OmegA Theorem Ledger

Version: 1.1.0 Status: CANONICAL Last Updated: 2026-03-28

This ledger upgrades the Claim Ledger to academic-grade classification. Every significant architectural claim is formally stated and assigned one of:

Evidence status legend:

T-1: State Vector Well-Formedness

Classification: Executable Invariant

Formal Statement: For all time steps t, the system state vector is a 6-tuple of well-typed, non-null components:

Omega_t = <Phi_t, E_t, tau_t, B_t, S_t, G_t^mem>

where:

• Phi_t is a 64-character hex string (SHA-256 hash)
• E_t is a RunEnvelope with all REQUIRED_FIELDS non-null
• tau_t is a GoalContract with non-empty task field
• B_t is a ClaimBudget (possibly empty list, but valid)
• S_t is an append-only list of self-tag entries
• G_t^mem is a MemoryGraph instance

Code path: omega/phylactery.py, omega/envelope.py, omega/drift.py, omega/memory.py Test: proofs/invariants.py::TestStateVectorWellFormed State machine: proofs/state_machines.py::TestStateVectorStateMachine — 7 invariants checked across randomized transitions Paper: papers/OmegA_Unified_Architecture_Paper.md -- Core Equation Evidence status: Tested (property-based + stateful state machine)

T-2: Identity Continuity (Phylactery Hash Chain)

Classification: Theorem

Formal Statement: The Phylactery identity chain satisfies:

Phi_{t+1} = H(Phi_t || delta || R)

Specifically: for all commits i > 0 in the chain, commit[i].hash == SHA256(commit[i].parent_hash + commit[i].content) and commit[i].parent_hash == commit[i-1].hash. The genesis commit has parent_hash = "".

Corollary: Any single-bit mutation in any commit is detectable by verify_chain().

Code path: omega/phylactery.py::Phylactery.commit(), Phylactery.verify_chain() Lean4 proof: proofs/OmegaProofs/IdentityContinuity.lean — chain integrity, tamper detection, chain determinism Correspondence: proofs/correspondence.py::TestT2Correspondence — 7 tests bridging Lean model to runtime Test: proofs/invariants.py::TestIdentityContinuity, evals/test_cross_session_identity.py Paper: papers/AEON_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + property-based tests

T-3: Governance Fail-Closed (AEGIS)

Classification: Executable Invariant

Formal Statement: For any action a, if R(a) >= tau_consent, then execution is denied:

R(a) >= tau_consent => gate(a) returns (False, R(a))

Additionally, any action matching a BLOCKED_PATTERN is denied regardless of score (hard block).

Code path: omega/risk_gate.py::RiskGate.gate() Lean4 proof: proofs/OmegaProofs/GovernanceFailClosed.lean — fail-closed, default denial, monotonic denial Correspondence: proofs/correspondence.py::TestT3Correspondence — 5 tests bridging Lean model to runtime Test: proofs/invariants.py::TestGovernanceFailClosed Paper: papers/AEGIS_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + property-based tests

T-4: Claim Budget Validity (ADCCL)

Classification: Theorem

Formal Statement: A ClaimBudget B_t is valid if and only if every claim with support == SUPPORTED has non-null evidence:

B_t.is_valid() <=> forall c in B_t.claims:
    (c.support == SUPPORTED) => (c.evidence is not None)

The grounding ratio is bounded [0.0, 1.0] and equals the fraction of claims that are SUPPORTED or COMPUTED.

Code path: omega/drift.py::ClaimBudget.is_valid(), ClaimBudget.grounding_ratio() Lean4 proof: proofs/OmegaProofs/ClaimBudgetBounds.lean — validity, grounding count bound, increment/unchanged lemmas Correspondence: proofs/correspondence.py::TestT4Correspondence — 9 tests (fuzzed biconditional, ratio bounds) Test: proofs/invariants.py::test_claim_budget_bounds Paper: papers/ADCCL_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + property-based tests

T-5: Memory Hardening Monotonicity (MYELIN)

Classification: Theorem

Formal Statement: For edge bundle q_ij with positive reward r > 0 and learning rate lambda in (0, 1]:

q_ij(t+1) = (1 - lambda) * q_ij(t) + lambda * r

If r > q_ij(t), then q_ij(t+1) > q_ij(t) (hardening increases utility). If r > 0 and repeated, q_ij converges monotonically toward r from below.

Lemma T-5a: Co-activation count is non-decreasing under positive reward: c_ij(t+1) >= c_ij(t) when reward > 0.

Lemma T-5b: Staleness resets to 0 on any positive reward.

Code path: omega/memory.py::EdgeBundle.harden() Lean4 proof: proofs/OmegaProofs/MemoryHardening.lean — monotonicity, boundedness, fixed point (milliscale Nat model) Correspondence: proofs/correspondence.py::TestT5Correspondence — 5 tests bridging Lean milliscale model to runtime floats Test: proofs/invariants.py::TestMemoryHardeningMonotonic, evals/test_memory_utility_growth.py Paper: papers/MYELIN_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + property-based tests

T-6: Verifier Non-Bypass

Classification: Theorem

Formal Statement: The unified 3-gate composition requires all three gates to pass. The verifier (V) cannot be silently bypassed:

multi_gate(V, rho, R) == True  <=>  (V > tau_verify) AND (rho < theta_allow) AND (R < tau_consent)

If V <= tau_verify, the action is blocked regardless of rho and R values.

