test(frost): cover the batch's failure modes and its crypto without a database

Phase 6 of docs/frost-batch-signing.md. 361 jvmTest and 227 testDebugUnitTest
pass.

## Inbound path (SignedGroupKeyStateTest)

Both drive the manager with a hand-built inner event rather than one the other
device queued, which is the only way to be a faulty or dishonest member in this
harness.

- A one-value nonce offered for a three-item batch does not count towards the
  threshold: the coordinator never reaches a signer set. The length check is all
  that stands between a batch and a signer whose contribution lines up against
  the wrong messages, so truncating or padding would produce partial signatures
  aggregated against events nobody agreed to. The test then pumps the real nonce
  and the batch completes -- it is a stall, not damage, which is
  FrostSignerMessage's composite key doing its job.
- A second proposal under the session's own id changes neither its event ids nor
  its seeds. Every seed is already committed to its item's message; a different
  batch under the same id would have those seeds produce a second partial
  signature over a second message, which is how a share is extracted.

## Real FROST, no database (FrostSigningRoundTest)

- A k=3 batch from one signer set, all three verifying against the room's key --
  the manager's shape with the database taken out of the way.
- Item 0's signature does not verify against item 1. Signing three events in
  lockstep must not make any of them interchangeable.
- Both halves of the no-shared-nonce property, because either alone is enough to
  be relied on by accident: SecretNonce.generate mixes the message in, so one
  seed under two messages already gives two nonces -- and the manager mints
  distinct seeds regardless.

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

composeApp/src/commonTest/kotlin/press/mantra/compose/managers/FrostSigningRoundTest.kt

@@ -160,6 +160,105 @@ class FrostSigningRoundTest {
         )
     }
 
+    @Test
+    fun `one signer set signs a batch of three, and every signature verifies`() {
+        // The manager's batch, with the database taken out of the way: one signer
+        // set and one tweak cache shared, and a nonce, a Session and a signature
+        // per event. If any of that is wired up wrongly the aggregate simply
+        // fails to verify, which is the whole reason this file exists.
+        val ids = listOf("first", "second", "third").map(::eventId)
+        val messages = ids.map { ByteVector(it.hexToByteArray()) }
+
+        val signerIds = listOf(0, 1)
+
+        // A seed per signer per item. The manager mints these independently; here
+        // they only have to differ, which is the property under test.
+        val nonces = signerIds.map { signerId ->
+            messages.mapIndexed { index, message ->
+                nonceOf(signerId, message, "f".repeat(62) + "$index${signerId + 1}")
+            }
+        }
+
+        val signatures = messages.mapIndexed { index, message ->
+            val session = sessionFor(signerIds, nonces.map { it[index].second }, message)
+
+            val partials = signerIds.mapIndexed { position, signerId ->
+                session.sign(
+                    nonces[position][index].first,
+                    keyMaterial.secretShares[signerId],
+                    signerId.toUInt()
+                ).right!!
+            }
+
+            session.aggregateSigs(partials).right!!
+        }
+
+        ids.forEachIndexed { index, id ->
+            assertTrue(
+                Nip01Crypto.verify(
+                    signature = signatures[index].toByteArray(),
+                    hash = id.hexToByteArray(),
+                    pubKey = groupPubKey.hexToByteArray()
+                ),
+                "item $index of the batch must verify against the room's own key"
+            )
+        }
+    }
+
+    @Test
+    fun `a batch signature does not carry to another item of the same batch`() {
+        // What keeps a batch k independent signatures rather than one loose one.
+        // Signing three events in lockstep must not make any of them
+        // interchangeable.
+        val ids = listOf("first", "second").map(::eventId)
+        val messages = ids.map { ByteVector(it.hexToByteArray()) }
+        val signerIds = listOf(0, 1)
+
+        val nonces = signerIds.map { signerId ->
+            messages.mapIndexed { index, message ->
+                nonceOf(signerId, message, "9".repeat(62) + "$index${signerId + 1}")
+            }
+        }
+
+        val session = sessionFor(signerIds, nonces.map { it[0].second }, messages[0])
+        val partials = signerIds.mapIndexed { position, signerId ->
+            session.sign(
+                nonces[position][0].first,
+                keyMaterial.secretShares[signerId],
+                signerId.toUInt()
+            ).right!!
+        }
+        val first = session.aggregateSigs(partials).right!!
+
+        assertFalse(
+            Nip01Crypto.verify(
+                signature = first.toByteArray(),
+                hash = ids[1].hexToByteArray(),
+                pubKey = groupPubKey.hexToByteArray()
+            ),
+            "item 0's signature must not verify against item 1"
+        )
+    }
+
+    @Test
+    fun `two items of a batch never share a nonce`() {
+        // The one mistake in this whole design that loses the key, asserted at the
+        // level where it would be made. `SecretNonce.generate` mixes the message
+        // in, so two items of a batch cannot collide even given the same seed --
+        // but the manager gives them distinct seeds as well, and both halves are
+        // checked here because either alone is enough to be relied on by accident.
+        val messages = listOf("first", "second").map { ByteVector(eventId(it).hexToByteArray()) }
+
+        val sameSeed = messages.map { nonceOf(0, it, "7".repeat(64)).second.data.toHex() }
+        assertEquals(2, sameSeed.toSet().size, "one seed under two messages must give two nonces")
+
+        val distinctSeeds = messages.mapIndexed { index, message ->
+            nonceOf(0, message, "8".repeat(63) + "$index").second.data.toHex()
+        }
+        assertEquals(2, distinctSeeds.toSet().size)
+        assertEquals(emptySet(), sameSeed.toSet().intersect(distinctSeeds.toSet()))
+    }
+
     @Test
     fun `a signature over one event does not verify against another`() {
         val id = eventId("the group agrees")