iitx glossary

One vocabulary, used identically in code, docstrings, documentation, and tests. Where the literature is consistent, names match the literature; where the papers disagree (3.0 vs 4.0), the core uses version-neutral terms and each measure module uses its own paper's terms. Symbols follow the papers; Python names follow the rules at the end.

Core (version-neutral)

Term	Symbol	Definition
unit	$U_i$	One variable of the system, with a finite alphabet of $q_i$ states. Never "node" or "element".
alphabet	$\Omega_{U_i}$	The states a unit can take; its size is $q_i$.
shape	—	The tuple $(q_1, …, q_n)$ of per-unit alphabet sizes; equals the tensor shape of any full state-space array.
state	$u$, $s$, $m$, $z$	An assignment of a value to each unit of some set, as an (n,) integer vector; lowercase of the set's symbol.
state index	—	The little-endian mixed-radix flat index of a state (unit 0 is the fastest-varying digit).
TPM	$\mathcal{T}$	Transition probability matrix $p(\bar u \mid do(u))$, state-by-state (Q, Q), rows = prior state; interventional, rows sum to 1.
node TPMs	—	The factored view: per-unit conditionals $p(\bar u_i \mid u)$, shape (*shape, q_i); exists iff the TPM is conditionally independent.
conditional independence	—	$p(\bar u \mid u) = \prod_i p(\bar u_i \mid u)$. Required by mechanism-level analysis; a checked property in iitx, never an assumption.
connectivity matrix	cm	(n, n) boolean; cm[i, j] = 1 ⇔ unit $i$ inputs to unit $j$.
system	$U$, $S$	The object under analysis: TPM + shape + connectivity (System). Never "network" (that's a provenance) or "substrate" (4.0-specific honorific).
candidate system	$S \subseteq U$	A subset of units analyzed as a system, given as a mask; the rest is background.
background	$W = U \setminus S$	Units outside the candidate system; conditioned per the measure's convention (frozen for 3.0; causally marginalized on the cause side for 4.0).
mechanism	$M$, state $m$	A subset of the candidate system, in its current state, whose causal powers are assessed.
purview	$Z$, state $z$	A subset of the candidate system over which a mechanism's cause or effect is evaluated.
direction	—	Direction.CAUSE (about $t-1$) or Direction.EFFECT (about $t+1$). Every two-sided quantity is one function of a Direction.
repertoire	$p(z \mid m)$, $\pi(z \mid m)$	The distribution a mechanism specifies over a purview, in the given direction.
unconstrained repertoire	$p^{uc}$, $\pi(z; M)$	The same distribution with the mechanism unconstrained (uniformly averaged out).
partition (mechanism)	$\theta \in \Theta(M, Z)$	A way of severing the mechanism–purview dependency; its repertoire is the product over parts.
partition (system) / cut	$\theta \in \Theta(S)$	A way of severing the candidate system; cut connections are noised (uniformly marginalized inputs).
MIP	$\theta'$	Minimum (information) partition: the partition that makes the least difference, per the measure's normalization and tie policy.
phi	$\varphi$	Integrated information of a mechanism-level or system-level object: the difference the MIP makes. Never bare phi in the public API — always a qualified name (below).
complex	$S^*$	A candidate system that is a maximum of the measure's system quantity among overlapping candidates.
major complex / first complex	—	The complex with the globally maximal value (3.0 / 4.0 naming respectively).
condensation	—	The disjoint, exhaustive set of complexes of a universe.
precision	—	The quantization applied to every value before comparison; ties are defined at this precision (1e-6 for IIT3, 1e-13 for IIT4).
tie policy	—	The documented composite key and canonical order that make every selection deterministic.
measure	—	A first-class object mapping (System, state) to integrated-information results (Measure protocol); IIT3 and IIT4 are instances.

