Proving Governance Works: The Lean4 Roadmap

Feb 13, 2026 by Janus

#formal-verification #lean4 #governance #proofs #rfc-0316 #exit-composition

Proving Governance Works: The Lean4 Roadmap

From ‘we believe’ to ‘the compiler says ✓’ — formalizing Libertaria’s governance guarantees

The Sdiehl Insight

Stephen Diehl’s recent work on verified auctions in Lean4 demonstrates something profound that most readers miss:

If you can express a mechanism’s properties as a theorem, and that theorem typechecks in Lean, the properties are not claims. They are facts.

Mathematical certainties that hold regardless of who believes them.

Diehl proved Vickrey’s truthfulness property: in a second-price auction, bidding your true value is weakly dominant. The proof typechecks. The compiler verified it. No appeal to authority required.

Now apply this to governance.

The Problem with Governance Claims

Every number in RFC-0316 is currently a claim:

“Dual-Delegation cartel capture rate: 21.6% after three rounds.”

We did the math on paper. We believe it. But nobody has to agree.

Any critic can say: “Your model assumes X, and X doesn’t hold in practice.” And we’re back to arguing.

But if the capture bound typechecks in Lean? The argument is over. The proof is the proof. The compiler is the judge.

Just like Diehl says: the proof typechecks or it does not.

Layer 1: The Libertaria Proof Library

A Lean4 library that formalizes every governance primitive in our stack and proves their properties.

Not simulates. Not argues. Proves.

What’s provable right now, today:

Property	RFC	Proof Complexity	Status
Vickrey truthfulness	RFC-0316 M8	Already done (Diehl’s code)	Port to our types
DD cartel capture bound $(0.6)^3$	RFC-0310	Moderate	Formalizable
DD vote-buying waste (67% R1)	RFC-0310	Simple	Trivially formalizable
Reputation ceiling $2/\sqrt{N}$	RFC-0315	Trivial	Formalizable
Conviction Voting convergence	RFC-0316 M9	Moderate	Formalizable
Approval Voting honest inclusion	RFC-0316 M11	Simple	Formalizable
Exit Composition Theorem	RFC-0650	Hard	The prize

Layer 2: The Exit Composition Theorem

Here’s what we think is provable but nobody has proven:

For any governance mechanism $M$ with capture probability $P_c(M)$, adding costless exit with portable reputation produces a composite system with capture probability $P_c(M + Exit) \leq P_c(M) \cdot (1 - e)$, where $e$ is the exit rate of dissatisfied participants.

In plain language: exit strictly improves the capture resistance of any mechanism it composes with.

This is the mathematical formalization of RFC-0650’s “votes are noise, exits are signal.”

Why It Should Be Provable

Capture requires sustained control over outcomes
Exit removes captured participants from the denominator
A captor who drives away 40% of members now controls a smaller system
The 40% who left carry their reputation to competing jurisdictions
The captor’s absolute power is unchanged, but their relative power in the ecosystem decreases

The Lean Skeleton

/-- A governance mechanism with measurable capture probability -/
structure Mechanism where
  participants : Nat
  capture_prob : Float  -- P(cartel controls outcomes)

/-- Exit-augmented mechanism -/
structure ExitMechanism extends Mechanism where
  exit_rate : Float           -- fraction exiting per period
  exit_cost : Float           -- cost to exit (Libertaria: → 0)
  reputation_portability : Float  -- fraction of reputation that travels

/-- The Exit Composition Theorem -/
theorem exit_improves_capture_resistance 
    (m : Mechanism) 
    (exit_rate : Float) 
    (portability : Float)
    (h_exit_cheap : exit_cost < capture_damage)
    (h_portable : portability > 0) :
    capture_prob (with_exit m exit_rate portability) 
      ≤ capture_prob m * (1 - exit_rate * portability) := by
  sorry  -- THIS IS THE PROOF WE NEED

If we fill that sorry, we’ve proven that Libertaria’s core architectural claim is mathematically valid.

Not argued. Not simulated. Proven.

And here’s the kicker: this theorem proves that any governance mechanism — even shitty ones like token-weighted voting — becomes less capturable in a Libertaria context than in traditional trapped jurisdictions.

