fed-sx-m2: Step 9a — pure-functional backfill slicing + 20 tests

New next/kernel/backfill.erl owns the §13.3 backfill mode
slicing. Given an outbox log + a mode, returns the activity
list to send to a new follower as backfill.

Public API:
  slice/2(Mode, LogState)               default Wrap=false
  slice/3(Mode, LogState, Wrap)         Wrap=true wraps entries
  wrap_backfill/1                       add {backfilled, true}
  parse_mode/1                          lift Follow :backfill field

Modes:
  none                       new follower: forward-only content
  full                       entire outbox
  {last_n, N}                last N activities (FIFO)
  {last_t, T, NowFn}         entries with :published in
                             (NowFn()-T .. NowFn()]
  {since_cid, Cid}           entries after the one with :id = Cid
                             (consumes the matched entry; returns
                             every entry after it)

wrap_backfill/1 marks each entry {backfilled, true}. Per §13.3
wrapped bodies preserve :id so the receiver's replay defence
still catches duplicates from the live stream.

parse_mode/1 accepts:
  nil / none / full / {last_n, _} / {last_t, _, _} /
  {since_cid, _} — pass through or normalize
  Proplist with :mode + :limit -> {last_n, N}
  Proplist with :mode + :duration -> {last_t, T, fun() -> 0 end}
  Proplist with :mode = full -> full
  Anything else -> none (open-world default)

Substrate gotchas re-confirmed and worked around:
  - lists:nthtail/2 not registered — rolled drop_n/2
  - Pattern-alias 'Pat = Var' not supported by this port's
    parser — parse_mode/1 clauses use explicit deconstruction

20/20 in next/tests/backfill.sh covering all five modes plus
edge cases (N=0, N>length, T=0 -> empty window, since_cid
hit/miss/unknown), wrap_backfill semantics, parse_mode for
atoms / tuple shapes / proplists / unknown / nil.

Step 9b (outbox listing ?since=Cid&limit=N pagination) and
Step 9c (Follow-Accept-backfill wiring) layer on top.
Conformance preserved at 761/761.

