Derived Yields

This document derives the expected yields from XPower's two primary value-accrual mechanisms — proof-of-work mining and NFT staking — and analyzes the combined mine→deposit→stake pipeline for optimal strategy selection.

Mining Yield

Expected Reward

For a uniformly random nonce search over the 256-bit space, the expected XPOW reward per hash attempt is the sum over all difficulty levels of the probability-weighted reward:

$$\mathbb{E}[R]=\sum _{z=1}^{\mathrm{\infty }}P(z)\times R(z)=\sum _{z=1}^{64}\frac{2^z-1}{{16}^z}$$

Since $P(z)=1/{16}^z$ and $R(z)=(2^z-1)\times {10}^{18}$:

$$\mathbb{E}[R]=(\sum _{z=1}^{64}{2}^{-3z}-\sum _{z=1}^{64}{2}^{-4z})\times {10}^{18}$$

Using geometric series with infinite upper bound (negligible error for $z>64$):

$$\sum _{z=1}^{\mathrm{\infty }}{2}^{-3z}=\frac{1/8}{1-1/8}=\frac{1}{7},\sum _{z=1}^{\mathrm{\infty }}{2}^{-4z}=\frac{1/16}{1-1/16}=\frac{1}{15}$$

Therefore:

$$\mathbb{E}[R]=(\frac{1}{7}-\frac{1}{15})\times {10}^{18}=\frac{8}{105}\times {10}^{18}\approx 0.07619\times {10}^{18}$$

In practical terms, a miner performing $H$ hashes expects approximately $0.076H$ XPOW, or equivalently, one XPOW per ~13.1 hash attempts.

Expected Hashing Time

Mining performance is bounded by browser JavaScript single-thread throughput. Typical consumer hardware achieves ${10}^4$ to ${10}^5$ Keccak-256 hashes per second. At ${10}^5$ H/s:

$$\mathbb{E}[\text{XPOW/hour}]=0.07619\times {10}^5\times 3600\approx 27.4\times {10}^{18}$$

The variance is high — most rewards come from low-difficulty solutions ($z\le 3$), while high-difficulty solutions are exponentially rare but lucrative. The reward distribution has a long right tail: a $z=8$ solution (1 in ~4.3B hashes) yields 255 XPOW but may require days of computation on consumer hardware.

Difficulty and Reward Table

z	Expected Hashes	P(z)	Reward (XPOW)	XPOW/Hash
1	16	6.250%	1	0.0625
2	256	0.391%	3	0.0117
3	4,096	0.0244%	7	0.00171
4	65,536	0.00153%	15	0.000229
5	1,048,576	0.0000954%	31	0.0000296
6	16,777,216	0.00000596%	63	0.00000376
7	268,435,456	0.000000373%	127	0.000000473
8	4,294,967,296	0.0000000233%	255	0.0000000594

Note that XPOW/hash decreases with $z$ — lower-difficulty solutions are the primary source of expected value, accounting for over 97% of total expected yield.

Staking Yield

Annualized APY

The annual percentage yield (APY) for a staked position is the sum of the level-based APR and the vintage-based APB:

$$\text{APY}(\ell ,y)=\text{APR}(\ell )+\text{APB}(y)$$

Using default production parameters:

APR (parameters $[0,3,\mathrm{3,375,000},256]$):

$$\text{APR}(\ell )=\frac{\ell \times \mathrm{3,375,000}}{3}\text{ PPM}=\mathrm{1,125,000}\times \ell \text{ PPM}=1.125\ell \mathrm{\%}$$

APB (parameters $[0,1,\mathrm{100,000},256]$):

$$\text{APB}(y)=y\times \mathrm{100,000}\text{ PPM}=0.1y\mathrm{\%}$$

Illustrative APY Table

Level	Denomination	APR (%)	APB 1yr (%)	APY 1yr (%)	APB 3yr (%)	APY 3yr (%)	APB 5yr (%)	APY 5yr (%)
0	${10}^0$	0.000	0.100	0.100	0.300	0.300	0.500	0.500
3	${10}^3$	3.375	0.100	3.475	0.300	3.675	0.500	3.875
6	${10}^6$	6.750	0.100	6.850	0.300	7.050	0.500	7.250
9	${10}^9$	10.125	0.100	10.225	0.300	10.425	0.500	10.625
15	${10}^{15}$	16.875	0.100	16.975	0.300	17.175	0.500	17.375
21	${10}^{21}$	23.625	0.100	23.725	0.300	23.925	0.500	24.125
27	${10}^{27}$	30.375	0.100	30.475	0.300	30.675	0.500	30.875
30	${10}^{30}$	33.750	0.100	33.850	0.300	34.050	0.500	34.250
45	${10}^{45}$	50.625	0.100	50.725	0.300	50.925	0.500	51.125
60	${10}^{60}$	67.500	0.100	67.600	0.300	67.800	0.500	68.000
81	${10}^{81}$	91.125	0.100	91.225	0.300	91.425	0.500	91.625
99	${10}^{99}$	111.375	0.100	111.475	0.300	111.675	0.500	111.875

The APB contribution is modest — at 0.1% per year, it takes 10 years of staking to add 1% to the effective rate. For high-level positions, APR dominates; for low-level positions (particularly level 0), APB is the primary yield driver.

Cumulative Reward Over Time

Rewards accrue per-second, with the total claimable amount growing linearly with time (since APR and APB are fixed under default linear parameters):

$$\text{reward}(t)=\frac{(\text{APR}+\text{APB})\times D\times {10}^{18}}{{10}^6\times C}\times t$$

For a level-30 NFT ($D={10}^{30}$, $\text{APR}=33.75\mathrm{\%}$):

$$\text{reward}(t)=\frac{\mathrm{33,750,000}\times {10}^{30}\times {10}^{18}}{{10}^6\times \mathrm{3,155,760,000}}\times t\approx 1.07\times {10}^{37}\times t$$

Over one year ($t=\mathrm{31,536,000}$ s), the raw reward is approximately $3.37\times {10}^{43}$ wei of APOW. Due to the rate-limited APOW minting, actual claimable amounts may be lower if the supply curve constrains output.

