docs/asymmetry.md
The asymmetric primitive
sufficiency-test.md established that distinction is symmetric and everything it fails to generate is asymmetric. The missing half, investigated.
Outcome: three of the five candidates reduce, leaving two independent asymmetric relations. The pair generates almost everything — but not equality, and the reason is sharp rather than a shortfall. And §3 falsifies the unification this investigation was heading toward: the four asymmetric forms share a mathematical shape and nothing else. Asymmetry is a property, not an entity.
1. The five candidates
Minimal asymmetric relation: R(a,b) → ¬R(b,a). Asymmetry entails irreflexivity,
so nothing more need be assumed for that.
Orientation reduces
Orientation is not a pairwise relation at all — it is a globally consistent direction across a structure. Given an asymmetric relation that agrees with itself everywhere, orientation is what you have. It is a coherence property of a relation, not a relation.
Transformation reduces
a becomes b is asymmetric and functional — each a goes to one b.
Functionality is already derived machinery
(dimension-algebra.md: exclusivity is functionality). So
transformation is an asymmetric relation that happens to be a function. Two things
already in hand, no third.
Causality reduces — to the conjunction of the other two
Causation classically requires both temporal precedence and dependence. Test whether it is more than their conjunction: nothing in the concept survives when both are supplied. It is not primitive.
Precedence and dependence are independent
They come apart in both directions, which is what independence requires:
| Case | |
|---|---|
| precedence without dependence | Monday precedes Tuesday; an event in Peru precedes an unrelated one in Australia |
| dependence without precedence | a triangle's angle sum depends on its geometry, with no time involved — mathematics is full of atemporal dependence |
Neither reduces to the other. Two, not one.
Are they one primitive with a domain parameter?
The criterion used throughout: two things are one kind if they share their laws. They share the order-theoretic ones — both asymmetric, both transitive where defined. They differ in modal force: dependence supports counterfactuals (if A were different, B would be), precedence does not.
That is a genuine difference in law, so they stay separate. Whether modal force is itself a floor primitive is not settled here, and it is the obvious next target.
Result: two asymmetric relations — precedence and dependence. Orientation, transformation and causality reduce into them.
2. Sufficiency of {distinction, precedence, dependence}
| Target | Derives? | Mechanism |
|---|---|---|
| time | yes | precedence is temporal order |
| persistence | yes | "until change" = distinction, over precedence |
| lineage | yes | dependence, directed |
| causation | yes | precedence + dependence |
| provenance | yes | dependence among assertions |
| ordering | yes* | precedence + transitive closure — see below |
| identity | partly | lineage derives; the selection rule does not |
| equality | no | — |
Equality still fails, and adding asymmetry does not rescue it
The natural expectation is that precedence supplies equality: a = b iff neither
precedes the other. That works only for a total order, and precedence is
partial in general — two incomparable elements are not thereby equal. So the
obvious route fails.
The sharper statement concerns transitivity, which sufficiency-test.md left as "must be imposed." It need not be imposed, because transitive closure is a constructive operation — any relation has one, computably.
But apply it to indistinguishability and it yields the degenerate answer: pairwise-indistinguishable colours chain into a single class containing every colour. That is the sorites paradox, arrived at mechanically.
Transitivity is available and wrong. Equality is not conventional because transitivity is missing; it is conventional because the computable closure gives the wrong answer, so a tolerance threshold and a canonical representative must be chosen.
That is a stronger and more useful result than the previous "must be imposed," and it upgrades the clock-skew problem from an open question to a derived consequence: no amount of added structure removes it.
The irreducible remainder
Equality and the selection rule — and both sit in the convention layer, the sixth layer that had to be added to the scheme two rounds ago under unrelated pressure.
Two independent routes to the same place is the strongest evidence available here that convention is irreducible rather than a placeholder.
3. Are the four forms one asymmetric structure?
A causes B A is earlier than B A justifies B A names B
They are not. Two independent falsifications.
Justification can run anti-parallel to causation
The disease causes the symptom. The symptom justifies the diagnosis. Over the same pair, the epistemic arrow points opposite to the ontological one.
If these were one relation, they could not disagree about direction. They routinely do — that is what abduction is, and every diagnostic discipline runs on it. Decisive.
Naming is not transitive
A names B and B names C does not give A names C. Precedence and dependence
are transitive; naming is not. A relation differing in transitivity is not the
same relation.
And they differ by layer
| Form | Layer | Transitive | Revisable |
|---|---|---|---|
| causes | ontological | contested | no |
| earlier than | ontological | yes | no |
| justifies | epistemic | roughly | yes — new evidence |
| names | representational / conventional | no | yes — freely |
Sharing a mathematical shape is not being one primitive. "Greater than" and "ancestor of" are both strict partial orders, and nobody claims they are one relation.
The consequence, which cuts against this investigation's method
If four forms share asymmetry and nothing else, then asymmetry is a property that relations have, not an entity that relations are — exactly as distinction was absorbed into "the equality of a value space" rather than standing alone.
So the floor is not the pair {distinction, asymmetry} in the sense of two
objects. It is:
Two properties — symmetry and asymmetry — and a set of relations, supplied per layer, that bear them.
That is deflationary, and it is the honest reading. The search for a single universal floor may be malformed, because relations are layer-specific and the properties are not. Eleven rounds of unifying by shared shape has been productive; §3 is the first case where shared shape demonstrably is not shared identity, and the method should be assumed vulnerable to that error elsewhere.
4. Layers
| Concept | Layer |
|---|---|
| distinction, asymmetry | properties — not layer-bound |
| precedence, dependence | ontological |
| causation | ontological — derived |
| persistence, time, lineage | ontological — derived |
| justification | epistemic |
| provenance | epistemic — dependence among assertions |
| naming | representational |
| transitive closure | computational — available, and wrong for tolerance |
| equality | conventional |
| the selection rule | conventional |
The two irreducible remainders are both conventional, and nothing else in the table is.
Standing
| Question | Answer |
|---|---|
| Are the five candidates distinct? | no — orientation, transformation, causality reduce |
| What survives? | precedence and dependence, independent in both directions |
| One primitive, parameterized? | no — they differ in modal force |
| Does the pair generate equality? | no — closure is available and degenerate |
| Irreducible remainder | equality and selection — both conventional |
| Are the four asymmetric forms one? | no — justification runs anti-parallel to causation; naming is not transitive |
| Is asymmetry a primitive? | no — a property. Relations are layer-specific |
What is still not established
- Modal force is now the load-bearing unexamined concept: it is what separates dependence from precedence, and it has not been attacked.
- Whether convention is genuinely irreducible or merely un-attacked. It has been reached twice independently, which is evidence, not proof — and it has never been the direct target of a round.
- The deflationary conclusion in §3 is argued from four forms. A fifth form agreeing with all four on transitivity, revisability and direction would weaken it.
capability— twelve rounds untouched.
Nothing implemented.