For AI agents: the complete documentation index is available at https://open-spaced-repetition.github.io/ts-fsrs/llms.txt, the full documentation bundle is available at https://open-spaced-repetition.github.io/ts-fsrs/llms-full.txt, and this page is available as Markdown at https://open-spaced-repetition.github.io/ts-fsrs/guide/model/fsrs-4dot5.md.
ts-fsrs
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      • FSRS-4.5

      FSRS-4.5 has 17 trainable weights. It changes the forgetting curve from FSRS-4 while retaining the same difficulty and successful-recall structure.

      See the Model overview for the shared symbols and Model contract.

      Formula source

      Reviewed against awesome-fsrs Wiki — FSRS-4.5.

      Forgetting curve and interval

      R(t,S)=(1+19t81S)−1/2R(t,S)=\left(1+\frac{19t}{81S}\right)^{-1/2}

      I(r,S)=81S19(r−2−1)I(r,S)=\frac{81S}{19}\left(r^{-2}-1\right)

      Initial memory

      S0(G)=clamp⁡(wG−1,Smin,Smax)S_0(G)=\operatorname{clamp}\left(w_{G-1},S_{min},S_{max}\right)

      D0(G)=clamp⁡(w4−(G−3)w5,Dmin,Dmax)D_0(G)=\operatorname{clamp}\left(w_4-(G-3)w_5,D_{min},D_{max}\right)

      Difficulty

      D∗=D−w6(G−3)D^*=D-w_6(G-3)

      D′=clamp⁡(w7w4+(1−w7)D∗,Dmin,Dmax)D'=\operatorname{clamp}\left(w_7w_4+(1-w_7)D^*,D_{min},D_{max}\right)

      Stability after review

      HG=w15H_G = w_{15} for Hard and 11 otherwise; EG=w16E_G = w_{16} for Easy and 11 otherwise.

      Sf=min⁡(w11D−w12((S+1)w13−1)ew14(1−R),S)S_f=\min\left(w_{11}D^{-w_{12}}\left((S+1)^{w_{13}}-1\right)e^{w_{14}(1-R)},S\right)

      Sr=S[1+ew8(11−D)S−w9(ew10(1−R)−1)HGEG]S_r=S\left[1+e^{w_8}(11-D)S^{-w_9}\left(e^{w_{10}(1-R)}-1\right)H_GE_G\right]

      Sraw′(G)={Sf,G=1Sr,G∈{2,3,4}S'_{raw}(G)=
\begin{cases}
S_f,&G=1\\
S_r,&G\in\{2,3,4\}
\end{cases}

      The Again branch prevents post-lapse stability from exceeding the previous SS. The model then clamps Sraw′S'_{\mathrm{raw}} to the range from Smin⁡S_{\min} to Smax⁡S_{\max}.