The Geometry of Interaction
What shape does reality have if reality is made of interactions and nothing else? No space assumed, no time assumed, no coordinates, no dimensions, no numbers beyond counting. The answer below is substantially settled — including a question five earlier attempts got wrong, and which this page previously got wrong in the opposite direction. It also gives the exact boundary: two thresholds below which time and space do not exist, several claims withdrawn under review, and one wall never crossed.
DERIVED — proved. NUMERICAL — established by running it; evidence, not theorem. WITHDRAWN — claimed, then refuted; the refutation is given. OPEN — known to be missing. Nine adversarial reviews have deleted every attempt to blur these — including three claims on an earlier version of this page.
The starting material
There is one time
Elementary interaction happens only at the floor — so there is exactly one informing relation, and therefore exactly one order. Nesting does not create new clocks; it creates nested windows onto that one order. DERIVED.
A thing's time is its window. Take everything a thing is built from, and carry over the order among those events: that is its time. Its duration is the longest chain of completions inside it — a pure count. (Careful: that chain must be built from informings inside the thing, or the number is not its own — see below.) DERIVED, with the intrinsic correction below.
Is that a universal background time — the thing the theory denies?
An earlier version of this page answered "no" with three tests. All three were worthless, and running the control showed it: a literal Newtonian absolute time — the very thing the tests were meant to exclude — passes them, with a wider spread than the theory's own order. WITHDRAWN.
The real question is not whether the order is total or shared, but whether a thing's time is its own. Taken from the global order it is not (see below), and that is a genuine backdrop. Taken from inside the thing it is. That is the test that discriminates, and it is why the intrinsic definition matters rather than being a technicality. OPEN in part — the ambient order fails it; the intrinsic one passes.
Space is where the order fails to relate
Every pair is either ordered (one before the other — time) or not (coexisting — space). Space is not a container; it is the incomparability inside the very same order that gives time. A "moment of space" is a maximal set of mutually coexisting events. DERIVED.
Composite time: five attempts, and the answer that was already written down
Five rounds of work tried to define when one composite thing precedes another — is this molecule before that one? The obvious rule ("everything it is made of comes before everything the other is made of") is a genuine order but relates under 1% of pairs. An earlier round then concluded the question itself was malformed — that extended things simply are not ordered.
That conclusion was wrong, and the evidence for it was misreported. The table published in support showed only that one demanding rule, under a column labelled "any rule." Other rules had already been proved to work. With the omitted columns restored, the contrast disappears:
| interaction density | the strict rule (published alone) | a proved weaker rule | the completion rule | ordering events |
|---|---|---|---|---|
| 0.25 | 0.3% | 21.0% | 11.2% | 32.1% |
| 0.40 | 0.8% | 34.4% | 29.4% | 41.7% |
| 0.60 | 1.1% | 42.3% | 40.9% | 46.9% |
And the right rule was in the foundations the whole time. The theory already says a thing finishes exactly when its parts finish, and that its existence is that last completion. So order things by where they complete:
That is a genuine order (verified over 200 structures, no violations), and it is dense wherever the floor is active. Composite time is solved, and it was never malformed — five rounds missed it by quantifying over everything a thing is made of, instead of over where it finishes. DERIVED + NUMERICAL.
A thing's own time must be built from inside it
One further correction, and it matters. If you take a thing's time from the global order, you get a number that is not really its own: a thing's duration can be changed by chains of interaction lying entirely outside it — its "own" time doubling with no change whatsoever inside. That violates the theory's founding requirement that local time be intrinsic, not borrowed from any backdrop. WITHDRAWN.
The repair: build a thing's time only from the informings inside it. That gives a genuinely intrinsic duration — smaller than the borrowed figure, and real. DERIVED.
What actually seals an interior
Two attempts failed here. The first claimed a thing's inner workings are simply invisible from outside — false, and it had been resting on an assumption retired several revisions earlier without anyone noticing. The second replaced it with "carried, but not addressed to you" — but "addressed" was a word the formalism had never attached to anything, so it was a posit wearing the clothes of a result. BOTH WITHDRAWN.
What is actually true is sharper than either, and it is asymmetric:
The proof is short: a thing's completion is, by definition, the last of it — so nothing inside comes after it, and therefore no chain leaving it can ever re-enter. Everything that thing shows the future is fixed by its completion alone, no matter how its insides are rearranged. But a chain arriving can pass through the interior, so what reaches it does depend on its structure. Verified by perturbation: outgoing relations invariant in 509 of 509 cases, incoming ones changing. DERIVED — and it is the theory's own doctrine: what a thing contributes upward is its completion.
Where space exists — and why the universe must be finite
Space is not everywhere. Two things are spatially related exactly where a chain of interaction connects them. Where none does, they are not "far apart" — they are not spatially related at all. Whether any given world is connected throughout is a contingent fact about that world, not something the theory should prove, exactly as connectedness is contingent for spacetime generally. The theory's job is to say what space is where it exists, and it does. DERIVED (what space is) + contingent (whether it is everywhere).
And the universe must be finite. This one is forced, and it surprised me. The theory says every interaction has finitely many parts, that nothing descends forever, and that a thing finishes only when all its parts finish. Those three together leave no room: if unboundedly many interactions ever completed, there would have to be an infinite descending chain of parts-within-parts — which the theory forbids. And you cannot escape by letting something have infinitely many parts, because then it never finishes, and what never finishes does not exist.
That is a real cosmological commitment, and it is forced rather than chosen — which makes it a result rather than an embarrassment. It also fits the theory's own arc: the Big Bang is the universe's mark being set, the end of the universe is its being received, and between them lies a finite interior. Anyone who wants an infinite universe has to give up either the bottoming-out of matter or the rule that a thing finishes when its parts do. DERIVED.
Time also has a condition
Temporal order at a level rises with how tightly the things at that level hold themselves together — scattered substrate gives almost none, bound substrate several times more. NUMERICAL — a behaviour of random structures, not a theorem.
The geometry, complete
The one wall
Everything above is structure: orders, counts, bounds, thresholds. What is absent is magnitude — no interval, no metric signature, no distance-with-units, and no derivation that space has any particular number of directions. The words "dimension," "worldline," "proper time" and "volume" are deliberately absent: each presupposes structure the theory has not earned, and each has been struck from an earlier draft of this page for exactly that reason.
All of it sits behind the single posit this project has hit from every direction and never crossed: superposition (see The Math, from the ground up). It is not renamed here.
Still open, and named: a horizon result that lost its support and has not regained it — an earlier attenuation argument turned out to give zero rather than decaying resolution. That is a question about what an agent can infer, so it belongs to the epistemics rather than to the geometry, but it is a real debt and it is not paid. The coherence effect on temporal order is numerical, not proved.
The statement of record is 08_GEOMETRY.md — the consolidated formulation, every claim tagged DERIVED / NUMERICAL / POSIT / OPEN, with all ten withdrawals listed. The derivation trail, including the full text of nine adversarial reviews, is formalism/cycle_1.md–cycle_16.md. Numerics: calc/one_time.py, footprint_order.py, cycle13_counterexamples.py. All readable in Docs.