Assuming r as the ratio of energy loss which is sponsored, then the equations become:

$\frac{d P_t}{d t} =K_t-q P_t=L+N_t-q P_t=L-(1-r)q P_t$

$N_t=r q P_t$

Here $N_t$ comes from $K_{N o n M o n e y}$ and $L$ comes from $K_{M o n e y}$. Therefore effectively with $(1-r)q$ as the energy loss rate,

$P=\frac{L}{(1-r)q}$

$N_{m a x}=\frac{r}{1-r}L$

$C=\frac{L}{e^{(1-r)q t}-1}$

$K=\frac{L}{1-e^{-(1-r)q t}}$

$\frac{K}{C}=e^{(1-r)q t}$

K and C are around $\frac{L}{(1-r) q t}$

The sponsored energy in charging,

$\int_0^t {N_t d t}=\left( K_t - P \right) \frac{r}{1-r}$

The sponsored energy in consuming,

$\int_0^t {N_t d t}=\left( P-C_t \right) \frac{r}{1-r}$

The average sponsored energy power,

$\overline{N}=(K-C)\frac{r}{1-r}=\frac{r L}{1-r}$

The total energy loss power,

$\overline{K_m}=\overline{N}+(K-C)=\frac{L}{1-r}$

With q=13.5% and effective energy loss rate 0.5%, r=96.3% and $\overline{N}=26L$

The battery P remains as $\frac{\overline{K_m}}{q}=\frac{L}{(1-r)q}$ while the unit cost becomes $\frac{L}{q}$. Based on market characteristics, this x26 surplus distributes between the producer surplus and the consumer surplus in any transaction of one battery dollar. For example of the battery money producers, he could produce 1000 battery dollars and keep 10 battery dollars for himself and sells the remaining 990 batteries which due to the competitive market so he could only sell at the cost and the buyers enjoy the surplus.

$\overline{N}=26L$ is inspiring because it means people can spend 1 Joule energy to collect 27 Joule on average thanks to the Mother Nature Sponsor. Define energy arbitrage ratio by the surplus energy amount that one can collect by spending 1 Joule. People sense fairness by comparison with this energy arbitrage ratio in the natural context. If one can spend 1 Joule to achieve 26 Joules surplus naturally then it is difficult to hire one for work in lower return. All civilization and creature are essentially a product of energy arbitrage such as hunters do not pay for the growing of the victims and people do not pay the sunshine for collecting sea salt. If a creature needs to spend more energy than what it could collect, i.e. negative energy arbitrage ratio, it ceases to exist sooner than later. There must be some energy arbitrage ratio for a specific planet where summer may be better than winter and a country with natural or unfair endowment may be better than the country otherwise. This is also the reason why a very same task has higher real wage in wealthier countries because wealth is defined in the sense of capability to demand energy usage. A person will be in luxury mode in wealthy environment. In countries where the energy arbitrage ratio is higher, the real wage level will be high and people's time preference will be low. Energy fairness is critical to welfare fairness. Should people around the globe have equal energy arbitrage ratio, there won't be drastically contrast living standards.

It is reported that the fusion is already done in the lab but the spent energy amount is much more than the energy amount collected. The high energy arbitrage ratio enjoyed by current civilization has been supported by fossil energy. If fusion or sustainable energy is not a thing and fossil energy is not an option, this energy arbitrage ratio will be lower which means lower through the above math and higher effective loss rate and higher time preference rate. For example, if the energy arbitrage ration becomes half, then the time preference rate would become 0.96%. Energy arbitrage is all about spending lower L and getting higher K, then the relationship of effective energy loss rate q', aka the time preference of the sound money system, and the physical energy loss rate q is $q'=\frac{L}{K}q$. A battery farm runs with K and q. A mining farm runs with L and q'. No arbitrage is allowed between the two business; mining business is to mirror the energy storage business with the help of Mother Nature sponsor so that the energy to fight the energy loss across time can be lower.

1 + energy arbitrage ratio is essentially energy gain factor which is, fairly, also the ratio of the energy quota reward over the cost in a money transaction. Say, in a planet of energy gain factor being 50 and Alice spend 1 unit money to boil 1 liter water by herself. Then alternatively Alice hires Bob only to boil 0.02 liter by 1 unit money because of the transfer of the energy quota and surplus from Alice to Bob.

While $\overline{K_m}$ is not totally in $K_{E c o n o m y}$ due to contribution from $K_{ N o n M o n e y}$, L is totally in $K_{ E c o n o m y}$ to be accountable. Rephrase L to be $K'_m$ so that $P=\frac{\overline{K_m}}{q}=\frac{K'_m}{q'}$

Then $K'_{M o n e y} \equiv V K'_m$ and $K_{E c o n o m y}=\frac{K'_{M o n e y}}{q' t}$. Let $K_{E n e r g y}$ be the energy consumed by the energy production sector and G be the energy gain factor, then the relationship of percentage of energy consumption appears:

$\frac{K'_{M o n e y}}{K_{E n e r g y}}=G q' t = q t$

$\frac{K_{M o n e y}}{K_{E c o n o m y}}=q' t$

See MacroEconomicsTheory for detail.