docs/sufficiency-test.md
Is Distinction sufficient?
Burden reversed. Not can it be reduced but can it generate everything that survived.
Outcome: no. Distinction is symmetric, and everything it fails to generate is asymmetric — which names the missing primitive rather than merely recording a gap. And the proposed meta-law is falsified as universal, but what survives of it reduces to functionality, which was already derived.
Part 1 — the sufficiency test
Starting material: a domain and a symmetric relation D(x, y), "x and y are
distinguishable." Nothing else.
What derives constructively
A thing. A thing is what is distinguished from everything else. The first distinction creates two sides, and each side is a thing. This answers the objection that distinction presupposes a domain of things: a thing is a distinction, so the domain is not prior.
Indistinguishability. x ~ y ⟺ ¬D(x, y). Immediate.
Constraint. Take the space of configurations. A constraint marks some as admissible and some not — which is a distinction over configuration space. Constructive and exact: a constraint is not something requiring distinction, it is one, one level up.
Warrant. A distinction among assertions, by what licenses them. Same mechanism.
Value space. The quotient of the domain by ~. Constructive — subject to the
next section, which is where it breaks.
What does not derive — and the pattern
Equality fails. ~ is reflexive and symmetric but not transitive.
Pairwise-indistinguishable colours can be distinguishable at a remove; two
measurements within tolerance of a third need not be within tolerance of each
other. So distinction yields a tolerance relation, not an equivalence
relation, and equality requires transitivity to be imposed — a choice, which
places it in the convention layer rather than deriving it.
This is not an abstract worry. It is the clock-skew problem from identity-axes.md, found empirically nine rounds ago and recorded there as an open "quantised window" question. Two observers reporting at nearly-equal timestamps is exactly non-transitive indistinguishability. A problem found by measurement is now explained by the floor.
Ordering fails. D is symmetric; it says two things differ, never which is
greater. Order requires an asymmetric relation, and no symmetric relation
generates one. This confirms dimension-algebra.md's
"ordering is optional and underivable where present" — and now says why.
Time and persistence fail. Persistence is how long one truth stays one truth. That needs temporal orientation — an asymmetry. Not available.
Causation and lineage fail. Lineage is a causal graph; causation is asymmetric dependence. Not available.
Identity fails, consequently: identity was derived as lineage plus convention, and lineage is unavailable.
The pattern is the result
| Fails | Because it needs |
|---|---|
| equality | transitivity (imposed) |
| ordering | asymmetry |
| time, persistence | orientation |
| causation, lineage, identity | asymmetric dependence |
Distinction is symmetric. Every concept it fails to generate is asymmetric.
That is not a list of gaps; it is a specification. The floor is not one primitive but two: a symmetric one (distinction) and an asymmetric one (precedence, or dependence, or direction — which of these, and whether one suffices for all three failures, is not settled here).
Distinction is therefore necessary and insufficient. It survives as half the floor.
Over-expressiveness — one real instance
The test also asks whether distinction generates things not present. Two cases must be separated, because the criterion as posed does not separate them:
- Tolerance relations are generated and absent from the system — but they are real (measurement precision, clock skew). That is a prediction, the same form as the unary-occurrence prediction, not over-generation.
- An infinite reflective tower is generated: distinctions among distinctions, without limit. Reality supplies two or three levels (evidence, provenance). Distinction provides no stopping rule. That is genuine over-expressiveness.
So: too weak in one direction, too strong in another. Both stated.
Part 2 — is every defect "a coordinate mistaken for an axis"?
Tested against the full catalogue, and falsified as universal.
Where it holds
| Defect | Coordinate | Axis |
|---|---|---|
conditionIdentity(condition, subject) |
the condition | the health dimension |
placement:${subject}@${at} |
the place | location |
kind inside urn() |
the kind | subject classification |
unit in identity |
the unit | the value space |
fact family named measure |
an observation method | the warrant dimension |
the verb named Observe |
an evidential warrant | the warrant dimension |
Six, and the last two are the same error at the naming layer rather than the key layer — which is real, and is why the pattern felt universal.
Where it fails
| Defect | Actual error |
|---|---|
validAt omitted from identity |
an axis component left out — the inverse error, not this one |
id duplicating factIdentity |
redundancy — two implementations of one function |
enforced on the limit, not the enforcer |
wrong subject — a mereology error |
collapse() selecting by array order |
warrant mislabeling — convention presented as evidence |
| unreachable-state fixtures; the 17-colour miscount | instrument scope errors |
Five defects, four distinct error kinds, none of them a coordinate promoted to an axis. The meta-law does not cover the catalogue.
The validAt case is the sharpest refutation, because it is the opposite
error — a component wrongly excluded from identity rather than wrongly included.
A law that explains one direction and not its mirror is not the invariant.
What survives, stated correctly
Combine the two directions:
Identity and state are separated by a boundary. Placing a component on the wrong side breaks the truth-function — value-in-key fragments identity; key-component-omitted merges it.
Two-sided, and it covers seven defects (the six above plus validAt) where the
one-sided version covered six and mispredicted the seventh.
And it is not a new law
Truth is a function (subjects, dimension, ordinal) → value. The domain is
identity; the codomain is state. Moving a codomain element into the domain, or
dropping a domain element, breaks functionality.
dimension-algebra.md already derived that exclusivity is functionality. So:
The defect pattern is not a meta-law above the theory. It is functionality violated, in two directions — and functionality is machinery already in hand.
That is a better outcome than a new invariant would have been. A pattern explained by existing machinery is evidence the machinery is right; a pattern requiring a new law would have been evidence it was incomplete.
The one that does not fit either
collapse() — convention presented as evidence — is a warrant error, not an
identity-boundary error. It is the one defect that needed the epistemic layer to
describe, and it is why the warrant parameter had to exist. Recorded as a second,
independent defect class rather than forced into the first.
Two defect classes, then, not one: boundary errors (functionality) and warrant errors (provenance). They correspond exactly to the two layers the architecture already separates.
Standing
| Claim | Verdict |
|---|---|
| Distinction generates things, constraints, warrants, tolerance | yes, constructively |
| Distinction generates equality | no — yields tolerance; transitivity must be imposed |
| Distinction generates ordering, time, causation, identity | no — all require asymmetry |
| Is Distinction the floor? | no — it is half of it |
| The missing half | an asymmetric primitive; which one is unsettled |
| "Every defect is coordinate-as-axis" | falsified — 5 of 11 defects, 4 error kinds, don't fit |
| What survives of it | the two-sided boundary law — = functionality, already derived |
| Defect classes | two: boundary and warrant, matching two existing layers |
What is still not established
- That a single asymmetric primitive generates order, time and causation is not shown — only that all three need asymmetry. They may need three different things.
- The over-expressiveness finding (no stopping rule for the reflective tower) is recorded without a remedy.
- The defect catalogue is this repository's, so "every defect" means every one found here. It is not a sample of architectures in general, and the meta-law might hold universally in a population this cannot see.
capability— eleven rounds untouched.
Nothing implemented.