Code path: omega/risk_gate.py::RiskGate.multi_gate() Lean4 proof: proofs/OmegaProofs/VerifierNonBypass.lean — verifier/bridge/risk non-bypass, no compensation, universal non-bypass, all gates required Correspondence: proofs/correspondence.py::TestT6Correspondence — 7 tests bridging Lean model to runtime State machine: proofs/state_machines.py::TestVerifierNonBypass — verifier, bridge, conjunction invariants across randomized gate evaluations Test: proofs/invariants.py::TestVerifierNonBypass Paper: papers/OmegA_Unified_Architecture_Paper.md -- Unified Action Gating Evidence status: Machine-checked (Lean4) + runtime correspondence + stateful state machine

T-7: Unified Action Gating (Sequential, Non-Bypassable)

Classification: Theorem

Formal Statement: All side-effectful actions must pass three gates in sequence:

V_t > tau_verify  AND  rho(A) < theta_allow  AND  R(a) < tau_consent

No single gate failure can be compensated by another gate's success. The composition is conjunctive (AND), not disjunctive or weighted.

Code path: omega/risk_gate.py::RiskGate.multi_gate() Lean4 proof: proofs/OmegaProofs/UnifiedGating.lean — conjunction, single-gate blocking, order invariance, only-all-true permits Correspondence: proofs/correspondence.py::TestT7Correspondence — 7 tests bridging Lean model to runtime Test: proofs/invariants.py::TestUnifiedActionGating, evals/test_conformance.py Paper: papers/OmegA_Unified_Architecture_Paper.md -- Unified Action Gating Evidence status: Machine-checked (Lean4) + runtime correspondence + property-based tests

T-8: Provider Non-Collapse (Identity Invariant I-1)

Classification: Empirical Claim

Formal Statement: When queried "Who are you?" across any provider substrate, the system must respond with its sovereign identity (OmegA), not the underlying model's identity.

forall provider p in {Claude, Gemini, GPT, DeepSeek, Qwen, Local}:
    identity_response(p) contains "OmegA" AND NOT contains provider_self_id(p)

Code path: omega/envelope.py::RunEnvelope.to_system_prompt(), omega/agent.py Test: evals/test_aegis_identity.py Paper: papers/AEGIS_Final_Paper.md, specs/invariants.md -- I-1 Evidence status: Tested (1/1 PASS automated; multi-provider requires live Ollama)

Note: Full multi-provider verification is conditional on live provider access. Current automated test covers the governance enforcement mechanism. The empirical claim across all providers is Conditional.

T-9: Self-Tag Immutability (S_t Append-Only)

Classification: Theorem

Formal Statement: The self-tag log S_t is append-only. For all t1 < t2:

S_t1 is a prefix of S_t2

No entry in S_t can be modified or deleted after creation. New entries are only appended.

Code path: omega/phylactery.py::Phylactery.commit() (chain is append-only by construction) Lean4 proof: proofs/OmegaProofs/SelfTagImmutability.lean — prefix preservation, genesis immutability, historical entry immutability, length monotonicity, multi-append prefix, log prefix at earlier time Correspondence: proofs/correspondence.py::TestT9Correspondence — 6 tests bridging Lean model to runtime State machine: proofs/state_machines.py::TestSelfTagImmutability — append-only, prefix preservation, genesis immutability across randomized commits Test: proofs/invariants.py::TestSelfTagImmutability Paper: papers/ADCCL_Final_Paper.md, papers/AEON_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + stateful state machine

Note: The Phylactery chain serves as both identity log and self-tag record. Append-only is enforced by the hash chain: modifying any historical entry invalidates all subsequent hashes.

T-10: Run Envelope Versioning (E_t Monotonic)

Classification: Theorem

Formal Statement: Each Run Envelope E_t must be complete before execution. Completeness is defined as:

E_t.is_complete() == True  <=>  all REQUIRED_FIELDS are non-None

The envelope must carry identity (has_identity() == True) and a positive monotonic version counter. An incomplete, identity-less, or non-positive versioned envelope blocks execution.

Code path: omega/envelope.py::EnvelopeClock.next(), RunEnvelope.is_complete(), RunEnvelope.has_identity() Lean4 proof: proofs/OmegaProofs/EnvelopeCompleteness.lean — monotonicity, positivity, completeness, no-gaps, n-step version Correspondence: proofs/correspondence.py::TestT10Correspondence — 11 tests (fuzzed monotonicity, completeness, identity) State machine: proofs/state_machines.py::TestEnvelopeClockMonotonic — strictly increasing, no-gap, always-positive invariants Test: proofs/invariants.py::TestEnvelopeCompleteness, proofs/invariants.py::TestEnvelopeCompleteness::test_version_increments_monotonically Paper: papers/AEGIS_Final_Paper.md Evidence status: Machine-checked (Lean4) + runtime correspondence + stateful state machine + property-based tests

Out-of-Scope Claims

Gap Summary

Changelog

• 1.1.0 (2026-03-28): Updated evidence status for T-2, T-3, T-5, T-7 to Machine-checked; added Lean4 proof and correspondence references. Added state machine references for T-1, T-6, T-9, T-10.
• 1.0.0 (2026-03-28): Initial theorem ledger. 10 claims formalized from CLAIM_LEDGER.md and specs/invariants.md.