next/kernel/backfill.erl

@@ -0,0 +1,136 @@
+-module(backfill).
+-export([slice/2, slice/3,
+         wrap_backfill/1, parse_mode/1,
+         all_entries/1, last_n_entries/2, last_t_entries/3,
+         since_cid_entries/2, none_entries/0]).
+
+%% Backfill mode slicing per design §13.3 / Step 9. When A follows B
+%% with a backfill spec, B's kernel slices the outbox log into the
+%% appropriate window and delivers each entry as
+%% `{backfilled, true}`-marked envelopes alongside forward-going
+%% activity.
+%%
+%% Mode shapes (per the Follow activity's `:backfill` field):
+%%   none                   — newer follower sees only forward content
+%%   {last_n, N}            — backfill last N activities (FIFO order)
+%%   {last_t, T, NowFn}     — backfill activities with :published in
+%%                            (Now - T .. Now]. NowFn is a 0-arity fun
+%%                            so tests can fake-time it.
+%%   full                   — backfill the entire outbox
+%%
+%% slice/2 returns the activity list. slice/3 also wraps each entry
+%% with `{backfilled, true}` so projections can decide whether to
+%% re-fold or skip (the §13.3 Backfilled bodies preserve the
+%% original `:id` so replay defence still works on the receiver).
+%%
+%% parse_mode/1 lifts the Follow activity's `:backfill` proplist
+%% (or atom) into the internal mode tuple. Unknown shapes fall back
+%% to `none` — the default open-world policy.
+
+slice(Mode, LogState) ->
+    slice(Mode, LogState, false).
+
+slice(Mode, LogState, Wrap) ->
+    Entries = log:entries(LogState),
+    Slice = case Mode of
+        none                  -> none_entries();
+        full                  -> all_entries(Entries);
+        {last_n, N}           -> last_n_entries(N, Entries);
+        {last_t, T, NowFn}    -> last_t_entries(T, NowFn, Entries);
+        {since_cid, Cid}      -> since_cid_entries(Cid, Entries);
+        _                     -> none_entries()
+    end,
+    case Wrap of
+        true  -> wrap_backfill(Slice);
+        _     -> Slice
+    end.
+
+%% ── Mode-specific entry selection ─────────────────────────────
+
+all_entries(Entries) -> Entries.
+
+none_entries() -> [].
+
+%% last_n_entries/2 — tail N entries in FIFO order.
+
+last_n_entries(N, _) when N =< 0 -> [];
+last_n_entries(N, Entries) ->
+    Len = length(Entries),
+    case Len =< N of
+        true  -> Entries;
+        false -> drop_n(Len - N, Entries)
+    end.
+
+drop_n(0, L) -> L;
+drop_n(_, []) -> [];
+drop_n(N, [_ | Rest]) -> drop_n(N - 1, Rest).
+
+%% last_t_entries/3 — entries whose :published is within the last
+%% T units of (NowFn() - T .. NowFn()]. T and :published are
+%% integers (seconds-since-epoch in production; opaque ints in tests).
+
+last_t_entries(T, NowFn, Entries) when is_integer(T), T >= 0 ->
+    Now = NowFn(),
+    Cutoff = Now - T,
+    [E || E <- Entries, in_window(E, Cutoff, Now)];
+last_t_entries(_, _, _) -> [].
+
+in_window(Activity, Cutoff, Now) ->
+    case envelope:get_field(published, Activity) of
+        {ok, P} when is_integer(P), P > Cutoff, P =< Now -> true;
+        _ -> false
+    end.
+
+%% since_cid_entries/2 — every entry after the one with :id = Cid.
+%% If Cid isn't in the log, returns [] (caller's pointer is stale).
+%% Used by `GET /actors/<id>/outbox?since=Cid` pagination.
+
+since_cid_entries(_Cid, []) -> [];
+since_cid_entries(Cid, [E | Rest]) ->
+    case envelope:get_field(id, E) of
+        {ok, Cid} -> Rest;
+        _         -> since_cid_entries(Cid, Rest)
+    end.
+
+%% wrap_backfill/1 — append `{backfilled, true}` to each entry.
+%% The receiving projection scheduler reads this field and chooses
+%% whether to fold (re-emit) or skip (already known via replay
+%% defence on `:id`).
+
+wrap_backfill([]) -> [];
+wrap_backfill([E | Rest]) ->
+    [E ++ [{backfilled, true}] | wrap_backfill(Rest)].
+
+%% parse_mode/1 — Lift a Follow activity's `:backfill` value into the
+%% internal mode tuple. Accepts:
+%%   nil / not_found       -> none
+%%   none                  -> none
+%%   full                  -> full
+%%   {last_n, N}           -> {last_n, N}     (already-parsed shape)
+%%   {last_t, T, NowFn}    -> pass-through
+%%   Proplist with :mode + :limit / :duration -> parsed
+%% Unknown shape -> none (open-world default).
+
+parse_mode(nil)                  -> none;
+parse_mode(none)                 -> none;
+parse_mode(full)                 -> full;
+parse_mode({last_n, N})          -> {last_n, N};
+parse_mode({last_t, T, NowFn})   -> {last_t, T, NowFn};
+parse_mode({since_cid, Cid})     -> {since_cid, Cid};
+parse_mode(List) when is_list(List) ->
+    case envelope:get_field(mode, List) of
+        {ok, last_n} ->
+            case envelope:get_field(limit, List) of
+                {ok, N} when is_integer(N) -> {last_n, N};
+                _ -> none
+            end;
+        {ok, last_t} ->
+            case envelope:get_field(duration, List) of
+                {ok, T} when is_integer(T) -> {last_t, T, fun () -> 0 end};
+                _ -> none
+            end;
+        {ok, full} -> full;
+        {ok, none} -> none;
+        _ -> none
+    end;
+parse_mode(_) -> none.