PIR Evolver Deep Dive: From Thue-Morse to Trading Signals

The Journey

This article traces the complete path from raw market data to a trading signal — through the PIR Evolver Engine. Every step is grounded in mathematics that was discovered in the 1800s and proven in our neural network research.

Step 1: The Thue-Morse Sequence

The Thue-Morse sequence is a binary sequence where the n-th term is determined by the parity of the number of 1s in the binary representation of n:

n:      0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15
binary: 0  1  10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
1s:     0  1  1  2  1   2   2   3   1    2    2    3    2    3    3    4
sign:   +  -  -  +  -   +   +   -   -    +    +    -    +    -    -    +

Key property: for n = 2^k, the signed sum is exactly zero. The sequence is at perfect equilibrium at powers of two. This is not an approximation — it is a mathematical fact.

Step 2: Applying Signs to Returns

Given a return sequence r₁, r₂, ..., rₙ, we assign Thue-Morse signs:

signed_sum = Σᵢ sign(i) × rᵢ

If the returns are balanced (equal positive and negative contributions), the signed sum is near zero. If there's a structural imbalance — e.g., all the positive returns happen at even indices — the signed sum deviates.

Step 3: Balance Gap

The balance gap is the absolute value of the signed sum, normalized by the number of terms:

balance_gap = |signed_sum| / n

• balance_gap ≈ 0: Market is in equilibrium (stable regime)
• balance_gap >> 0: Market has shifted (regime transition)

This is computed in O(n) time — one pass through the return sequence. No matrix operations, no optimization, no parameter tuning.

Step 4: Pattern Area (Entropy)

The pattern area is the shoelace area of the signed cumulative path:

path(i) = Σⱼ₌₀ⁱ sign(j) × rⱼ
area = ½ |Σᵢ (path(i) × path(i+1) - path(i+1) × path(i))|

This measures the "shape" of the return sequence:

• Low area: Clean trend — the path goes in one direction
• High area: Scattered — the path zigzags, indicating uncertainty

Step 5: Regime Classification

Combining balance gap and pattern area:

Balance Gap	Pattern Area	Returns	Regime
Low	Low	Positive	Bullish
Low	Low	Flat	Range
Low	Low	Negative	Bearish
High	High	Any	Transition

The "Transition" state is the most valuable signal — it warns that the current regime is breaking down before the new regime is established.

Step 6: Position Sizing

Each regime has a confidence score derived from how far the balance gap is from the historical baseline:

confidence = 1 - (current_gap / max_historical_gap)

Position sizing:

• Bullish + high confidence: Full long position
• Bullish + low confidence: Reduced position (half)
• Transition: Exit to cash
• Bearish: Short position (if enabled) or cash
• Range: Minimal position, wait for breakout

Step 7: The Rebalancer

The top-16 crypto momentum rebalancer:

1. Compute PIR metrics for all 16 assets every tick
2. Classify each asset's regime
3. Compute confidence scores
4. Allocate capital proportional to confidence × expected return
5. Rebalance on regime transitions (not on fixed time intervals)

This means the system trades when the math says the regime has changed — not when a timer fires. Fewer trades, better timing.

Why This Works

Markets have structural equilibria. Trends, ranges, and consolidation patterns are all forms of equilibrium. PIR detects when that equilibrium breaks — when the "shape" of returns stops being balanced.

This is the same math that detects when a neural network's weights lose equilibrium (TPNN structural health). The domain is different (financial returns vs. network weights), but the math is identical:

PIR balance gap = |Σᵢ sign(i) × xᵢ| / n

where x can be anything — returns, weights, mastery scores, or cell states.

SignalO

The PIR Evolver Engine powers SignalO, our live signal feed product. SignalO exposes the engine's output as a real-time subscription service:

• Per-asset regime classification (updated every tick)
• Balance gap and pattern area values
• Regime transition alerts (push + email)
• Historical replay for backtesting
• API access for programmatic integration

Educational disclaimer: SignalO is for educational purposes only. PIR regime detection is experimental. No signal can predict future market performance. Always do your own research.