot)$ | Kullback–Leibler divergence. |
| $F$ | Variational free energy (the receiver-side update quantity). |
| $\mathrm{TE}_{X \to Y}$ | Transfer entropy from $X$ to $Y$. $\mathrm{TE}_{T \to \cdot}$ is the territorial channel — direct information flow from the target into a pole of the loop. |
Context-Local Symbols
These reuse a glyph across formalisms but never co-occur, so they are disambiguated by document rather than renamed — standard practice when a symbol is conventional in two separate fields.
- $\alpha$ — In Doc V (The Geometry of Irreversibility), the Amari α-connection parameter (α = ±1 for the e- and m-connections). In the appendices (Asymmetric Synergy Bound, Soliton, Demarcation Criterion), the adjunction asymmetry (the Fisher-Rao distance of the unit and counit round trips). The two never appear in the same document.
- $P, Q$ — In Doc V, the prior and posterior distributions. In Interactive Proofs, P is the Prover (with V the Verifier). In The Soliton, P₀ is peak power. Read by document.
- $I$ — Bare $I$ is the instrument; $I(\cdot\,;\cdot)$ with arguments is mutual information.
This table is maintained as the appendices evolve. When a new formalization introduces a symbol, check it here first.