The Math, From the Ground Up
Start with a completed interaction. Learn to name it, relate it to others, count ordered completions, and construct a candidate ruler. Each step shows what the notation means, why a result follows, and what remains to be connected to the theory.
How to read the status labels. A commitment comes from the ontology. An assumption supplies mathematical structure. A definition names a construction. A theorem proves something about it under stated assumptions. Open means that the needed connection has not been established. A useful definition is not yet a proof that nature uses it.
1. The interaction comes first
An interaction is the theory's six-item structure. It is not merely an arrow drawn between two independently existing objects.
A sets the mark. Propagation carries it through the substrate. B becomes aware, and the interaction exists. The actual mark has value m(e) = 1: “I exist.” Its value is not computed from other marks. Awareness is primitive and changes the receiving agent; it is not another detection interaction.
Commitment. An unfinished interaction does not exist from the higher interaction's point of view. Its sub-interactions can finish while it remains unfinished. Their completed marks contribute to its internal time. An observer can interact with completed constituents at different scales; smallness does not make them inaccessible.
A Form is a different kind of concept: an ever-changing agent maintained across many interactions and categorized by its Protocols. A human is a Form; a particular interaction in which that human receives a mark is an event. Action maintains a Form. Propagation carries a particular interaction's mark.
2. Sets: descriptions and completed interactions
A set is a collection. Membership, e ∈ E, means “e belongs to E.” To describe a parent whose children have finished before it has, we need separate collections:
Definition. D is bookkeeping, not a collection of extra existing things. It can contain a description of the unfinished One Interaction. Its completed descendants belong to E without making their parent a completed event.
The notation |S| counts members of a finite set S. Naming a set gives us neither a ruler nor a clock. This build uses finite examples and counts; it does not assume the universe is finite. Richer number systems can be constructed with additional mathematics, but that would not by itself establish a physical role for them.
3. Two relations with different jobs
| Notation | Read it aloud | What it records |
|---|---|---|
c ⊏ p | “c belongs to p's Propagation” | Immediate nesting; a parent relation on descriptions. |
x ⇀ y | “x informs y” | A supplied elementary dependency between completed Mass-interactions. |
x ≺ y | “a chain of informing leads from x to y” | The order generated by following one or more arrows. |
Nesting does not mean “earlier than.” It also does not, by itself, route a mark. The down–across–up account of Propagation and its FILO stack must eventually be represented without turning the nesting relation into a second kind of informing.
Definition. L is the completed Mass-interactions: leaves with no sub-interactions. Their A-pole is the Source, by the ontology. This does not yet identify a Mass-interaction with an electron or a numerical mass. Sharing the Source also does not draw an informing path between every pair.
4. Order and local time
Assumption. There is no directed cycle of informing: following arrows never returns to its starting event. Under this assumption, reachability is a strict partial order.
Why is this a theorem?
A self-path would be a cycle, which the assumption excludes. Joining a path from x to y to a path from y to z gives a path from x to z. These are irreflexivity and transitivity, the two required properties.
This is a candidate order of completed elementary interactions. “Partial” means pairs may be unrelated. It does not mean some pair must be unrelated: a single chain also qualifies. A universal clock has not been introduced, but its impossibility has not been proved.
Local means: keep the internal arrows first
Write R for the supplied arrows and S for a specified finite context of completed elementary interactions. The superscript “+” means “follow one or more arrows.”
Read the first line as “keep only arrows whose two ends are inside S.” Then take their closure. This gives an order whose paths stay inside S.
Try three events
The full history orders a before b. Inside S there is no arrow and no path: u is outside. If we first followed the full history and then removed u, we would import an external connection into the internal order.
Result: internal ordering can be weaker than the ordering inherited from a larger context. This describes the difference; it does not say external influence is forbidden.
Open bridge. A parent's completed immediate children and the elementary interactions in their substrate are different sets. We still owe a construction showing how the floor order produces the ordering of the child marks that the parent receives.
5. Count, depth, and breadth
Three numbers answer three questions. A chain consists of events ordered one after another. An antichain consists of pairwise unordered events.
| History | N | h | w |
|---|---|---|---|
| a ⇀ b ⇀ c | 3 | 3 | 1 |
| a, b, c with no arrows | 3 | 1 | 3 |
Both histories have three completed interactions. Only one has three successive completions. Thus “more interaction, more time” needs its ordering specified: adding in parallel need not lengthen a chain.
