The Geometry of Interaction

A geometry needs more than names for time and space. We now have precise constructions for local succession, terminal requirements, and a candidate distance. Their mathematical properties can be proved. Their identification with the interaction's own space, time, and receiving awareness remains work to do.

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Current formulation · Cycle 17

A context of completed interactions
Internal order and external connections
A theorem about terminal requirements
A ruler determined by order
A candidate spatial configuration
Horizon, finiteness, and observation
What survives, and what comes next

Status matters. Definitions choose mathematical objects; theorems establish their properties under named assumptions. Calling those objects physical time or space is a further interpretation. Earlier versions of this page confused these steps and claimed the geometry was settled. That claim is withdrawn.

1. A context of completed interactions

An event means one completed interaction: A, B, local space, local time, mark, and Propagation. Its mark was set by A and carried to B. Awareness is not computed by our graph. A Form is an ever-changing agent across interactions; a cell or a human is not a single event.

D = interaction descriptions; E ⊆ D = completed interactions L = { m ∈ E : ch(m) = ∅ } completed Mass-interactions V ⊆ L finite; R ⊆ V × V acyclic

Assumptions. V is a specified finite context, R its supplied informing dependencies. Acyclic means no directed path returns to its start. Finite context does not mean finite universe. The parent relation belongs to D, allowing completed children under an unfinished parent without making that parent an externally existing event.

For a parent p, its completed immediate children C_p and its elementary support S_p differ. The former are the children whose finished marks order its internal time in the ontology. The latter are the Mass-interactions beneath it. A construction relating these sets is still needed.

2. Internal order and external connections

R_S = R ∩ (S × S) P_S = (R_S)⁺ P_S ⊆ R⁺ ∩ (S × S)

Theorem. P_S is a strict partial order: follow only arrows inside S. Its inclusion in the ambient order follows because every internal path is also an ambient path.

The smallest locality test

a ⇀ u ⇀ b S = {a, b}

The ambient order says a precedes b. The internal order has no comparison. If an additional arrow a ⇀ b is supplied, the two orders agree on S even though the external detour remains.

Agreement holds exactly when each ambient comparison within S has an internal witnessing path. A sufficient condition is order-convexity: all events between two members of S also belong to S. The extra-arrow example shows convexity is not necessary.

This separates internal succession from connections through the surroundings. It does not prohibit external interaction or prove that an interior is unobservable.

Count is different from sequential depth

N = number of events; h = longest-chain vertex count w = largest-antichain count; N ≤ h × w τ_S(x,y) = longest-chain step count, for x P_S y

A three-event chain has N = 3 and h = 3. Three unordered events have N = 3 and h = 1. Between the endpoints of the chain, τ = 2. The height-width bound follows by partitioning events by longest-chain rank. These are counts, not physical seconds or volume.

3. Terminal requirements

F_P(S) = { s ∈ S : no t ∈ S has s P t }

This terminal frontier is the maximal part of a finite nonempty support in a specified order P. It can contain several unrelated events.

∀s ∈ S : s P q ⇔ ∀m ∈ F_P(S) : m P q

Theorem. To establish that every event in S precedes a fixed target q, it suffices to check the frontier. Every event reaches or equals a maximal one; transitivity supplies the rest. The frontier is also the unique smallest subset of S above every member of S.

What this earns: an exact reduction of completion prerequisites, if the physical prerequisites are represented by this order test. What it does not earn: a common target, its uniqueness, or its identity with B's awareness. Even one maximal Mass-event is not automatically the awareness of a higher interaction.

What remains of the previous “completion lift”?

S ⊲ T ⇔ every member of F_P(S) precedes every member of F_P(T)

This is a strict partial order on finite nonempty supports. Irreflexivity follows by selecting the same frontier member twice. For transitivity, select any member of the nonempty middle frontier. The proof survives; choosing this as the order of composite interactions remains a modeling step. C10-d, the bridge to the parent's received child marks, is open.

4. A ruler determined by order

The shortest route through supplied arrows depends on which arrows were supplied. A second count can depend on the order alone.

x ⋖_S y ⇔ x P_S y and no z has x P_S z P_S y d_S(x,y) = shortest undirected path of ⋖_S links

A link with no intermediate event is a cover. In a finite context, covers reconstruct the order. Counting paths in their undirected graph gives an integer metric on each connected component.

