Stage 4 — Registers & RAM Val

Degree 3

Rounds $\log_2 K_{\text{reg}} + \log_2 T$

$\displaystyle\sum$ over: $X_k \in \{0,1\}^{\log_2 K_{\text{reg}}}, X_t \in \{0,1\}^{\log_2 T}$

Opening point $(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})$

\textcolor{BurntOrange}{\textsf{RdWriteValue}}(r_{\text{cycle}}^{(3)}) + \gamma \cdot \textcolor{BurntOrange}{\textsf{Rs1Value}}(r_{\text{cycle}}^{(3)}) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{Rs2Value}}(r_{\text{cycle}}^{(3)})

\widetilde{\text{eq}}(r_{\text{cycle}}^{(3)}, X_t) \cdot \left(\textcolor{BurntOrange}{\textsf{RdWa}}(X_k, X_t) \cdot \left(\textcolor{ForestGreen}{\textsf{RdInc}}(X_t) + \textcolor{BurntOrange}{\textsf{RegistersVal}}(X_k, X_t)\right) + \gamma \cdot \textcolor{BurntOrange}{\textsf{Rs1Ra}}(X_k, X_t) \cdot \textcolor{BurntOrange}{\textsf{RegistersVal}}(X_k, X_t) + \gamma^{2} \cdot \textcolor{BurntOrange}{\textsf{Rs2Ra}}(X_k, X_t) \cdot \textcolor{BurntOrange}{\textsf{RegistersVal}}(X_k, X_t)\right)

$\textcolor{BurntOrange}{\textsf{RegistersVal}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})$
$\textcolor{BurntOrange}{\textsf{Rs1Ra}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})$
$\textcolor{BurntOrange}{\textsf{Rs2Ra}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})$
$\textcolor{BurntOrange}{\textsf{RdWa}}(r_{K_{\text{reg}}}^{(4)}, r_{\text{cycle}}^{(4)})$
$\textcolor{ForestGreen}{\textsf{RdInc}}(r_{\text{cycle}}^{(4)})$

Degree 3

Rounds $\log_2 T$

$\displaystyle\sum$ over: $X_t \in \{0,1\}^{\log_2 T}$

Opening point $(r_{\text{cycle}}^{(4)})$

\textcolor{BurntOrange}{\textsf{RamVal}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(2)}) - \textcolor{BurntOrange}{\textsf{RamValInit}}(r_{K_{\text{ram}}}^{(2)}) + \gamma \cdot \left(\textcolor{BurntOrange}{\textsf{RamValFinal}}(r_{K_{\text{ram}}}^{(2)}) - \textcolor{BurntOrange}{\textsf{RamValInit}}(r_{K_{\text{ram}}}^{(2)})\right)

\textcolor{ForestGreen}{\textsf{RamInc}}(X_t) \cdot \textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, X_t) \cdot \left(\widetilde{\textsf{LT}}(X_t, r_{\text{cycle}}^{(2)}) + \gamma\right)

$\textcolor{ForestGreen}{\textsf{RamInc}}(r_{\text{cycle}}^{(4)})$
$\textcolor{BurntOrange}{\textsf{RamRa}}(r_{K_{\text{ram}}}^{(2)}, r_{\text{cycle}}^{(4)})$