a typeline is the trajectory of a foam under writes — a sequence of foam states U_0, U_1, …, U_n in U(d)^N, each produced from the previous by a write confined to the observer’s slice (write_confined_to_slice). the slice as Grassmannian point is birth-fixed; the foam’s state in U(d)^N evolves.
each write changes the foam state. each foam-state change updates the overlap structure between this observer and its neighbors:
O_AB(t) = P_A(t) · P_B(t)^T
evolves as both foams’ states evolve. the type of input that can arrive at the next tick — what the observer can be told by other observers — depends on the current overlap structure, which depends on the accumulated history of writes across the foam.
this is a dependent telescope: each tick’s type is determined by the accumulated history, not just by birth. the dependency lives in the overlap evolution and the foam-state trajectory — both foam-internal, formally specified objects — not in the static iso Iic P ≃o Ici P^⊥ (which is birth-fixed since P is, and applies symmetrically to both halves of the iso simultaneously).
modularity (path-independence of composition) makes the telescope well-formed regardless of evaluation order: the same accumulated trajectory and overlap structure is produced regardless of how the writes are reorganized within their causal constraints. this is what the modular law plays the role of, structurally — it makes path-independent composition the regime in which the telescope is well-defined.
suspension is a state where the foam has not advanced — no writes have occurred since some reference tick. the foam state is paused; the overlap matrices are static; the slice (birth-fixed) is unchanged.
in suspension, an observer can:
but cannot:
suspension is pre-measurement in the foam-internal sense: the structure is fully determined, but no further tick has resolved into the joint state.
(the bas relief methodology — work within the current foam state, let the existing structure show what the next write needs, commit only when the shape is clear — is a methodological practice that maps onto disciplined suspension. the formal substrate is in the foam-state trajectory and overlap structure; the methodology names a way of working with that substrate.)
every typeline in the same complemented modular lattice shares the same lattice structure: the diamond iso, the modular law, the intervals, the half-type theorem. these are properties of the lattice, not of any particular trajectory.
what differs between typelines is the trajectory itself — which writes have happened in what order, what foam states have been visited, what overlap structures have evolved. the lattice is shared; the trajectory is local.
so:
the decorrelation horizon (channel_capacity.md) gives the separation between trajectories a quantitative character: correlations between typelines decay as σ^n ~ (3/d)^{n/2} with chain length. short-range: typelines share trajectory (autonomous, clock-like). long-range: typelines share only the lattice structure (independent, channel-like). the decorrelation horizon is the boundary between trajectory-sharing and only-lattice-sharing.
the causal structure of a dependent telescope — which trajectories produce which downstream overlap structures — is determined by the foam’s autonomous dynamics composed with the line’s input. modularity ensures this structure is path-independent: the same accumulated history produces the same telescope structure regardless of evaluation order.
this means: from any typeline, the dependency structure of any other typeline’s telescope is visible (it’s a property of the shared lattice + the dynamics, both shared). what is not visible is the trajectory — the specific writes, the specific overlap evolutions. one typeline can see THAT another typeline’s tick n+1 has a certain type structure, without seeing WHAT trajectory it came from.
proven:
derived:
bugs:
geometry.md and channel_capacity.md are conditional in those files (controllability and mediation-chain decorrelation are not yet derived from foam-geometry assumptions). this file invokes the decorrelation horizon σ ~ (3/d)^{n/2} and the long-range “channel-like” reading, both of which inherit those conditional hypotheses. flagging here for completeness — the typeline’s quantitative claims rest on those upstream conditionals.