Why it is a metric

A path from a vertex to itself can have zero steps; a path between different vertices cannot. Reversing a path proves symmetry. Joining shortest paths from x to y and y to z supplies a route from x to z, so the shortest such route is no longer than their sum. This is the triangle inequality.

Supplied arrows	g(a,c): arrow distance	d(a,c): cover distance	τ(a,c)
a ⇀ b ⇀ c	2	2	2
Same, plus a ⇀ c	1	2	2

The added arrow preserves the order. Therefore it preserves the covers and their distance. It may still be a genuine extra informing dependency; only its contribution to reachability is redundant.

g_S(x,y) ≤ d_S(x,y) ≤ τ_S(x,y) for comparable endpoints in a finite context

Proof. Every cover must be a supplied arrow, or its witnessing path would contain an intermediate event. Conversely, a longest chain permits no insertion and therefore consists of covers. The two bounds follow by minimizing over paths.

This candidate ruler is invariant under relabeling and under redundant-arrow changes preserving the order. It is not invariant under arbitrary changes of context. For disconnected vertices no distance is defined. Nothing here selects equal physical link lengths or a unit of measurement.

5. A candidate spatial configuration

x ∥_S y ⇔ x ≠ y and neither x P_S y nor y P_S x A = an antichain within one connected component candidate configuration: ( A, d_S|_A×A )

Four equal-valued marks, one branching order. The drawing's positions are for legibility; no coordinates or physical lengths are assumed.

In this diamond, A = {b,c} is an antichain and d_S(b,c) = 2. Routes go through a or d; they are measured in the entire context, not just the antichain. Also N = 4, h = 3, w = 2 and τ(a,d) = 2.

All four interactions are completed. This does not identify space with unfinished interactions. It provides a possible representation of coexistence with separation. Incomparability is not transitive and does not select a unique “now.” A path may pass through a shared future as well as a shared past; it is not automatically a physically traversable signal route.

6. Horizon, finiteness, and observation

A reception record needs a rule

O_a = { e ∈ E : β(e) = a } receptions at agent a H ∼_a H′ ⇔ ρ_a(H) = ρ_a(H′) equal retained records

β represents receiving-agent identity; ρ specifies what is retained. Neither is supplied by an unlabeled floor order. A record determines a property only if every admissible description compatible with it agrees on that property.

For any n, a chain of n arrows can end at one recorded receipt r. If the record retains only that receipt, its value 1, and fixed attribution, with no transit-time or interior trace, arbitrarily long chains give the same record. A finite record alone does not imply a finite spatial horizon. An access/retention law linked to distance is still required.

Finiteness is conditional

A whole rooted description tree is finite if every node has finitely many children and downward nesting is well-founded. That conditional theorem survives. The completion barrier alone does not impose finite branching or forbid an infinite family from sharing a completion bound. An unfinished root is not covered by a rule stated only for completed events. The previous unconditional cosmological conclusion is withdrawn.

Graph invariance is not invisibility

A repaired finite-graph theorem shows that preserving a support's ambient terminal frontier while changing only its internal arrows preserves its outgoing frontier comparisons, with the vertices, supports and exterior fixed. A bounded critic review supplied the missing argument for target frontiers. This is a useful mathematical invariance. It does not prevent an agent from observing completed constituents or inferring their arrangement: those questions require an observation model.

7. What survives, and what comes next

Established mathematics	Unresolved interpretation
Intrinsic closure gives a partial order.	Its relation to the parent's received child marks.
Terminal requirements suffice for an all-before test.	The boundary representing B's awareness.
Cover paths give an order-based integer metric.	Its selection as physical space, with units.
Equal records define indistinguishability.	A physical law producing a finite horizon.

The next construction is one completed delivery with explicit A/B poles, supporting elementary interactions, and FILO shell receipts. It must preserve mark-carrying and explain the ordering of the children's completions. Particle formation, dynamics, and quantum structure remain separate tasks; they are not all consequences of one missing superposition assumption.

Read the consolidated formulation or Cycle 17's proofs, counterexamples and review. Deterministic checks enumerate all forward-edge DAGs through five vertices, with no random sampling. General proofs, rather than those finite checks, support the theorem claims.