Research-Stack/6-Documentation/docs/plans/SOVEREIGN_PROCEED_PLAN_V1.md

# Sovereign Proceed Plan: 48-Hour Execution

## 1. Theorem Hardening (Burgers.lean)

### Current State
The `Burgers.lean` file contains three `sorry` markers in the `complexityOmega_nonneg` and `nu_eff_ge_nu0` lemmas, and one in `witnessComplexity_nonneg`.

### Approach
We can use the BFS-Prover (Breadth-First Search Prover) to automatically prove these lemmas. The BFS-Prover is a tactic that uses a breadth-first search to find a proof by applying a set of rules.

However, note that the `witnessComplexity_nonneg` and `complexityOmega_nonneg` lemmas are straightforward because the fixed-point arithmetic in Q16_16 is non-negative and the operations (multiplication and addition) preserve non-negativity. Similarly, the `nu_eff_ge_nu0` theorem is a simple inequality.

### Steps
1. **Prove `witnessComplexity_nonneg`**:
   - The witness complexity contribution is `n^2 * |a|^2`. Since `n` and `a` are fixed-point numbers, and the operations are non-negative, the result is non-negative.
   - We can use `norm_num` or `field_simp` to prove this.

2. **Prove `complexityOmega_nonneg`**:
   - The complexity metric is half the sum of non-negative terms. Since the sum is non-negative, half of it is non-negative.

3. **Prove `nu_eff_ge_nu0`**:
   - We know that `effectiveViscosity = nu0 * (1 + Omega)`. Since `Omega` is non-negative (as proven above) and `nu0` is non-negative, then `1 + Omega` is at least 1, so `nu0 * (1 + Omega)` is at least `nu0`.

### Proposed Actions
- Replace the `sorry` markers with proofs using `norm_num` or `field_simp` for the first two lemmas.
- For `nu_eff_ge_nu0`, use `linarith` or `ring` to prove the inequality.

## 2. Hardware Loopback

### Current State
We need to design a UART packet structure and test harness for the Tang Nano 9K.

### Approach
We will design a simple packet format that includes:
- A start byte (e.g., 0xAA)
- The state (18-bit integer)
- A checksum (1-byte, XOR of all bytes)

The host (using Python) will send a packet to the FPGA, and the FPGA will echo it back with a response (e.g., the state and a computed value from the kernel).

### Steps
1. **Define the packet structure**:
   - Start: 0xAA (1 byte)
   - State: 18-bit integer (3 bytes, big-endian)
   - Checksum: 1 byte (XOR of all bytes except the start byte)

2. **Design the FPGA UART receiver**:
   - Read the start byte.
   - Read the state (3 bytes).
   - Compute the checksum and verify it.
   - If valid, compute the kernel output (e.g., Burgers equation) and send it back.

3. **Design the host test script**:
   - Use `pyserial` to send packets and receive responses.
   - Compare the response with a known value (computed in Python).

### Proposed Actions
- Define the packet structure in the FPGA code (UART receiver).
- Write a Python script to send and receive packets.

## 3. AVM-R Integration

### Current State
We are moving from isolated kernels to a hierarchical vector roll-up (AVM-R).

### Approach
The AVM-R (Abstract Vector Machine - Roll-Up) is a hierarchical representation of the vector space. We need to integrate the existing kernels (like Burgers) into this structure.

### Steps
1. **Define the AVM-R data structure**:
   - A vector is represented as a tree of smaller vectors (roll-up).
   - Each node in the tree is a vector of a certain dimension.

2. **Integrate the Burgers kernel**:
   - The Burgers kernel should be represented as a function that operates on vectors in the AVM-R format.

3. **Test the integration**:
   - Run the Burgers kernel on a small vector and verify the result.

### Proposed Actions
- Define the AVM-R data structure and operations.
- Modify the Burgers kernel to work on AVM-R vectors.

## 4. Visual Surface

### Current State
We are transitioning from static plots to a live WebSocket-driven dashboard.

### Approach
We will create a FastAPI backend that serves a WebSocket endpoint for real-time updates and a frontend that uses WebSockets to receive data and update a visualization.

### Steps
1. **Backend**:
   - Create a FastAPI app with a WebSocket endpoint.
   - The endpoint will send updates (e.g., vector states) to the frontend.

2. **Frontend**:
   - Use JavaScript to connect to the WebSocket.
   - Update a canvas or chart with the received data.

### Proposed Actions
- Set up the FastAPI backend with a WebSocket endpoint.
- Develop the frontend HTML/JS to connect to the WebSocket.

## Detailed 48-Hour Plan

### Day 1 (Morning)

1. **Theorem Hardening**:
   - Prove `witnessComplexity_nonneg` and `complexityOmega_nonneg` in `Burgers.lean`.
   - Replace `sorry` markers.

2. **Hardware Loopback**:
   - Define the packet structure in the FPGA code.
   - Write a Python script to send and receive packets.

### Day 1 (Afternoon)

1. **Theorem Hardening**:
   - Prove `nu_eff_ge_nu0`.

2. **Hardware Loopback**:
   - Complete the FPGA UART receiver for the packet.
   - Test the UART communication with a simple echo.

### Day 2 (Morning)

1. **AVM-R Integration**:
   - Define the AVM-R data structure and basic operations.

2. **Visual Surface**:
   - Set up the FastAPI backend with a WebSocket endpoint.

### Day 2 (Afternoon)

1. **AVM-R Integration**:
   - Integrate the Burgers kernel with AVM-R.

2. **Visual Surface**:
   - Develop the frontend to connect to the WebSocket and display data.

### Day 2 (Evening)

1. **Integration Testing**:
   - Run the AVM-R Burgers kernel and visualize the results.

## Conclusion

This plan focuses on making progress in four key areas within the next 48 hours. By following these steps, we can systematically address the remaining `sorry` markers, set up hardware verification, integrate the AVM-R structure, and build a live visualization dashboard.

Note: The actual implementation details may require adjustments based on the specific environment and tools available.