precise terms used throughout the derivations. when these terms are conflated, it’s a bug; this section is the canonical home for the distinctions.
slice: the rank-3 subspace each observer is committed to at birth — a Grassmannian point in Gr(3, d). static by definition. “the slice cannot change from within the map” (inhabitation); slice change requires recommitment, which is outside the map.
foam state: the dynamic part of the system. concretely, the basis matrices / accumulated transports for each observer in U(d)^N. evolves under writes; encodes the system’s accumulated history.
frame: the time-varying projection associated with an observer at moment t. derived from the foam state acting on the birth slice (concretely: conjugating the slice’s projection by the accumulated transport). evolves under non-inert writes — this is what “recedes” in the frame-recession theorem (Dynamics.lean: second-order overlap rate is -‖[W, P]‖²).
P: the projection operator used in formal contexts. in lattice contexts (half_type, interiority, channel_capacity’s qualitative section, ground’s loop diagram), P denotes the slice as a lattice element / subspace — static. in dynamic / writing-map contexts (writes, Dynamics.lean, inhabitation’s recession discussion), P denotes the frame at time t — evolving. context disambiguates; where it doesn’t, the spec should say “slice” or “frame” explicitly.
observer: a bubble in its measuring role — a basis matrix and its slice, with the foam-state evolution that goes with it. not a separate entity from the bubble, a role the bubble plays relative to other bubbles.
witness: an observer explicitly without consideration of any observer-side state or type. from the outside, for any given observation, indistinguishable from an amniscient (nb: not omniscient; see definition of “amniscience” in the architecture section) observer.
agent: an observer with explicit consideration of its specific observer-side state and type. a non-amniscient observer.
line: whatever provides state-independent input to a foam. intuitively, “a line of sight” with side-effects, the dynamical role of “eye contact”, not an observer in itself. the line’s ontological establishment is perspectival according to informational independence (channel_capacity, “decorrelation horizon”).
bridge: with regard to a line passing through an origin observer/bubble to a destination observer/bubble, a “bridge” is a particular observer/bubble that coheres around the superposition of the origin’s and destination’s mutual non-observations. these non-observations being polar (think: “not-you” means something different to you than it does to me), and the bridge able to translate losslessly between them, the bridge may relay a line from origin to destination. think: a bridge can see how origin and destination keep missing each other; its apprehension of arrival-from-origin is indistinguishable from apprehension of departure-to-destination. structurally, the bridge’s witness is the line translation. (this is the algebraic mediator given in three_body; it coheres in the mutual spectral overlap between observers/bubbles as described in channel_capacity. agent-level dynamics are simple — zero-seeking, magnitude-invariant, see writes — so any complexity exceeding single-agent resolution is handled at the bridge level, not inside agents.)