docs/dimension-algebra.md
Dimensions are not names
Dimensions attacked as structures rather than labels, and the binary persistence claim attacked with eight interval phenomena.
Outcome: a dimension reduces to two primitives — a value space and a persistence behaviour. Exclusivity is not a property of dimensions at all. And all eight interval phenomena reduce; the binary survives, at a stated cost.
Part 1 — the algebra
Exclusivity is not a property. It is functionality.
minimal-ontology.md derived that mutually exclusive predicates share a dimension, and treated exclusivity as the criterion individuating dimensions. That was one step short.
The truth structure is
truth: (subjects, dimension, ordinal) → value
and the arrow is a function. A function assigns exactly one value. So a subject having one temperature, one location, one health-status is not an extra axiom about dimensions — it is already entailed by truth being a function to a value.
Exclusivity is functionality. It is not a property of a dimension, not a criterion applied to dimensions, and not something an observer must supply.
This retires the open question left at the end of minimal-ontology.md — "the exclusivity law needs a source; nothing says how exclusivity is known." It needs no source. Nothing knows it; the structure enforces it.
And it re-describes both live defects more sharply. conditionIdentity and
placement:${subject}@${at} do not merely violate a law about dimensions — they
break functionality by moving the value into the key, which is what allows one
package to be in two places at once. A function cannot do that. The code stopped
being a function.
Equality, and where units actually live
A value domain is a set of representations; 60 min and 1 h are two
representations of one value. So equality is not automatic — it is the quotient
saying which representations denote the same value.
But define the dimension over the quotient — the value space, not the representation set — and equality is built in. What then becomes of units?
unitbelongs to the record, not the truth. It is how an observer expressed the value, not part of what is true.
Two independent derivations converge here: identity-axes.md
found by unit conversion that unit must not be in identity, and the truth/record
split in minimal-ontology.md puts observer-supplied detail
in the record. Units are a presentation of the value, and they are misfiled in
MeasureFact.
Ordering, and a naming collision I introduced
Ordering is not required — place, colour and condition-class have none. Where
it exists it is not derivable: nothing in {healthy, degraded, failed} as a
set says healthy > degraded. It must be supplied as domain knowledge. So
ordering is an optional additional structure, primitive when present.
Interpolation derives from it: computing a value between two observations requires an ordering and a dense domain. Temperature has both. Integer counts are ordered but not dense — interpolating yields non-values. Categorical dimensions have neither.
A correction on vocabulary: I used interpolability in truth-kinds.md for "does the truth hold in the gap?", which was reduced to persistence. Value interpolation here is a different question — "can the value between two observations be computed?" Two concepts, one word, mine. They are independent: a state may be non-interpolable in value (location) while holding continuously in time.
Persistence is carried by the dimension
Temperature is always until-change. Transfer is always zero. Persistence is a property of the dimension, not of individual truths — no dimension is sometimes one and sometimes the other.
It does not derive from the other properties. Nothing in a value space says
whether its truths persist; {0…100} serves temperature (state) and quantity
transferred (occurrence) equally. Primitive.
The result
| Property | Standing |
|---|---|
| value space | primitive — the quotient, not the representation set |
| persistence behaviour | primitive — carried by the dimension, underivable |
| exclusivity | not a property — entailed by truth being a function |
| equality | absorbed into the value space |
| units | record layer — a presentation of the value |
| ordering | optional additional structure; primitive where present |
| value interpolation | derived — ordering + density |
A dimension is a value space with a persistence behaviour. Two components. Nothing else is primitive, and one listed property was not a property at all.
Dimensions are indeed not names. A name is what remains once everything
structural has been extracted — which is why open-ended metric: string has been
workable and also why nothing can be checked about it.
Part 2 — do interval phenomena break the binary?
Each has finite, non-zero duration: neither instantaneous nor indefinite. If a third persistence value exists, this is where it is.
The reduction: an interval is an episode of a state.
| Phenomenon | Dimension | Value during the interval |
|---|---|---|
| meeting | meeting-status | in-progress |
| storm | storm-status | active |
| conversation | conversation-status | open |
| chemical reaction | reaction-status | proceeding |
| CPU process | process-state | running |
| radio transmission | transmission-state | transmitting |
| journey | journey-status | in-transit |
| infection | infection-status | infected |
All eight reduce. The CPU process is the confirming case: operating systems
already model it exactly this way, as a state machine over running, blocked,
zombie — arrived at independently, under production pressure, for the same
phenomenon.
Attacking the reduction
"The end may be scheduled in advance." A meeting ends at 15:00 whether or not anyone observes it, so absence-means-unchanged gives the wrong answer afterwards. — The scheduled end is itself a truth. This is the TTL case from truth-reduction.md: two states plus a computation.
"Intervals have interior structure." A conversation has turns, a reaction has steps, a journey has legs. — The parts are their own truths. Containment is a relation, already eliminated.
"The end is vague." A storm dissipates gradually; there is no instant it stopped. — Vagueness is an observation problem. The episode ends when someone observes a differing value. Nothing ontological is at stake.
The strongest: "duration is intrinsic." A 3-hour meeting differs from a 1-hour meeting in a way that belongs to the meeting itself, so interval phenomena have a finite non-zero extent while occurrences have zero and states have none.
— This rests on a misreading of until-change, and it is worth stating precisely
because I could have made it myself. until-change does not mean unbounded.
A state episode ends at the next differing value, so state episodes already
have finite duration. Interval phenomena are not a middle case between zero and
infinite; they are the ordinary case of the state value. Duration is derived as
end − start, and nothing is lost.
The binary survives. All eight reduce, and the strongest objection dissolves on a correct reading of the existing value.
What the reduction costs — stated, not waved past
It is not free, and two things fall out.
It multiplies subjects. Every interval phenomenon needs a subject to bear its status: the meeting, the storm, the journey must each be reified. The ontology is minimal in kinds and generous in subjects. That is a real trade, and it is the same move as occurrences becoming subjects — consistent, but it means "how many subjects exist" is now doing work that "how many kinds exist" used to do.
Interval identity is episode identity, one level up. A storm has no subject until someone judges "this is one storm, not two." That judgement is the same question as "is this one episode or two?" — asked about the subject rather than about its values. So the hardest cases push the identity problem from the value layer to the subject layer rather than solving it.
That is a genuine limit, and it is where I would look next.
Standing
| Concept | Standing |
|---|---|
| persistence law | promoted; binary survives eight interval phenomena |
| dimension = (value space, persistence) | promoted |
| exclusivity | eliminated — entailed by functionality |
| units | relocated to the record layer |
| equality | absorbed into the value space |
| ordering, value interpolation | optional structure; derived |
| the open exclusivity-source question | closed — it needed no source |
What is still not established
capabilityremains untouched — five rounds now, and still the oldest unexamined assumption.- The interval reduction was tested on eight phenomena, not derived. The duration argument is structural; the eight are inductive support for it.
- Subject identity is now the load-bearing unexamined concept, having absorbed work the kind taxonomy used to carry.
Nothing implemented.