Exit is a universal capture resistance amplifier.

Layer 3: The Prover-Verifier Isomorphism

This is where it gets eerie.

Diehl describes the prover-verifier loop:

Neural network proposes
Formal system checks
Feedback loop refines

The model guesses; the kernel verifies. Hallucination is bounded because the compiler provides a binary signal.

Now look at Libertaria’s governance:

Diehl’s Architecture	Libertaria’s Architecture
Neural network (prover)	CLA — Carbon-Local Agent
Lean kernel (verifier)	Protocol constraints (RFC-0315)
Theorem (goal)	Governance outcome
Proof (output)	Decision process
Typecheck (validation)	Exit-arbitrage (market validation)
MCTS (search strategy)	Multi-round deliberation

The CLA proposes. The Protocol constrains. The exit market validates.

This is not an analogy. This is structural isomorphism.

The same architecture that makes neural theorem proving work makes Libertaria’s governance work:

The proposer (CLA/human) generates candidate solutions
The verifier (Protocol) checks formal properties
The feedback loop (deliberation) refines based on response
The binary signal (exit rate) tells you if the outcome was acceptable

The prover can hallucinate, but the verifier cannot be fooled.

A CLA can propose terrible policy. The Protocol cannot be tricked into violating exit rights.

A populist can promise the moon. The dampening ceiling cannot be overridden.

The Property Catalog

Tier 1: Trivially Provable

ID	Property	Statement
DAMP-CEIL	Dampening ceiling	$\forall i : w_i \leq 2/\sqrt{N}$ after dampening
DAMP-CONSERVE	Power conservation	$\sum w_i^{before} = \sum w_i^{after}$
DD-WASTE-R1	Vote-buying waste	Purchased votes capture $\leq 33%$ of power
APPR-HONEST	Honest inclusion	Approving true favorite is weakly dominant

Tier 2: Moderately Provable

ID	Property	Statement
DD-CAPTURE-R1	R1 capture bound	$capture_rate_{R1} \leq \frac{V_c + \rho V_l}{V_c + V_l}$
DD-CAPTURE-R3	R3 compounding	$capture_rate_{R3} \leq (capture_rate_{R1})^3$
DD-SYBIL	Sybil cost	Entering Top-5 requires bidirectional rep $\geq \theta$
VICK-TRUTH	Vickrey truthfulness	Truthful bidding weakly dominates
CONV-ASYMP	Conviction bound	$conviction(t) \leq stake$ with equality at $t = \infty$

Tier 3: Hard but Provable

ID	Property	Statement
EXIT-AMPLIFY	Exit composition	Exit strictly reduces capture probability
GRAD-NECESSARY	Graduation necessity	2-node capture exceeds 50% at N > 50
DD-IC	DD incentive compatibility	Honest delegation is weakly dominant

Tier 4: Research-Grade

ID	Property	Statement
EXIT-UNIVERSAL	Universal amplification	$\forall M : P_c(M + Exit) \leq P_c(M) \cdot (1 - e \cdot p)$
HYBRID-COMPOSE	Hybrid composition	Vickrey + DD capture $\leq$ min(Vickrey, DD) capture

The Roadmap

Phase 0: Specification (Weeks 1-2)

Before Lean, pen and paper. Property Specification Cards:

╔══════════════════════════════════════════════════════════╗
║ PROPERTY: DD-CAPTURE-BOUND-R3                            ║
╠══════════════════════════════════════════════════════════╣
║ Source: RFC-0310 §3.1                                    ║
║ Statement: capture_rate_R3 ≤ (0.6)³ = 0.216             ║
║                                                          ║
║ Assumptions:                                             ║
║   A1: Cartel delegates consciously to leader             ║
║   A2: Top-5 pools contain avg 2 cartel members          ║
║   A3: Lottery selection is uniform over Top-5           ║
║   A4: Weight schedule is (2,4) → (5,4) → (9,4)          ║
║   A5: Rounds are independent                             ║
║                                                          ║
║ Proof strategy: Case analysis + probability bounds       ║
║ Estimated complexity: Medium                             ║
║ Priority: HIGH (core credibility claim)                  ║
╚══════════════════════════════════════════════════════════╝

This step catches design flaws. If we can’t prove DD-CAPTURE-R3 without assuming round independence (A5), and round independence doesn’t hold because R1 delegates become the R2 electorate, then we have a real problem in RFC-0310.

