feat(ui): port retail selected-stack quantity

Bind the authored stack count entry and horizontal slider to one Core split-quantity owner, preserve retail count-first naming and exact 1000-step rounding, refresh on stack changes, and consume the selected amount during merges. Conformance covers the production DAT fixture and retained pointer/focus paths.

Co-Authored-By: Codex <codex@openai.com>

docs/research/2026-07-10-retail-toolbar-interaction-pseudocode.md

@@ -463,10 +463,10 @@ AttemptMerge(sourceId, targetId, quiet):                    // 0x005878F0
     return true
 ```
 
-acdream's selected-object stack entry/slider is still absent under AP-101, so the
-current inventory merge path requests the whole source stack and lets the same
-retail capacity clamp limit the transfer. Once the shared split-quantity owner
-lands, it supplies `requested` without changing legality or wire ordering.
+acdream now has one Core-owned split-quantity state shared by the selected-object
+entry/slider and inventory merge. A selected source supplies that quantity to
+`AttemptMerge`; an unselected source retains retail's full-source default. Legality,
+capacity clamping, and wire ordering are unchanged.
 
 ---
 
@@ -507,7 +507,7 @@ HandleSelectionChanged():                                 // 0x004BF380
     if object missing: return
 
     set selected name:
-        normal object: name, or formatted name + count when stacked
+        normal object: name, or `"%d %hs"` (count then name) when stacked
         owned coinstack: formatted coin total/name path
 
     splitSize = 1
@@ -567,13 +567,22 @@ RecvNotice_UpdateItemMana(id, fraction, valid):             // 0x004BD0C0
 - On entry deactivation, parse unsigned text, clamp to `1..maxSplitSize`, rewrite
   the text if clamped, update `GenItemHolder::splitSize`, update slider attribute
   `0x86 = splitSize/maxSplitSize`, and broadcast the stack-slider notice.
-- Slider message `0xA` maps its normalized value to an integer in
-  `1..maxSplitSize`, stores it, rewrites the entry, and broadcasts the notice.
+- `UIElement_Scrollbar::SetScrollbarPosition @ 0x00470EC0` clamps the normalized
+  position to `[0,1]`, multiplies by `1000.0f`, truncates through `_ftol2`, and
+  broadcasts that integer as message `0xA` parameter 1.
+- `gmToolbarUI::ListenToElementMessage @ 0x004BEFE0..0x004BF041` converts the
+  message exactly as follows, stores it, rewrites the entry, and broadcasts the
+  stack-slider notice:
 
-**Unresolved:** Binary Ninja lost the exact floating expression immediately before
-the slider's `1 - _ftol2()` conversion at `0x004BF02D`. The clamp, endpoints, and
-text update are exact, but the intermediate rounding direction must be recovered
-from Ghidra/disassembly before writing conformance vectors.
+```text
+positionMillis = truncate(clamp01(sliderRatio) * 1000)
+splitSize = clamp(1 + floor(positionMillis * maxSplitSize / 1000),
+                  1, maxSplitSize)
+```
+
+The apparently odd extra one is verbatim retail behavior. Static cdb disassembly
+of the matching v11.4186 binary also pins the stack-name format string at
+`0x007B4738` to `"%d %hs"` and the numeric-entry format at `0x007A0184` to `"%d"`.
 
 ---