The following documentation establishes the formal proofs, deterministic replay evidence, and cryptographic anchoring mechanisms for the Irreducible Vector System (IVS), prioritizing the strengthening of invariant semantics, replay completeness, and fork resistance.

I. Formal Invariant Proofs (Canonical Invariance Guarantees)

Architectural invariants in the IVS are transitioned from behavioral assertions to canonical requirements to ensure precision under hostile review.

1. $\Omega$ Mathematical Invariance ($V_0$)

The core operator $\Omega = (\text{state} + \text{bias}) \times \alpha$ serves as the axiomatic upstream for all transformations and must remain mathematically immutable.

• Axiomatic Independence: $\Omega$ exists as a pure algebraic operator and is independent of kernel execution, SE44 enforcement, or mesh consensus.
• Canonical Requirements: Invariance is enforced by defining a strict numeric input domain (e.g., SoftFloat sf64), disabling Fused Multiply-Add (FMA) contraction, and mandating a round-to-nearest-ties-to-even policy.
• Executable Assertion: Deterministic governance requires that for any triple of $(\text{state}, \text{bias}, \alpha)$, the function returns a bit-identical result across all runtime calls.

2. Deterministic Drift ($V_1$)

The Drift Engine ($\Psi_\ell$) must function as a purely deterministic temporal evolution mapping $\Omega_n \to \Omega_{n+1}$.

• Proof Logic: Drift must operate solely as a function of the prior fossilized state and explicitly declared inputs, prohibiting hidden randomness, implicit entropy, or unseeded RNG access.
• Environmental Constraints: Systemic determinism is guaranteed by the absence of implicit time dependencies, unordered iterations (e.g., dictionary traversal), or hardware flag dependencies.

3. SE44 Mechanical Refusal ($V_2$)

SE44 functions as a mechanical safety gate enforcing coherence ($C \ge 0.985$), entropy ($S \le 0.01$), and RMS drift ($\le 0.001$).

• Integer-Domain Comparison: To eliminate ULP divergence caused by floating-point comparisons near thresholds, the system scales values into a fixed-point integer domain (e.g., Q32.32) for all gate evaluations.
• Refusal Invariant: If any constraint is violated, the system must trigger a REBIND operation, reverting the runtime pointer to the last valid fossilized state and discarding the candidate.

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II. Replay Harness Evidence (Bitwise Completeness)

The replay harness establishes empirical determinism by demonstrating bitwise equivalence across divergent hardware environments.

1. Canonical Deterministic Substrate (V_{15})

To close proof gaps, the system utilizes a SoftFloat backend that replaces hardware-dependent floating-point behavior with a fully deterministic IEEE-754 software implementation. This ensures that hardware-specific features like AVX or x87 extended precision do not implicitly alter trajectories.

2. Deterministic Execution Procedure

• Canonical Seeding: The execution loop is initialized with a fixed seed (e.g., 0x5A17C3D9B842ULL) for all pseudo-random generators.
• Sequential Reduction: All vector reductions must use a canonical left-fold ordering to prevent non-deterministic parallel summation errors.
• Trace Comparison: Each execution step produces a SHA-256 hash of the serialized state. Bitwise identity is proven when binary traces from different architectures (e.g., x86-64 Linux and ARM64 macOS) are identical under a cmp utility.

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III. RFC-3161 Timestamp Integration Example

Integration with RFC-3161 elevates the IVS from a deterministic machine to an externally anchored identity system by embedding verifiable temporal tokens into the fossil hash.

1. Cryptographic Binding Policy

The timestamp token must be cryptographically inseparable from the fossil payload. During the fossilization of $\Omega_{n+1}$, the system generates an SHA-256 hash of the entry and submits it to a Time Stamping Authority (TSA).

2. OpenSSL-Based Implementation

def rfc3161_timestamp(data_bytes, tsa_url="http://timestamp.digicert.com"):
    # Generate timestamp query (SHA-256) via OpenSSL
    subprocess.run(["openssl", "ts", "-query", "-data", data_file_path,
                    "-sha256", "-cert", "-out", tsq_path], check=True)

    # Receive and store the signed reply from the TSA
    subprocess.run(["openssl", "ts", "-reply", "-queryfile", tsq_path,
                    "-out", tsr_path, "-text", "-url", tsa_url], check=True)
    # The resulting token is embedded directly in the Fossil Ledger entry.

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IV. Fork Detection Attack Scenario Demonstration

Fork resistance is maintained through strict chain linearity enforcement and genesis anchoring.

1. Scenario A: Non-Head Append (Branching Attempt)

An attacker attempts to insert a state entry that references an historical hash rather than the current latest_hash.

• Refusal: The FossilLedger validates that previous_hash matches the current chain head; a mismatch triggers an immediate exception and append denial.

2. Scenario B: Genesis Substitution Attack

An attacker attempts to present an entirely parallel shadow chain starting from a fraudulent block zero.

• Mitigation: Governance-grade hardening requires hard-coding the genesis hash into the runtime specification and binding the chain ID to the compiled infrastructure. The system rejects any chain that does not match the hard-coded root.

3. Scenario C: Metadata Detachment

An attacker attempts to modify a fossilized state while retaining the original timestamp metadata.

• Detection: The verify_chain() logic recomputes every block hash, validating the linkage and the embedded TSA token. Because the TSA token is part of the hashed payload, any mutation of the state results in a hash mismatch, rendering the alternative history detectable and invalid.