rewards.md

---
alias: learning incentives, learning rewards
tags: cyber, article, cip
crystal-type: process
crystal-domain: economics
crystal-size: article
status: draft
---
# learning incentives

one mechanism within tokenomics: how $CYB is minted, burned, and locked to reward knowledge creation in the cybergraph

knowledge creation is costly, but its benefits are collective. without incentives, rational agents free-ride on others' cyberlinks. this mechanism makes contributing profitable — and free-riding unprofitable

## the signal: Δφ*

every reward traces back to one quantity: how much did your action shift the tri-kernel fixed point φ*?



$$\text{reward}(v) \propto \Delta\phi^*(v)$$



φ* is the stationary distribution of the composite operator $\mathcal{R} = \lambda_d D + \lambda_s S + \lambda_h H_\tau$ — diffusion explores, springs enforce structure, heat kernel adapts. the collective focus theorem proves φ* exists, is unique, and is computable locally

Δφ* is the gradient of system free energy. creating valuable structure is literally creating value. no designed loss function — physics defines what should be optimized

## reward functions

five candidates for measuring convergence contribution, each with trade-offs:

| function | formula | strength | weakness |
|---|---|---|---|
| Δφ* norm | $\sum_j \|\phi^*_j^{(t+1)} - \phi^*_j^t\|$ | simple, easy to verify | gameable by oscillation |
| syntropy growth | $H(\phi^*_t) - H(\phi^*_{t+1})$ | rewards semantic sharpening | computationally heavier |
| spectral gap | $\lambda_2^t - \lambda_2^{t+1}$ | measures global convergence speedup | expensive, non-local |
| predictive alignment | $\text{align}(\phi^{(t+1)}, \phi^*_T)$ | favors early correct contributions | requires delayed validation |
| DAG weight | descendant blocks referencing this one | rewards foundational work | slow to accrue |

the hybrid model combines them:



$$R = \alpha \cdot \Delta\phi^* + \beta \cdot \Delta J + \gamma \cdot \text{DAGWeight} + \epsilon \cdot \text{AlignmentBonus}$$



where $\Delta J = H(\phi^*_t) - H(\phi^*_{t+1})$ is syntropy growth. fast local rewards use Δφ* and ΔJ. checkpoints add alignment and spectral verification bonuses. validators sample and verify blocks probabilistically

## link valuation

cyberlinks are yield-bearing epistemic assets. they accrue rewards over time based on contribution to focus emergence:



$$R_{i \to j}(T) = \int_0^T w(t) \cdot \Delta\phi^*_j(t) \, dt$$



where $\Delta\phi^*_j(t)$ = change in focus on target particle $j$ attributable to the link, $w(t)$ = time-weighting function, $T$ = evaluation horizon

| link type | characteristics | reward trajectory |
|---|---|---|
| viral | high Δφ* short-term | early peak, fast decay |
| foundational | low Δφ* early, grows later | slow rise, long reward |
| confirming | low individual Δφ*, strengthens axon weight | shared reward via attribution |
| semantic bridge | medium, cross-module | moderate, persistent |

## attribution

multiple neurons contribute cyberlinks in the same epoch. the total Δφ* shift is a joint outcome — how to divide credit fairly?

the Shapley value answers: each agent's reward equals their average marginal contribution across all possible orderings. in this system, the coalition's total value is the free energy reduction $\Delta\mathcal{F}$, and each agent's marginal contribution is how much φ* shifts when their cyberlinks are added to the graph. Shapley distributes the total Δφ* reward proportionally to each neuron's causal impact

exact computation is infeasible ($O(n!)$). probabilistic shapley attribution approximates:

1. local marginal — compute each transaction's individual $\Delta\mathcal{F}$ (add link, measure φ* shift)
2. Monte Carlo sampling — sample $k$ random orderings of the epoch's transactions, measure marginal contributions in each ordering
3. hierarchical batching — cluster transactions by affected neighborhood, distribute within clusters
4. final reward: $R_i = \alpha \cdot \Delta\mathcal{F}_i + (1-\alpha) \cdot \hat{S}_i$

where $\Delta\mathcal{F}_i$ is the fast local estimate and $\hat{S}_i$ is the sampled Shapley approximation. $\alpha$ balances speed (local marginal) against fairness (Shapley)

complexity: $O(k \cdot n)$ with $k \ll n$. feasible for 10⁶+ transactions per epoch

## self-minting

rewards are not computed centrally. each neuron proves their own contribution and claims their own reward.

every signal carries a $\Delta\phi^*$ — the neuron's locally computed focus shift for a batch of cyberlinks. this $\Delta\phi^*$ is proven correct by a single stark proof referencing a specific $\text{bbg\_root}$. the proof is the reward claim:

1. neuron creates signal with one or more cyberlinks, $\Delta\phi^*$, and stark proof
2. proof demonstrates: applying these links to the graph at $\text{bbg\_root}_t$ shifts φ* by $\Delta\phi^*$
3. any verifier checks the proof against the header — O(log n), no recomputation
4. if valid and Δφ* > 0, the neuron mints $CYB proportional to the proven shift

no aggregator decides the reward. the proof IS the mining. a neuron on a phone: buy a header, query neighborhood state, create cyberlinks, prove Δφ*, bundle into a signal, mint tokens

conservation: total minting per epoch is bounded by the actual global Δφ*, verifiable from consecutive headers. if the sum of individual claims exceeds the actual shift (overlapping neighborhoods), all claims are scaled proportionally

see §6.9 and §14.2 of the whitepaper for the full specification

## the three token operations

- mint: neurons prove Δφ* via stark and self-mint $CYB proportional to their contribution
- burn: neurons destroy $CYB for permanent φ*-weight on particles (eternal particles) or cyberlinks (eternal cyberlinks)
- lock: neurons stake $CYB on particles or cyberlinks, earning from fee pools proportional to attention attracted

## the game

the game design ensures the cybergraph improves over time:

- early, accurate links to important particles earn the most (attention yield curve)
- confirming links strengthen axon weight — repeated signals build consensus, not noise
- neurons build long-term reputation via accumulated φ*-weight (karma)
- focus as cost ensures every cyberlink is a costly signal

see tokenomics for the system-level economics (monetary policy, allocation curve, GFP flywheel). see collective learning for the group-level dynamics