Canonical Specification for the Adaptive Virtual Machine (AVM)

State: CALIBRATED_PENDING_VALIDATION

1. AVM Instruction Set Architecture (ISA)

The AVM ISA defines a language-agnostic instruction set that serves as the bridge between mathematical languages and Python execution. The ISA consists of:

Core Instruction Set

# Stack-based operations
PUSH(value: Any)  # Push a value onto the stack
POP()             # Pop the top value from the stack
APPLY(func: Any)   # Apply a function to the top N arguments
JUMP(label: int)   # Unconditional jump
JUMP_IF(condition: bool, label: int)  # Conditional jump
CALL(method: str)  # Call a Python method
IMPORT(module: str) # Import a Python module
RETURN()          # Return from function

# Data operations
STORE(name: str)   # Store a value in the symbol table
LOAD(name: str)    # Load a value from the symbol table
DUMP()            # Dump execution state for debugging

# Control flow
BEGIN(label: int)  # Mark a label
END()             # End of function

Data Representation

Values: All values are represented as Python-compatible objects or fixed-point atoms
Types: Minimal type system (int, Q16_16, Q0_16, bool, list, dict, function)
Constants: Predefined constants for math operations (π, e, etc.) expressed in Q16_16

Execution Model

Stack-based virtual machine
Type-checking during execution
Error handling via Python exceptions
Symbol table for cross-language data sharing

2. Semantic Stripping Algorithm

def strip_semantics(source: Any, threshold: float = 0.5) -> Any:
    """
    Strips language-specific semantics while preserving functionality
    """
    if is_invariant_root(source):  # Check if already a fundamental structure
        return source

    delta = calculate_delta(source)  # 0-1 score of language dependency
    
    if delta < threshold:
        # Decompose into invariant components
        components = decompose(source)
        stripped_components = [strip_semantics(c, threshold) for c in components]
        return reconstruct(stripped_components)
    else:
        # Preserve as invariant root
        return source

def calculate_delta(node: Any) -> float:
    """
    Computes language dependency score (0-1)
    - 0: Pure invariant structure
    - 1: Heavily language-specific
    """
    # Implementation would analyze type signatures, syntax, etc.
    pass

3. AVM Binary Interface (ABI)

Calling Convention

Stack layout: CPython-compatible stack layout
Argument passing: All arguments pushed in order
Return value: Single value on stack
Error handling: Python exceptions

Data Format

class AVMValue:
    def __init__(self, value: Any):
        self.value = value
        self.type = get_type(value)

def get_type(value: Any) -> str:
    """
    Maps to CPython's internal type representation
    """
    if isinstance(value, int): return "int"
    if hasattr(value, 'is_q16_16'): return "Q16_16"
    if hasattr(value, 'is_q0_16'): return "Q0_16"
    # ... other types

Registration Protocol

def register_primitive(name: str, func: Callable):
    """
    Registers a primitive function for direct AVM execution
    """
    _registry[name] = func

_registry = {}

4. Invariant Root Extraction Process

def extract_invariants(source: Any) -> List[Any]:
    """
    Recursively extracts fundamental mathematical structures
    """
    if is_primitive(source): return [source]
    
    if is_container(source):
        children = []
        for child in source.children:
            children.extend(extract_invariants(child))
        return children
    
    raise ValueError(f"Unsupported type: {type(source)}")

5. Formal Specification in Lean 4

namespace Semantics.AVM

inductive Value where | int : Int → Value | q16 : Q16_16 → Value | q0 : Q0_16 → Value | bool : Bool → Value

instance : informational_bind State Value where isLawful s := true cost s := 0 -- Base transition cost extract s := "AVM_STATE"

def compile (source : SourceLang) : Value := match source with | int n => Value.int n | fixed n => Value.q16 n | ratio n => Value.q0 n | bool b => Value.bool b


This specification provides a foundation for proving equivalence between source language semantics and AVM execution. The full implementation would include:

1. Type soundness proof
2. Preservation theorem
3. Progress theorem
4. Adequacy proof

## 6. CALIBRATED State Requirements
The AVM is considered **CALIBRATED** once the following conditions are met:

1. **Float Elimination**: All core primitive float references have been removed from the AVM ISA and Lean core.
2. **Type Definition**: `Q0_16` and `Q16_16` are explicitly defined and used as the primary numeric types.
3. **Behavioral Declaration**: Arithmetic, rounding (floor), and overflow (saturating) behaviors are formally declared and implemented.
4. **Boundary Conversion**: Explicit paths for converting external numeric formats (Int, Float-input) into `Q16_16` are defined.
5. **Determinism Invariant**: The AVM must achieve bit-exact reproducibility across all execution environments (Lean, Python-AVM, FPGA).
6. **Bind Semantics**: Composition of AVM instructions must follow the `informational_bind` metric laws.
7. **Validator Enforcement**: A pre-commit validator rejects any `f32`, `f64`, or `double` references in the `Semantics/AVM/` path.

## 7. Determinism Invariant
The AVM state transition function $f(S, I) \rightarrow S'$ must satisfy:
$\forall env_1, env_2 \in \{Lean, Python, FPGA\}, f_{env_1}(S, I) = f_{env_2}(S, I)$
This ensures that "Formal Drift" is mathematically impossible.

## 8. Boundary Conversion
```python
def to_q16_16(val: Any) -> int:
    if isinstance(val, int):
        return val << 16
    if isinstance(val, float):
        # Explicit boundary clip and scale
        clamped = max(-32768.0, min(32767.9999, val))
        return int(clamped * 65536)
    raise ValueError("Invalid Boundary Format")

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