> For the complete documentation index, see [llms.txt](https://v2tech.noahlabs.io/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://v2tech.noahlabs.io/2.mathematical-derivation-of-hmm-model-and-quantum-entropy-reduction/2.1-hmm-protocol-liquidity-is-permanent-infrastructure.md).
# 2.1 HMM Protocol: Liquidity is Permanent Infrastructure
The essence of HMM (Holders Market Making) is to treat assets in the liquidity pool as permanent infrastructure for the Pulsar-1 ecosystem. Once funds enter the pool, all holders automatically become governance participants, with every buy/sell transaction contributing to the planet's economic governance.
**2.2.1 AMM invariant and HMM correction**
The Standard Constant Product Market Maker (CPMM) adheres to:
where x is the number of XPULS, y is the USDT amount, and K is a constant.
The core of the HMM model is that the K 
-value 
is a function of time, and
We introduce a liquidity injection 
rate, which is derived 2.5% from the liquidity 
tax levied on each transaction.
Set 
as trading volume. Within the extremely short time dt, the newly added liquidity pool constant dK is derived from the tax increments dx and dy on x and y:
 According to the CPMM definition, the change 
in dK is approximated as:
 Substituting 
into the above formula:
Key conclusion: If 
there exists a trading volume, dK 
will definitely be greater than zero.
Conclusion: The HMM protocol mathematically demonstrates that the pool constant K increases monotonically over time through its tax structure. As K is a function of the price floor, this ensures the XPULS value floor continues to rise and cannot be withdrawn by liquidity providers, thereby achieving quantum entropy reduction in system value.