Count the ranks to see why
Give each event the number of events in a longest chain ending there. Events with the same rank cannot precede each other, so each rank contains at most w events. There are at most h ranks. Counting their members gives the bound. These ranks are not a privileged global clock.
Between comparable endpoints use τS(x,y) for the number of steps in a longest chain. Three vertices in a chain give two steps. A singleton has height 1 and zero steps from itself to itself. No seconds have been introduced.
6. What the last completions tell us
In a finite support S, collect the events with no later member of S. Call this the terminal frontier F(S). “Last” may name several unrelated events.
Theorem. Every member of S reaches or equals a terminal member. Therefore, for a fixed later target q:
Checking b and c suffices in the example: if both precede q, a does too. This makes the terminal frontier useful for expressing completion requirements.
Limit. The theorem does not make q exist, identify it with B's awareness, or prove that awareness is one of the Mass-events. The ontology says all required sub-interactions finish before the parent completes. Connecting that statement to this order test still needs a representation of the receiving boundary. The earlier “composite time solved” claim was too strong.
7. Constructing a candidate spatial ruler
For distinct events, x ∥S y means neither precedes the other in the selected internal order. This is a candidate relation of coexistence between completed interactions. It supplies no distance by itself.
The symbol ⋖ marks an irreducible order link, called a cover. On each connected component, counting these links produces a metric: distance is symmetric, is zero only from an event to itself, and cannot exceed the length of a route through a third event.
Here b and c are unordered. Their distance in the cover graph is 2, via a or d. From a to d, the longest ordered chain also has 2 steps. Every displayed vertex is a completed interaction with mark value 1.
Why use covers? In a ⇀ b ⇀ c, adding a ⇀ c creates a shortcut among supplied arrows without changing the order. The cover distance remains 2. It is determined by the order, so the redundant arrow cannot shorten it.
Here g counts the shortest route using supplied arrows, d the shortest route using covers, and τ the longest ordered chain. The first inequality follows because covers are arrows; the second because a longest chain is itself a cover path.
Candidate, not established physics. We have constructed an integer ruler. We have not shown that it measures physical space, selected units, or derived a spacetime signature. On an antichain A in one component, (A, dS|A×A) is a possible spatial configuration. Its routes may pass through events outside A. For disconnected pairs, this ruler has no distance.
Continue with the precise geometry and its limits →
8. An agent's record and its horizon
Let β(e) identify the agent at e's B-pole. Then Oa = {e : β(e) = a} is a's set of received interactions. This requires a mathematical representation of agent identity. It is not the same as a retained memory.
Read this as: two possible descriptions cannot be distinguished by this record when they yield the same record. A fact is determined only if it is true throughout the descriptions compatible with it.
One receipt does not bound its history
Suppose the record contains only “received a mark of value 1 at r,” with the same recorded attribution and no transit-time or interior trace. Both histories fit. So does a chain of any finite length.
A finite receipt count alone therefore gives no fixed spatial cutoff. To derive a horizon, we need a rule specifying what reaches the agent, what is retained, and how that access relates to the proposed ruler.
This also avoids saying a mark of value 1 conveys nothing: its arrival, attribution and ordering may matter, even though its payload value is constant.
9. The path toward larger concepts
| Stage | Available now | What must be earned next |
|---|---|---|
| Elementary succession | Acyclic arrows generate an order. | Justify their exact connection to delivered marks. |
| Parent time | Internal order and terminal requirements. | Attach shell receipts and B's awareness without confusing nesting with informing. |
| Spatial configuration | An antichain with a candidate integer metric. | Choose its physical meaning, context, and units. |
| Agent horizon | Equality of specified records. | An access/retention law that entails a cutoff. |
| Forms and particles | The ontology's Protocol-profile and Action. | A law of allowed deliveries and persistence; show which patterns it selects. |
| Quantum and gravitational structure | A separate reconstruction explores richer assumptions. | Derive or explicitly add the needed mathematics and test its consequences. |
These are separate questions. They have not been reduced to a single superposition assumption. The top-down reconstruction uses richer mathematical inputs; its results cannot be imported as conclusions of this elementary build.
The next concrete construction: one completed delivery, with its A/B poles, supporting Mass-interactions, and FILO shell receipts. It must reproduce the parent's ordering of child marks before we claim a geometry of higher interactions.
Current proofs: Cycle 17. Consolidated notation: 08 — Geometry. Earlier cycles remain in Docs as a research trail, including withdrawn claims. Checks enumerate finite graphs; they do not stand in for proofs or physical evidence.