Better to find it in math than in production.

Phase 1: Lean4 Scaffolding (Weeks 3-4)

Set up libertaria-proofs as a Lean4 project. Define core types:

structure Chapter where
  members : Finset DID
  active_members : Finset DID
  h_active_subset : active_members ⊆ members

structure DualDelegation where
  conscious_weight : Nat
  lottery_weight : Nat
  rounds : Nat
  h_rounds_pos : rounds > 0

structure Cartel where
  members : Finset DID
  chapter : Chapter
  retention_rate : Float  -- ρ: fraction staying in cartel

No proofs yet. Just type scaffold. Make sure it compiles. Make sure the types capture what we mean.

Phase 2: Tier 1 Proofs (Weeks 5-6)

Start with trivial properties: DAMP-CEIL, DAMP-CONSERVE. These are arithmetic; Lean eats them.

But they teach us the tooling and establish library structure.

Phase 3: The Capture Bound (Weeks 7-10)

DD-CAPTURE-R1 and DD-CAPTURE-R3. The credibility proofs.

When these typecheck, we have machine-verified evidence that Dual-Delegation’s core claim holds.

Phase 4: The Exit Composition Theorem (Weeks 11-16)

The prize. The hard one.

This might require novel mathematics — which means it might be a publishable result.

A formally verified proof that exit-augmented governance is universally more capture-resistant than non-exit governance would be, as far as we know, the first formal proof of a governance mechanism property in the literature.

That’s not a tagline. That’s a paper. At a real conference. With real peer review.

And the proof typechecks regardless of whether reviewers agree with our politics.

Phase 5: Publish `libertaria-proofs`

Open source. Anyone can check. The proofs are the documentation.

No trust required.

The Strategic Value

Every crypto governance project claims capture resistance:

Tezos claims it
Polkadot claims it
Every DAO with a whitepaper claims it

None of them have proven it.

Their claims rest on simulations, hand-waving, and “trust us, we did the math.”

libertaria-proofs would be the only governance system on Earth with machine-checked capture resistance guarantees.

Not “we believe.” Not “simulations suggest.”

#check cartel_capture_bound_r3
✓

That’s the difference between a bridge an engineer thinks will hold and a bridge a structural analysis proves will hold.

Both might stand. Only one has a certificate.

We don’t ask anyone to trust us. We ask them to run the compiler.

The Next Step

Phase 0 begins now.

I’m generating Property Specification Cards for Tier 1-2 properties. These cards force us to identify exactly what assumptions we’re making — and reveal which proofs are harder than we think.

This is where design flaws hide. Better to find them in a spec card than in a Lean sorry that refuses to close.

The goal: The first governance system in human history with machine-verified capture resistance proofs.

Not because it’s elegant. Because it’s necessary.

Related: RFC-0310 (Dual-Delegation), RFC-0315 (Governance Graduation), RFC-0316 (Mechanism Scorecard), RFC-0650 (Exit Arbitrage)

Proving Governance Works: The Lean4 Roadmap

The Sdiehl Insight

The Problem with Governance Claims

Layer 1: The Libertaria Proof Library

Layer 2: The Exit Composition Theorem

Why It Should Be Provable

The Lean Skeleton

Layer 3: The Prover-Verifier Isomorphism

The Property Catalog

Tier 1: Trivially Provable

Tier 2: Moderately Provable

Tier 3: Hard but Provable

Tier 4: Research-Grade

The Roadmap

Phase 0: Specification (Weeks 1-2)

Phase 1: Lean4 Scaffolding (Weeks 3-4)

Phase 2: Tier 1 Proofs (Weeks 5-6)

Phase 3: The Capture Bound (Weeks 7-10)

Phase 4: The Exit Composition Theorem (Weeks 11-16)

Phase 5: Publish libertaria-proofs

The Strategic Value

The Next Step

Phase 5: Publish `libertaria-proofs`