## 3 Proposed Method
### Platform Chain as Sidechain
It is also important to note that \(t_{M}\) operates the platform for economic benefit, typically through fees or value captured from user participation. Therefore, the long-term sustainability of the platform depends heavily on maintaining user trust and ensuring that the operational logic encoded in \(sc_{M}\) and \(sc_{P}\) is both reliable and fair. If users suspect that the rules are biased or insecure, they may avoid participating, undermining both the platform's integrity and \(t_{M}\)'s own incentives. However, evaluating the technical adequacy of \(sc_{M}\) and \(sc_{P}\) solely by \(t_{M}\) or general users presents challenges. Since \(t_{M}\) has direct influence over these contracts, conflicts of interest may arise. On the other hand, general users often lack the expertise needed to assess the complexity of smart contracts. Thus, an impartial technical evaluation of \(sc_{M}\) and \(sc_{P}\) by external experts is essential. Accordingly, the platform manager \(t_{M}\) is expected to rely on a neutral external entity to ensure a fair assessment of the validity of these smart contracts. The specific methods for obtaining this impartial evaluation are outside the scope of this paper, as it focuses on proposing a general framework rather than detailing external validation procedures. As a sidechain of \(\mathbb{RC}\), \(\mathbb{PC}_{M}\) operates independently under normal conditions but periodically establishes connections to \(\mathbb{RC}\) based on the rules immutably coded in \(sc_{M}\). Let \(B^{[n_{1}]},B^{[n_{2}]},\ldots\) denote consecutive \(\mathbb{RC}\) blocks connected to \(\mathbb{PC}_{M}\) after \(B^{[n_{0}]}\). We denote the sequence of consecutive blocks within \(\mathbb{PC}_{M}\) from \(B^{[n_{i}]}\) to \(B^{[n_{i+1}]}\) as \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) (hereafter referred to as the _platform chain segment_). The platform chain segment \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) starting at \(B^{[n_{i}]}\) indicates that the header of \(b^{[i,1]}\) includes the hash value of \(B^{[n_{i}]}\):  
\[H(B^{[n_{i}]})\in\text{header}(b^{[i,1]}). \tag{7}\]  
Similarly, the fact that the platform chain segment \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) ends at \(B^{[n_{i+1}]}\) implies that the following transaction \(TX_{1}\) is included in \(B^{[n_{i+1}]}\):  
\[TX_{1}=t_{M}\to sc_{M}:H(b^{[i,N_{i}]}). \tag{8}\]  
Blocks within the platform chain segment \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) also follow the standard hash value chaining rules. Once \(B^{[n_{i+1}]}\) is securely recorded on \(\mathbb{RC}\), modifying transactions within \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) becomes virtually impossible, even for \(t_{M}\). In this case, we say that \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) are connected to \(\mathbb{RC}\). This framework is autonomous, as \(\mathbb{RC}\) does not need to perform explicit actions to initiate or finalize a \(\mathbb{PC}_{M}\) segment within its own blocks. The operations on \(\mathbb{PC}_{M}\) proceed automatically, following the predefined rules in \(sc_{M}\). Using a well-established blockchain such as Bitcoin or Ethereum as \(\mathbb{RC}\) provides significant advantages, as it allows the system to inherit its stability and security without requiring active intervention from these well-known blockchains. Figure 1 illustrates these concepts; segment \(b^{[i,1]}\), \(b^{[i,2]}\), \(\ldots\), \(b^{[i,N_{i}]}\) is finalized when its commitment is recorded in block \(B^{[n_{i+1}]}\) on \(\mathbb{RC}\), ensuring immutability and trust inheritance.