docs(dox): build out the DOX AGENTS.md hierarchy

Root Child DOX Index assembled, plus a per-module AGENTS.md across the tree
(core/*, infrastructure/*, apps/*, testing/*, and docs/examples/frontend/etc),
each following the DOX section shape: Purpose, Ownership, Local Contracts,
Work Guidance, Verification, Child DOX Index.

core/approvals/AGENTS.md

@@ -0,0 +1,36 @@
+# core/approvals — AGENTS.md
+
+## Purpose
+
+Approval gate logic: evaluates tool-execution requests against active grants and session approval mode, records decisions as events, and exposes an engine interface for the kernel to query.
+
+## Ownership
+
+CORREX kernel team. Changes to approval semantics must be validated against `testing/approvals/`.
+
+## Local Contracts
+
+- `ApprovalEngine` — primary interface; `DefaultApprovalEngine` evaluates `DomainApprovalRequest` against `ApprovalState`.
+- `NoOpApprovalEngine` — always grants; used in test/replay contexts.
+- `ApprovalState` rebuilt from events via `ApprovalReducer` + `ApprovalProjector`.
+- `ApprovalGrant` holds scope identity (`ApprovalScopeIdentity`), mode, and expiry. Grants are recorded as events in `core:events` (`ApprovalEvents.kt`).
+- Hard Invariant #4 applies: approvals cannot override policy denial. Policy failure is terminal.
+
+## Work Guidance
+
+- Follow the standard Events→State→Reducer→Projector→Repository pattern (see `core/AGENTS.md`).
+- `DefaultApprovalReducer` only does `state.copy(...)`. No grant-evaluation logic in the reducer.
+- Grant evaluation lives in `DefaultApprovalEngine`, not in the reducer or projector.
+- `ApprovalContext` carries per-request context; `ApprovalDecision` is a sealed result type.
+
+## Verification
+
+```bash
+./gradlew :core:approvals:test --rerun-tasks
+```
+
+Extended tests in `testing/approvals/`: edge cases, fuzz, grant security, tier immutability.
+
+## Child DOX Index
+
+No child AGENTS.md (leaf module).