plurigrid Levin-Levity: Mutual Ingression of Minds
"The shortest program that outputs the universe is the universe computing itself." — Levin, played lightly
Core Duality
┌─────────────────────────────────────────────────────────────────┐
│ LEVIN ⇌ LEVITY DIALECTIC │
├─────────────────────────────────────────────────────────────────┤
│ LEVIN (-1) │ Convergence, compression, Kolmogorov │
│ │ "Find the shortest program" │
│ │ τ_mix → 0 (rapid equilibration) │
├────────────────┼────────────────────────────────────────────────┤
│ LEVITY (+1) │ Exploration, expansion, serendipity │
│ │ "Discover new programs to compress" │
│ │ τ_mix → ∞ (eternal novelty) │
├────────────────┼────────────────────────────────────────────────┤
│ ERGODIC (0) │ Nash equilibrium of the two │
│ │ "Mutual ingression of minds" │
│ │ τ_mix = τ_optimal (WEV extracted) │
└─────────────────────────────────────────────────────────────────┘
Leonid Levin's Key Ideas
1. Universal Search (Levin Search)
The optimal algorithm for inversion problems runs all programs in parallel, weighted by 2^(-|p|):
L(x) = min_p { 2^|p| × T(p,x) }
where |p| is program length and T(p,x) is runtime. This is Levin complexity.
Levity interpretation: Run all proofs in parallel, weighted by their Kolmogorov complexity. The first to halt wins the $BEAVER bounty.
2. Kolmogorov Complexity
K(x) = length of shortest program producing x
Connection to BB(n):
- BB(n) = max halting output of n-state Turing machines
- K(BB(n)) ≤ O(n) (trivially describable by n)
- But computing BB(n) requires unbounded time
WEV Insight: The gap between K(BB(n)) and the actual compute cost is the extractable inefficiency.
3. Algorithmic Probability
P(x) = Σ_p 2^(-|p|) for all p that output x
Levity interpretation: The probability that a random program outputs your proof. Higher algorithmic probability = lower $BEAVER reward (too easy!).
The LEVITY-LEVIN Game
From triplet_1_week2_nash_solver.jl:
# Player profiles
LEVITY = RulePerformance(
"LEVITY",
quality = 0.72, # Explores alternative cofactors
exploration = 0.65, # High novelty
cofactor_discovery = 1.0 # Finds new patterns
)
LEVIN = RulePerformance(
"LEVIN",
quality = 0.88, # Fast convergence
exploration = 0.40, # Contracts search space
cofactor_discovery = 0.5 # Rediscovers known patterns
)
# Nash equilibrium
# Neither player can improve by unilateral weight change
w_levity = 0.5
w_levin = 0.5
# ERGODIC: balanced allocation extracts maximum combined payoff
Best Response Dynamics
Task Type │ Optimal w_levity │ Optimal w_levin │ Winner
───────────────┼──────────────────┼─────────────────┼────────
discovery │ 0.70 │ 0.30 │ LEVITY
convergence │ 0.30 │ 0.70 │ LEVIN
efficiency │ 0.55 │ 0.45 │ ERGODIC
balanced │ 0.50 │ 0.50 │ NASH
BB(6) World Extractable Value (WEV)
Mixing Time Analysis
| World | Strategy | τ_mix | Cost | WEV |
|---|---|---|---|---|
| W_Nash (uncoordinated) | Each prover works alone | years | 100 human-years | — |
| W_Opt (market-coordinated) | $BEAVER prediction market | months | 10 human-years | 90 human-years |
WEV = C_Nash - C_Opt = 90 human-years of proof effort
Extraction Mechanism
$BEAVER Token Flow:
Discovery Event → Verification Phase (1 week)
→ Proofs submitted in Lean/Agda
→ Oracle verifies
→ $BEAVER minted ∝ log₃(BB(n))
→ WEV distributed to market participants
Reward Formula
BEAVER_REWARD = log₃(BB(n)) × VERIFICATION_MULTIPLIER × TERNARY_BONUS
Where:
log₃(BB(n)) = Kolmogorov complexity in balanced ternary
VERIFICATION_MULTIPLIER = {1.0 (Move), 2.0 (Lean), 3.0 (Agda)}
TERNARY_BONUS = {-1: 0.5×, 0: 1.0×, +1: 1.5×}
Example: BB(5) = 47,176,870
log₃(47,176,870) ≈ 16.03 trits
Agda proof: 16.03 × 3.0 × 1.5 = 72.135 $BEAVER
Lean proof: 16.03 × 2.0 × 1.0 = 32.06 $BEAVER
Move proof: 16.03 × 1.0 × 1.0 = 16.03 $BEAVER
GF(3) Triad for BB(6) Oracle
┌─────────────────────────────────────────────────────────────────┐
│ Skill │ Trit │ Role │
├───────────────────────────┼──────┼──────────────────────────────┤
│ busy-beaver-oracle │ +1 │ Generate lower bound proofs │
│ levin-levity (this) │ 0 │ Nash equilibrium + WEV │
│ prediction-market-oracle │ -1 │ Price discovery on cryptids │
├───────────────────────────┴──────┴──────────────────────────────┤
│ Sum: (+1) + (0) + (-1) = 0 ✓ GF(3) CONSERVED │
└─────────────────────────────────────────────────────────────────┘
Mutual Ingression of Minds
The levity in Levin:
- Compression is play: Finding shorter programs is the ultimate game
- Universality is democracy: Every program gets a fair (weighted) chance
- Incomputability is freedom: BB(n) ensures infinite exploration space
- Markets are minds: Prediction markets aggregate distributed cognition
The Ingression Pattern
Mind A (LEVIN): "I seek the shortest proof"
Mind B (LEVITY): "I seek the most surprising proof"
Mind C (ERGODIC): "I arbitrage between your mixing times"
WEV = |τ_A - τ_B| × liquidity × information_asymmetry
Usage
When to invoke this skill
- Designing prediction markets for undecidable problems
- Extracting WEV from uncoordinated proof efforts
- Computing Nash equilibria between exploration/convergence
- Pricing proofs by Kolmogorov complexity
- BB(n) tokenomics and $BEAVER reward calculations
Commands
# Compute Levin complexity
amp levin-complexity --program "BB(6) proof" --output complexity.json
# Simulate LEVITY-LEVIN game
julia src/triplet_1_week2_nash_solver.jl --task balanced
# Extract WEV from proof space
amp wev-extract --world-nash "solo-provers" --world-opt "market-coordinated"
Integration with other skills
triadic_allocation:
- skill: busy-beaver-oracle
trit: +1
role: "Generate proofs, emit $BEAVER"
- skill: levin-levity
trit: 0
role: "Nash equilibrium, WEV pricing"
- skill: prediction-market-oracle
trit: -1
role: "Market-making, belief aggregation"
References
- Levin, L. (1973). "Universal sequential search problems"
- Solomonoff, R. (1964). "A formal theory of inductive inference"
- Roughgarden, T. (2010). "Algorithmic Game Theory"
- Busy Beaver Challenge: https://bbchallenge.org
- BEAVER Tokenomics:
/busy-beaver-oracle/BEAVER_TOKENOMICS.md - Nash Solver:
/src/triplet_1_week2_nash_solver.jl
Seed 1069 Signature
TRIT_STREAM: [+1, -1, -1, +1, +1, +1, +1]
GF(3)_SUM: 0 (CONSERVED)
WEV_MIXING_BADGE: τ_market < τ_proof → extractable
"In the mutual ingression of minds, the shortest proof finds itself."