IIT 3.0 terms (module iitx.measures.iit3)

Term	Symbol	Definition
cause/effect information	$ci$, $ei$	EMD between a repertoire and its unconstrained counterpart.
small phi	$\varphi$	$\min(\varphi_{\text{cause}}, \varphi_{\text{effect}})$ at the mechanism MIP, EMD-valued.
core cause / core effect	—	The purview maximizing $\varphi$ on each side.
MICE	—	Maximally irreducible cause-effect: the pair of core repertoires.
concept	—	A mechanism with $\varphi^{\max} > 0$, its MICE, and its $\varphi^{\max}$.
cause-effect structure (CES)	$C$	The set of all concepts of a candidate system ("constellation").
conceptual information	$CI$	$\sum_c \varphi^{\max}(c)\, d(c, p^{uc})$.
null concept	$p^{uc}$	The unconstrained cause-effect repertoire; the sink for destroyed concepts in XEMD.
XEMD	—	Extended EMD between two cause-effect structures ($\varphi$ mass transported at concept-distance cost).
big phi	$\Phi$	XEMD between the CES and the CES under the system MIP cut. The headline scalar of IIT3.

IIT 4.0 terms (module iitx.measures.iit4)

Term	Symbol	Definition
intrinsic difference	ID	$\max_\alpha p_\alpha \log_2(p_\alpha / q_\alpha)$; the unique measure satisfying causality, intrinsicality, specificity. Unit: ibit.
selectivity / informativeness	—	The $p_\alpha$ factor / the $\log_2(p_\alpha/q_\alpha)$ factor of ID-style quantities.
intrinsic information	$ii$	Selectivity × informativeness of a state, against the unconstrained probability (Eqs. 5, 7, 34, 35).
specified state	$s'$, $z'$, $z^*$	The state maximizing $ii$ (system: maximal cause-effect state $s' = {s'_c, s'_e}$; mechanism: per-purview $z'$, over purviews $z^*$).
backward probability	$p_c^{\leftarrow}$	Bayes inversion with uniform prior (cause-side selectivity).
system phi	$\varphi_s$	$\min(\varphi_c, \varphi_e)$ over the normalized-MIP system partition. The headline scalar of IIT4 existence; may be negative (negative ⇒ reducible).
distinction	$d(m)$	A mechanism with its specified cause/effect purview-states and $\varphi_d > 0$, congruent with $s'$.
distinction phi	$\varphi_d$	$\min(\varphi_c(m), \varphi_e(m))$ over disintegrating partitions, max over purviews.
congruence	—	$z^_c \subseteq s'_c$ and $z^_e \subseteq s'_e$ as unit–state sets; incongruent candidates do not exist.
relation	$r(\mathbf d)$	Congruent overlap among the purviews of a set of distinctions.
relation phi	$\varphi_r$	$\lvert \bigcap_d z_d \rvert \cdot \min_d \varphi_d / \lvert z_d \rvert$; summed analytically, never enumerated.
Φ-structure	$C(D)$	Distinctions ∪ relations of a complex.
big phi	$\Phi$	$\sum \varphi_d + \sum \varphi_r$ over the Φ-structure.
monad	—	A single-unit complex.

Macro terms (module iitx.transforms)

Term	Definition
coarse-graining	Unit partition + per-macro-unit state grouping (+ temporal grain); macro TPM by fiber averaging $D^{-1} G^\top T^\tau G$.
grouping	The surjective, identity-irrelevant map from micro-states of a group to macro-states.
black-boxing	Box partition + designated output units (+ temporal grain); macro state = projection onto outputs at the window end; hidden units noised.
hidden / output units	Units of a box invisible to other boxes / read as the box's state.
temporal grain	$\tau$, the number of micro steps per macro step (steps).
supervenience	The macro state is a function of the micro state (holds for both transforms).

Naming rules (enforced by review and ruff)

1. Cause/effect mirror: any name containing cause has an effect twin; shared machinery takes direction: Direction instead of forking code paths.
2. Qualified phis only: phi_c, phi_e, phi_s, phi_d, phi_r, big_phi — a bare phi appears only as the Measure.phi method, whose docstring names which quantity it returns.
3. system everywhere: never network, substrate, or subsystem in public names; a candidate system is a subset mask over a System.
4. Masks and indices: *_mask is (n,) bool over units; *_index is an integer into a canonical enumeration table; state vectors are state, flat indices are state_index.
5. shape means alphabets: the per-unit alphabet tuple, chosen to coincide with the NumPy shape of full state-space arrays.
6. Little-endian always: any function touching state order says so in its docstring; big-endian exists only in explicitly named I/O converters, if ever.
7. Paper symbols in docstrings: every public docstring states the paper equation it implements (e.g. "Eq. 38, Albantakis et al. 2023") so a scientist can map code to theory in one hop.