fix(memories): search the photo provider by the local day, not the UTC one

"This day" sent the entry's calendar date to the provider as both bounds, and
both providers turned that date into a UTC window — Immich by concatenating a
Z suffix, Synology by relying on a date-only string parsing as UTC midnight.
For a reader at UTC+10 the searched window was 10:00 on the day to 09:59 the
next morning, so every breakfast photo was missing while the next morning's
turned up in its place, and nothing said the day was incomplete. The date
headings had the mirror defect, filing a 07:32 photo under the previous day.

Immich publishes the right anchor already: localDateTime is required on every
search result and is what its own timeline groups by. The window is padded by
a day on each side and the page narrowed again by each photo's own local date,
carried through as localTakenAt; a server that does not send it falls back to
the timestamp used until now, so nothing changes there. Because the padding
means raw rows and answered rows are no longer the same thing, the page is
filled server-side until it holds what the caller asked for, bounded so a
pathological album cannot spin.

Synology has no local stamp, so its anchor comes from the caller as a UTC
offset, resolved for the day being searched rather than for today — a day
looked up across a DST boundary would otherwise start an hour out. The MCP
tool takes the same optional input, defaulting to the behaviour it has now.

The picker's pagination sentinel moved out of the non-empty branch: a page the
new filter empties has to keep paging, and it now says so instead of claiming
there are no photos.

Closes #2336

server/tests/unit/nest/memories.helpers.test.ts

@@ -8,7 +8,12 @@
  * inside either provider suite.
  */
 import { describe, it, expect } from 'vitest';
-import { sortAssetsByTakenAtDesc } from '../../../src/nest/memories/memories.helpers';
+import {
+  dayStartEpochSeconds,
+  isWithinLocalDayRange,
+  shiftCalendarDay,
+  sortAssetsByTakenAtDesc,
+} from '../../../src/nest/memories/memories.helpers';
 
 const asset = (id: string, takenAt?: string | null) => ({ id, takenAt });
 
@@ -78,3 +83,69 @@ describe('sortAssetsByTakenAtDesc', () => {
     expect(ids(out)).toEqual(['utc', 'offset']);
   });
 });
+
+
+describe('shiftCalendarDay', () => {
+  it('MEM-DAY-001: pads a day outwards, crossing month and year ends', () => {
+    expect(shiftCalendarDay('2026-03-15', -1)).toBe('2026-03-14');
+    expect(shiftCalendarDay('2026-03-15', 1)).toBe('2026-03-16');
+    expect(shiftCalendarDay('2026-03-01', -1)).toBe('2026-02-28');
+    expect(shiftCalendarDay('2026-12-31', 1)).toBe('2027-01-01');
+  });
+
+  it('MEM-DAY-002: hands a bound it cannot read straight back, so it fails upstream as before', () => {
+    expect(shiftCalendarDay('not-a-day', -1)).toBe('not-a-day');
+    expect(shiftCalendarDay('', 1)).toBe('');
+  });
+});
+
+
+describe('isWithinLocalDayRange', () => {
+  const at = (takenAt: string | null, localTakenAt?: string | null) => ({ takenAt, localTakenAt });
+
+  it('MEM-DAY-010: reads the day off the local capture stamp when the provider sends one', () => {
+    // 21:00 UTC on the 14th is already the 15th in Sydney, and Immich says so.
+    expect(isWithinLocalDayRange(at('2026-03-14T21:00:00Z', '2026-03-15T08:00:00.000Z'), '2026-03-15', '2026-03-15')).toBe(true);
+    // And the mirror: 22:00 UTC on the 15th is already the 16th there, so it
+    // does not belong to the 15th however the UTC day reads.
+    expect(isWithinLocalDayRange(at('2026-03-15T22:00:00Z', '2026-03-16T09:00:00.000Z'), '2026-03-15', '2026-03-15')).toBe(false);
+  });
+
+  it('MEM-DAY-011: falls back to the capture instant, which is the UTC day it always used', () => {
+    expect(isWithinLocalDayRange(at('2026-03-15T23:30:00Z'), '2026-03-15', '2026-03-15')).toBe(true);
+    expect(isWithinLocalDayRange(at('2026-03-16T00:30:00Z'), '2026-03-15', '2026-03-15')).toBe(false);
+  });
+
+  it('MEM-DAY-012: narrows only the bound it was given', () => {
+    expect(isWithinLocalDayRange(at('2026-03-10T09:00:00Z'), '2026-03-15', undefined)).toBe(false);
+    expect(isWithinLocalDayRange(at('2026-03-20T09:00:00Z'), '2026-03-15', undefined)).toBe(true);
+    expect(isWithinLocalDayRange(at('2026-03-10T09:00:00Z'), undefined, '2026-03-15')).toBe(true);
+    expect(isWithinLocalDayRange(at('2026-03-20T09:00:00Z'), undefined, '2026-03-15')).toBe(false);
+    expect(isWithinLocalDayRange(at('2026-03-20T09:00:00Z'))).toBe(true);
+  });
+
+  it('MEM-DAY-013: keeps an asset with no usable timestamp rather than losing it', () => {
+    expect(isWithinLocalDayRange(at(null), '2026-03-15', '2026-03-15')).toBe(true);
+    expect(isWithinLocalDayRange(at('nope'), '2026-03-15', '2026-03-15')).toBe(true);
+  });
+});
+
+
+describe('dayStartEpochSeconds', () => {
+  it('MEM-DAY-020: with no offset it is the UTC midnight the window always used', () => {
+    expect(dayStartEpochSeconds('2026-03-15')).toBe(Math.floor(new Date('2026-03-15').getTime() / 1000));
+  });
+
+  it('MEM-DAY-021: an eastern offset starts the day earlier in UTC, a western one later', () => {
+    const utc = dayStartEpochSeconds('2026-03-15');
+    // UTC+10 reaches midnight ten hours before UTC does.
+    expect(dayStartEpochSeconds('2026-03-15', 600)).toBe(utc - 600 * 60);
+    expect(dayStartEpochSeconds('2026-03-15', -480)).toBe(utc + 480 * 60);
+  });
+
+  it('MEM-DAY-022: a bound that is not a calendar day is parsed as before, offset ignored', () => {
+    const instant = '2026-03-15T06:00:00Z';
+    expect(dayStartEpochSeconds(instant, 600)).toBe(Math.floor(Date.parse(instant) / 1000));
+    expect(Number.isNaN(dayStartEpochSeconds('nope'))).toBe(true);
+  });
+});