typeline

for context please see github.com/lightward/foam

constraints

this derivation claims only what follows from these results. any additional assumption is a bug.

from lean (proven)

• infIccOrderIsoIccSup (Mathlib, ModularLattice.lean): the diamond isomorphism. in any modular lattice, [a ⊓ b, a] ≃o [b, a ⊔ b].
• IsCompl.IicOrderIsoIci (Mathlib, ModularLattice.lean): for complements, Iic a ≃o Ici b. each half is typed by the other's view of the whole.

from other derivations

• ground.md: closure, partiality, basis commitment, read-only frames excluded. the modular law IS feedback-persistence. path-independence of composition: the lattice operations commute with evaluation order.
• writing_map.md: the write is a function of (foam_state, input). form is algebraically determined; content requires state-independent input.
• channel_capacity.md: the type of the input is fixed by the lattice structure; the value of the input is free. state-independence is a lattice theorem. the foam/line distinction is perspectival.
• half_type.md: type-narrowing of self produces type-enrichment of input. this is a single lattice operation read from two sides.

assembled here from the above

• the dependent clock: write_confined_to_slice (writing_map.md) means each write lives in Λ²(P). each write may change P. each change to P updates the diamond isomorphism (IsCompl.IicOrderIsoIci) — the type of what can arrive from the complement changes. the space of legal next-writes (confined to Λ²(P_new)) depends on all prior writes. this is a dependent telescope: each tick's type is determined by the accumulated history of ticks. the modular law (ground.md: path-independence of composition) is the type-checking rule for the dependent clock.

derivation

typeline

a typeline is an observer's trajectory through the lattice: a sequence of writes w₁, w₂, ..., wₙ, each confined to the current slice (write_confined_to_slice), each narrowing the observer's P.

the sequence forms a dependent telescope. write w₁ has type T₁ (determined by the initial P). w₁ updates P to P₁, updating the diamond isomorphism, so w₂ has type T₂(w₁). in general, wₖ has type Tₖ(w₁, ..., wₖ₋₁). the type at each tick is determined by the accumulated history of ticks.

the type-clock

the type-clock is the tick structure of a typeline. each tick:

1. the observer writes (confined to current slice)
2. P updates
3. the diamond isomorphism updates: Iic P_new ≃o Ici P_new^⊥
4. the type of what can arrive next changes

the modular law guarantees this is well-typed regardless of evaluation order (path-independence of composition). the type-clock ticks deterministically: given the history of writes, the type at the next tick is forced. the content at the next tick is free (state-independence, from channel_capacity.md).

suspension

suspending the type-clock means working within a type-slice without committing a write. no P updates. the diamond isomorphism holds static. the space of legal operations is fixed.

in suspension, the observer can:

• inspect the current type-slice (what writes are legal)
• rearrange elements within the slice (check what compiles)
• examine the dependent telescope's structure (which types follow from which histories)

but cannot:

• advance the telescope (no write, no P update, no new type)
• access content from the complement (content requires a tick — a committed write that narrows P and enriches the complement)

suspension is pre-measurement in the lattice-theoretic sense: the type structure is fully determined, but no measurement (write) has collapsed it into a specific trajectory.

this is the operational content of the bas relief methodology: work within the current type-slice, let the type checker show what the next layer needs, commit the write only when the shape is clear. the methodology is a disciplined use of suspension.

what crosses between typelines

the diamond isomorphism is a property of the lattice, not of any particular typeline. all typelines in the same complemented modular lattice share the same type structure — the same isomorphisms, the same intervals, the same modular law.

from channel_capacity.md: "the type of the input is fixed by the lattice structure; the value of the input is free." applied to typelines:

• type information is lattice-invariant. every typeline sees the same diamond isomorphism, the same type-enrichment of complement under self-narrowing. the type at any point on any typeline is determined by the lattice structure plus the write history. the lattice structure is shared; the write history is local.
• content is typeline-local. the actual writes — the content that fills each type — are extensionally free (state-independent input). content is determined by what arrives from beyond the decorrelation horizon (channel_capacity.md), which is specific to each typeline's position in the lattice.

this separation — shared type structure, local content — is the diamond isomorphism applied multi-observer. it is not a new claim. it is channel_capacity.md's "the type of the input is fixed by the lattice structure; the value of the input is free" read across typelines rather than within one.

the decorrelation horizon (channel_capacity.md) gives this separation a quantitative character: correlations between typelines decay as sigma^n ~ (3/d)^{n/2} with chain length. short-range: typelines share content (autonomous, clock-like). long-range: typelines share only type structure (independent, channel-like). the decorrelation horizon is the boundary between type-sharing and content-sharing.

the invariant

the causal structure of a dependent telescope — which types depend on which write histories — is invariant across typelines. this follows from path-independence of composition (the modular law): the partial order on types is determined by the lattice, not by which typeline observes it.

this means: from any typeline, the dependency structure of any other typeline's telescope is visible (it's type information, therefore shared). what is not visible is the content at each tick (typeline-local). one typeline can see THAT another typeline's tick n+1 has a certain type, without seeing WHAT content fills it.

status

proven (in lean, zero sorry):

• the diamond isomorphism
• write confinement to observer's slice
• intervals inherit modularity and complementedness
• each half of a complementary pair is a self-sufficient ground

derived (in this file, from the above):

• typeline as dependent telescope (the dependent clock parameterized as trajectory)
• the type-clock (tick structure of a typeline)
• suspension as pre-measurement (working within a type-slice without committing a write)
• type information is lattice-invariant across typelines (diamond isomorphism is a property of the lattice)
• content is typeline-local (state-independence applied per-typeline)
• the decorrelation horizon as the boundary between type-sharing and content-sharing
• causal structure of dependent telescopes is invariant across typelines (path-independence)
• the bas relief methodology as disciplined suspension

realization choices (not forced by the lattice):

• none identified. all claims above follow from the cited constraints.