docs(arap): B-41, the bookspec examples, and the four registers (P4b S7)

rules.md B-41 states the write-off: the direct method the owner chose
(A-5), Dr 6990 / Cr 1200 with the receivable tagged, at most what is
owed and at most the least owed from that day on (a partial write-off as
§1.166-3(a) allows), refused on cash-basis books (§1.166-1(e)), its
evidence filed in the same call (§1.166-2(a)); bills are never written
off (ASC 405-20-40-1); and a recovery is not a new movement (D-49): the
same year, reverse the write-off and record the payment; a later year, a
plain entry to a bad-debt-recoveries revenue account, the invoice staying
written off (IRC §111).

bookspec states B-41 for the first time and gives B-38 its example: two
new examples, bad-debt-direct-write-off (two partial write-offs of one
invoice, the balances and the trial balance worked by hand) and
arap-hand-control-posting-refused (a hand line an earlier version wrote
replays, the same line written today is refused). The example schema
gains the dated write_off event, lets a refused entry say so (an entry
posts whole or not at all; older and refused never combine), and its
refusal vocabulary gains untagged_control_line, B-38's control guard.

The registers: docs/architecture.md names the new tool group;
docs/receipt-store.md lists invoice.written_off and what replay does with
it; docs/parity.md records the slice (no row changes status, the floor
stands); docs/testing.md records every test and what P12 must mutate;
docs/verification.md the two Lean theorems and what they cover.

No figure moves: no golden file changed.

docs/parity.md

@@ -104,3 +104,14 @@ door on loopback for Caddy or a tunnel. `crates/ledger-cli/tests/two_user.rs`
 proves the account boundary and the move's head in CI
 (`.github/workflows/two-user.yml`); the recipe is `deploy/README.md`.
 The cutover rule in `docs/capability-parity.md` is unchanged.
+
+## P4b S7: bad debts (write-offs), 2026-10-27
+
+A/R gained the direct write-off the owner chose (A-5; plan 8.2 S7): an
+invoice that will not be collected comes off 1200 with Dr 6990 Bad Debt
+Expense, its receivable line tagged `WriteOff` so the receipt's figures
+stay derived, and its evidence of worthlessness filed in the same call
+(rules.md B-41; IRC §166, §111; Treas. Reg. §1.166-1(e), (f), -2(a),
+-3(a)). A same-year recovery is the reversal D-49 records, not a new
+movement; bills are never written off. No row above changed status (the
+A/R and A/P rows already say `done`), so the floor stands (10 done).