Broad Study
Win carrying spells from five or more different schools.

School means firing pattern, not element. The game ships eight: bolt, spread, nova, orbit, aura, chain, cleave and sentry. Two lightning spells can share one school, and two spells of different elements can too — every entry in Codex → Spells is tagged with its school.

Winning route

Start a new expedition as the Warden-Mage and march on its first node — 80 seconds, and the only mission that starts with an empty build. In the war-march panel set the stage to Stage 1 / 12; the picker opens on the stage you last started, and later stages are harder on every node. Arc Bolt supplies bolt. The ground is drawn at random; this run drew Skordaveth.
Collect gems and take new spells from other schools: Line Breaker (chain), Undertow (spread), Sentry Ward (sentry), and Frost Sigil (nova). These five are enough.
If those cards do not appear, take another new school, such as Ward Orbs (orbit). Finish the mission carrying at least five schools.

Full run

One continuous mission, real draft choices, the winning build and the earned achievement ledger. Original game music and effects; no commentary. The run was played by a scripted test player using only the game's normal controls, not by a person; failed attempts were discarded.

Sentry Ward shown as a new spell on the level-up card — Sentry Ward adds the sentry school; Anchor Throw on the same card is another bolt.

Undertow appears as a new spell on a level-up card — Undertow adds the spread school.

Frost Sigil appears as a new spell on a level-up card — Frost Sigil adds nova, the fifth school.

Nine spell icons in the lower-right build bar near the win — Nine spells at the win, every one still at Lv 1 — well past the five schools the writ needs.

Game achievement ledger showing Broad Study earned — Broad Study unlocked in this run. Open the image to read the ledger.