<think>
Okay, let's tackle this problem. The goal is to implement the ring_slot function according to the given specifications. Let me start by understanding the requirements.

First, the function needs to determine which slot holds the k-th oldest survivor in a telemetry recorder. The recorder has slots numbered 0 to capacity-1. Each new reading is written into slot n mod capacity, where n is the reading number starting from 0. After writes, the survivors are the newest min(writes, capacity) readings. So, if there are more writes than capacity, only the latest capacity readings are kept.

The task is to find the slot number for the k-th oldest survivor. The input parameters are capacity, writes, and k. We need to return the slot number. But if any of the parameters are invalid, we should raise a ValueError.

Let me break down the steps:

1. **Validation**: Check that capacity is a positive integer, writes is a non-negative integer, and k is a non-negative integer. Also, ensure that k is less than the survivor count, which is min(writes, capacity). If not, raise an error.

2. **Determine the survivor count**: The number of survivors is min(writes, capacity). So, if writes is less than capacity, all writes are survivors. If writes is more, only the last capacity writes are survivors.

3. **Find the k-th oldest survivor**: The oldest survivor is the first one (k=0), then next (k=1), etc. So, the k-th oldest is the (k)th in the list of survivors when ordered by their writing order. Wait, but the survivors are the latest ones. Wait, the problem says that after writes, the survivors are the newest min(writes, capacity) readings. So, if writes is 6 and capacity is 4, the survivors are the last 4 readings (writes 2,3,4,5). The oldest among them is the first one (k=0) which is writing 2, then k=1 is 3, etc.

So, the survivors are the last 'capacity' readings if writes >= capacity. Otherwise, all writes are survivors.

So, the k-th oldest survivor is the (k)th in the list of survivors when ordered by their writing order. Wait, but the survivors are the latest ones. So, the first survivor (k=0) is the earliest of the survivors. For example, if writes is 6 and capacity is 4, the survivors are writes 2,3,4,5. So, the oldest is 2 (k=0), then 3 (k=1), then 4 (k=2), then 5 (k=3). So, for k=3, it's 5.

But the problem says that in the example, with capacity 4 and 6 writes, the k=3 is 5 in slot 1. Wait, how does that work?

Wait, the example says: capacity 4, writes 6. The survivors are the last 4 readings (writes 2,3,4,5). The k=3 is the 3rd oldest, which is 5. The slot for that is 1. How is that calculated?

Each reading is written into slot n mod capacity. So, for n=0, slot 0; n=1, slot 1; n=2, slot 2; n=3, slot 3; n=4, slot 0; n=5, slot 1; n=6, slot 2.

But the survivors are the last 4 readings (n=2,3,4,5). So, for