Elliptic Curve AI (ECAI) is built on a simple but radical premise:

Intelligence is not predicted. It is deterministically encoded and retrieved.

The following article walks through a Python implementation that demonstrates five core proof properties:

1. Canonical Collision (Deterministic Convergence)
2. Subfield Isolation (Domain Separation)
3. Deterministic Hash-to-Curve Mapping
4. Cryptographic Binding Proof (Signature Integrity)
5. Strict Retrieval (Zero-Knowledge Discovery)

Each section explains both the theory and the corresponding Python implementation.

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Goal: Different surface representations of the same fact must converge to the same encoded state.

For example:

• "Tax Rate: 10%"
• " Tax Rate:\n\t10% "

These should resolve to the same canonical representation before encoding.

Code Section

_ws_re = re.compile(r"\s+")

def canonical_fact(s: str) -> str:
    """
    Minimal canonicalization:
    - strip
    - collapse whitespace runs
    """
    s = s.strip()
    s = _ws_re.sub(" ", s)
    return s

Explanation

• Leading and trailing whitespace is removed.
• All whitespace runs collapse to a single space.
• The result is deterministic.
• Only canonicalized data is allowed into the curve layer.

This guarantees:

Same truth → Same canonical string → Same curve coordinate.

Without canonicalization, deterministic recovery collapses.

—

ECAI prevents “semantic bleed” across domains.

The same phrase in different legal or logical subfields must map to different coordinates.

Example:

• AU | "Tax Rate: 10%"
• UK | "Tax Rate: 10%"

These must not collide.

Code Section

def domain_separate(domain: str, canonical: str) -> bytes:
    """
    Subfield isolation.
    """
    return (domain + "|" + canonical).encode("utf-8")

Explanation

By prefixing the canonical fact with a domain tag, we guarantee:

• AU|Tax Rate: 10% ≠ UK|Tax Rate: 10%
• Even identical canonical strings remain mathematically isolated.

This is classical cryptographic domain separation applied to knowledge indexing.

—

We now map canonical, domain-separated knowledge into an actual elliptic curve point.

Important correction:

Random hash bytes are NOT automatically valid curve points.

Instead, we use deterministic try-and-increment.

Code Section

def hash_to_curve_point(domain: str, fact: str, max_tries: int = 4096):
    canon = canonical_fact(fact)
    seed = sha256(domain_separate(domain, canon))

    for i in range(max_tries):
        i_bytes = i.to_bytes(4, "big")
        x32 = sha256(seed + i_bytes)[:32]
        pk = _attempt_point_from_x32(x32, y_even=True)
        if pk is not None:
            return pk, i

    raise RuntimeError("Failed to find curve point")

Explanation

Process:

1. Canonicalize fact
2. Domain separate
3. Hash to produce seed
4. Iterate counter i
5. Attempt to interpret derived 32-byte value as x-coordinate
6. If valid → return curve point

This produces:

Deterministic, reproducible curve coordinates for structured knowledge.

Given the same input, the same point is always recovered.

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Encoding alone is not enough. We must prove:

• This domain
• This canonical fact
• This exact curve coordinate

were bound intentionally.

We achieve this with ECDSA signatures.

Code Section

def proof_message(domain: str, canonical: str, point_bytes: bytes) -> bytes:
    return b"|".join([
        b"ECAI_PROOF_v1",
        domain.encode("utf-8"),
        canonical.encode("utf-8"),
        point_bytes,
    ])

sig = signer_private_key.sign(msg, ec.ECDSA(hashes.SHA256()))

signer_public_key.verify(signature, msg, ec.ECDSA(hashes.SHA256()))

Explanation

The signature proves:

• The signer attested to this exact mapping.
• The binding cannot be altered without invalidating the signature.
• The curve coordinate is cryptographically anchored.

This gives:

Integrity + Authenticity + Non-Repudiation.

Note: ECC signatures are not post-quantum secure under Shor’s algorithm.

—

ECAI is not a prediction engine.

If knowledge is not stored, it is not inferred.

Code Section

index = {}
index[("AU", "Tax Rate")] = au.point_uncompressed

def get(term_domain: str, term: str):
    return index.get((term_domain, term), None)

Explanation

• Retrieval is exact.
• Missing entries return None.
• No fuzzy matching.
• No probabilistic guessing.

This enforces the philosophical principle:

If it was not encoded, it does not exist in the system.

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The demo function validates:

• Canonical convergence
• Subfield isolation
• Signature verification
• Deterministic mapping
• Strict retrieval behavior

if __name__ == "__main__":
    demo()

Full Source: ecai_proof.py

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1. ECC (secp256r1) is not post-quantum safe.
2. Try-and-increment is deterministic but not a formally proven uniform hash-to-curve.
3. Production systems should consider:
   • RFC 9380 hash-to-curve
   • Merkle inclusion proofs
   • Post-quantum signature schemes (Dilithium, SPHINCS+)

—

The system now demonstrates:

This transforms elliptic curves from:

• Key exchange primitives

into:

• Deterministic knowledge coordinates.

—

ECAI does not compute meaning probabilistically.

It:

1. Canonicalizes truth
2. Encodes deterministically
3. Anchors cryptographically
4. Retrieves strictly

No hallucination. No approximation. Only structured recovery.