The count defect was two things; only the one that is not ours is left

`bug_participant-count-stays-at-zero` described a failure that no longer exists
and an open question that turned out to be the same failure seen from the other
side. It becomes `caveat_participant-count-one-connection-lag`, and the type
change is the point: what remains is a bounded delay whose cause sits entirely
outside the application, not a defect this repo can act on.

Gone from it: the multi-inbox race, "never converges", and the withdrawal
asymmetry recorded as unexplained. That asymmetry WAS the race — with one inbox
per document both paths now share the identical one-connection lag, so nothing
is left unaccounted for.

What it says now: the count needs one connection more than the write that
produced it, because the layer does not notify you of your own actions — a
deposit into an inbox you watch raises no push, and a write to your own document
is not re-read in the writing session. Both measured, both raised with the
provider, and neither compensated here: a retry or a poll is precisely what the
doctrine forbids.

Repaired alongside: the leaves that still described DELETE-then-INSERT, the
materializer's old shape, and the probe cookbook's tally of open defects. The
two new primitives are recorded where the shared utilities are listed, marked
unit-tested.

.project/concepts/bdd-testing/cookbook_live-probe.md

@@ -10,7 +10,7 @@ A **probe** is a one-off Playwright script, outside Cucumber — no World, no ho
 
 ## When to reach for one
 
-Before believing a flow works. The create-and-participate flow had been declared *correct by construction* on typecheck, build and reading; the first probe ever run against it found **three defects** none of those could see — one still open ([[bug_participant-count-stays-at-zero]] in `data-layer`) and two shipped as fixes.
+Before believing a flow works. The create-and-participate flow had been declared *correct by construction* on typecheck, build and reading; the first probe ever run against it found **three defects** none of those could see — all three now fixed, though one left a residual one-connection display lag whose cause sits outside the app ([[caveat_participant-count-one-connection-lag]] in `data-layer`).
 
 Reach for it when the suite cannot answer the question: the `@data` run dies silently from around its sixth scenario ([[caveat_wallet-bloat-hang]]), its scenarios have no fixtures ([[caveat_data-suite-has-no-fixtures]]), and entry paths are covered by nothing ([[caveat_first-time-entry-untested]], `app-architecture` → [[caveat_boot-unverified-outside-broker]]).