Effect of the Supply Curve

The APower mint() function enforces a long-term supply rate of approximately one APOW per minute. Excess claims accumulate but produce zero marginal minting output beyond the supply ceiling. This means:

• Early claims: With low total supply, claims are minted at close to 1:1 ratio
• High-frequency claims: A burst of claims in quick succession sees diminishing APOW output as the running average catches up
• Steady-state: At equilibrium, the system mints approximately 1 APOW/min regardless of aggregate staked XPOW

The effective yield therefore depends not only on the APR+APB formula but also on the distribution of claims across the staker population and the current position of the supply curve's exponential moving average.

Combined Mine→Stake Pipeline

Pipeline Analysis

A complete wealth-maximization cycle proceeds through three stages:

$$\text{mine}\to \text{deposit}\to \text{stake}\to \text{claim}$$

1. Mine XPOW at expected rate of 0.076 XPOW/hash, netting 50% of minted tokens (the other 50% goes to treasury)
2. Deposit XPOW into XPowerNft at chosen level $\ell$, paying gas for the deposit transaction
3. Stake XPowerNft into NftTreasury, receiving APowerNft with age = 0
4. Hold until desired maturity, accumulating APR and APB rewards
5. Claim APOW, paying gas and subject to the supply curve

Breakeven Analysis

The lowest economic level is $\ell =3$, requiring an XPowerNft with denomination $D={10}^3=\mathrm{1,000}$ XPOW. At 0.038 XPOW/hash (after 50% treasury split), a miner needs approximately 26,300 hashes to accumulate enough XPOW for a single level-3 deposit.

At the default web miner throughput of ${10}^5$ H/s, this requires about 0.26 seconds of computation — trivial. However, the real bottleneck is the gas cost of the deposit and stake transactions, which on Avalanche C-Chain typically range from 0.01 to 0.05 AVAX depending on network congestion.

Yield Attribution

For a level-30 stake of ${10}^{30}$ XPOW held for 3 years:

Component	Rate	Annual Yield (XPOW equiv.)	Fraction
APR(30)	33.750%	$0.3375\times {10}^{30}$	99.12%
APB(3)	0.300%	$0.003\times {10}^{30}$	0.88%

The APR dominates for all levels ≥ 3. Level 0 (unit NFT with no APR base) relies entirely on the APB for yield — 0.1% per year on a single XPOW, approximately 0.001 XPOW annually.

Optimal Strategies

Level Selection by Horizon

The choice of NFT level involves a tradeoff between earning power and liquidity:

Strategy	Level Range	Horizon	Rationale
Liquid	0–6	< 1 year	Short maturity (0–3 years), quick redemption
Balanced	9–21	1–5 years	Moderate APR (10–24%) with 7–128 year maturity
Long	27–45	5–25 years	Strong APR (30–50%), effectively permanent lock
Permanent	60–99	Generational	Maximum APR (67–111%), million+ year maturity

The maturity lock grows exponentially: $2^{\lfloor \ell /3\rfloor }$ years. A level-27 NFT ($\ell /3=9$) has a $2^9=512$-year maturity — effectively permanent for any individual staker. Levels above 30 can be considered non-redeemable within a human lifetime.

When to Upgrade

Upgrading burns 1,000 NFTs at level $\ell$ to mint 1 NFT at level $\ell +3$. The tradeoff:

Benefit: 10× denomination increase yields 10× absolute reward, and the APR increases from $1.125\ell \mathrm{\%}$ to $1.125(\ell +3)\mathrm{\%}$ — a 3.375 percentage point boost.

Cost: Maturity extends from $2^{\lfloor \ell /3\rfloor }$ to $2^{\lfloor \ell /3\rfloor +1}$ years (doubles). Gas cost of the upgrade transaction (~100k–200k gas).

Decision rule: Upgrade when the expected incremental rewards over the planned holding period exceed the gas cost, and the extended maturity does not conflict with liquidity needs. For a staker with a 10+ year horizon, level 9 → 15 upgrade is almost always optimal. For a staker needing liquidity within 3 years, levels above 6 should be avoided.

Gas Efficiency

Transaction costs create a minimum viable position size. Assuming 0.02 AVAX (~$0.50 at typical prices) per stake or claim transaction, the minimum XPOW deposit for gas to represent less than 1% of annual yield depends on the APY:

$$D_{\text{min}}>\frac{\text{gas\_cost}}{\text{APY}\times 0.01}$$

For level 3 (APY = 3.375%), this implies $D_{\text{min}}>30\times D$ in XPOW-denominated gas — meaning the position must earn enough to cover the transaction cost within a reasonable timeframe. Batch operations and gas price monitoring can reduce this overhead.

Remarks

The yields presented here are illustrative and based on default polynomial parameters and the assumption of static rates. Actual returns vary based on:

• Dynamic scalar adjustments from per-level share distribution (can boost or reduce effective APR by significant margins)
• Supply curve constraints on APOW minting (yield may be deferred during high-claim periods)
• Parameter changes by governance within Rpp bounds ($\pm 50\mathrm{\%}$ per change, 1-month minimum interval)
• Gas price volatility on Avalanche C-Chain affecting transaction economics
• Claim frequency and the running exponential moving average in the APOW minting formula

This analysis is not financial advice. Past mathematical derivations do not guarantee future on-chain outcomes.