15.13.6 What the grammar constructors mean experimentally

15.13.6 What the grammar constructors mean experimentally

The URL–3 and URL–4 constructors are operational abstractions of observable defects:

Base

the current observation–action interface is sufficient for the registered one-step behavior;

Delay(E)

a lagged witness remains after fitting \(E\), so an additional layer of observable history is required;

Time(E)

behavior depends on non-unit holding time rather than only on the next-state sequence;

Mix(E)

a registered mixture witness indicates branching or probabilistic effect structure not represented by \(E\).

These are not definitions of delay, time, or probabilistic mixture in every coalgebraic category. They are the semantics of the finite experimental instrument. A lag-\(k\) probe returns one with probability 0.72 when at least \(k\) delay layers are present and 0.08 otherwise. The corresponding rates are 0.70 versus 0.04 for time and 0.68 versus 0.05 for mixture. The asymmetry makes the probes informative but fallible.

URL–2 has a similarly explicit intervention menu. An obs probe chooses a persistent reset, an alias-separating action, and a history-sensitive readout; a time probe selects a duration-sensitive reset and action; a joint probe combines them. When the observation constructor is truly required, the obs probe reports a history witness with probability 0.75, compared with 0.08 when it is absent. The time probe reports a duration witness with probability 0.70 when required and 0.03 otherwise. The active learner chooses the instrument with greatest expected posterior information gain; the passive control cycles through all four.

URL–4 adds ordered experiments. If Time and Mix are both present but commute under the registered instrument, either order returns a compound witness with probability 0.18. If the hidden theory contains \(\chi _{\mathrm{Time},\mathrm{Mix}}\), the matching Time>Mix probe fires with probability 0.78 and the reverse probe with probability 0.10. If either constructor is absent, the rate is 0.05. The claimed discovery of a comparison cell is therefore grounded in a repeatable directional asymmetry, not in the name assigned to the world.

Boundary: One URL–4 discovery trace. In a representative Time(Mix(Base)) world, positive marginal time and mixture observations first moved posterior mass to that base expression. Repeated Time>Mix experiments then produced the event pattern \(0,1,1,1,0,1,1\). After the sixth positive directional witness, the posterior of Law(Time>Mix) reached 0.956 and crossed the preregistered admission threshold. The reverse orientation and the no-law composite remained explicit competitors until the evidence separated them.