The Mathematics of
Never Forgetting.
For 37 years, spaced repetition software relied on Piotr Woźniak’s SM-2 algorithm from 1987. FSRS-5 revolutionizes cognitive acquisition by modeling memory as a 3-dimensional dynamic system: Difficulty, Stability, and Retrievability.
Forgetting Curves: FSRS-5 vs SM-2
The Three Variables of Memory
In traditional flashcard systems, a card is simply 'due' or 'not due'. FSRS-5 models every sentence as a continuous point in 3-dimensional memory space:
Represents the intrinsic complexity of the linguistic item. Measured on a continuous scale from 1 (trivial cognate) to 10 (intense cognitive load, e.g. irregular subjunctive exceptions).
The number of days required for recall probability to drop from 100% to your desired threshold (90%). As stability grows, review intervals expand exponentially without forgetting risk.
The instantaneous probability that you will successfully recall the target sentence blank at any given moment t days since your last review.
Core Mathematical Formulas of FSRS-5
The mathematical engine operates via state transition equations across four user feedback ratings: Again (G=1), Hard (G=2), Good (G=3), Easy (G=4).
Where Factor is normalized such that R(S, S) = 0.90. As elapsed time $t$ increases, recall probability drops following a generalized power forgetting curve.
Initial stability is directly parametrized by the first grade. Initial difficulty is inversely bounded so that 'Easy' ratings immediately lower cognitive friction.
Notice the term e^(w₁₀(1-R)) - 1: when you review a card at very low retrievability R and still get it right ('desirable difficulty'), stability receives a massive cognitive bonus.
Unlike SM-2 which mercilessly resets intervals back to 1 day on any mistake, FSRS-5 preserves a fraction of your previously acquired neural stability trace.
Decoding the 17 Weights of FSRS-5
These weights are trained on over 200,000,000 empirical review logs using maximum likelihood estimation (MLE).
| Weight | Parameter Name | Mathematical Role | Default Value |
|---|---|---|---|
| w₀ - w₃ | Initial Stability S₀(G) | Determines initial stability after 1st review for grades: Again (1), Hard (2), Good (3), Easy (4). | 0.40, 1.18, 3.17, 15.69 |
| w₄ | Initial Difficulty D₀ | Baseline difficulty anchor for new items before user rating adjustments. | 7.21 |
| w₅ | Difficulty Grade Scaling | Step size for difficulty adjustment based on first response grade. | 0.53 |
| w₆ | Difficulty Delta | Linear rate of difficulty adjustment on subsequent reviews. | 1.10 |
| w₇ | Mean Reversion Weight | Damps extreme difficulty shifts by pulling Difficulty back toward the global mean. | 0.15 |
| w₈ | Stability Base Recall Gain | Exponential scaling factor for stability increase upon successful recall. | 1.65 |
| w₉ | Difficulty Damping on Stability | Controls how much high card difficulty reduces stability growth. | 0.14 |
| w₁₀ | Retrievability Decay Sensitivity | Higher values reward successful recall when retrievability R was low (desirable difficulty). | 0.94 |
| w₁₁ | Post-Lapse Stability Base | Initial stability floor assigned immediately after a card lapse (forgetting). | 2.18 |
| w₁₂ | Lapse Difficulty Scaling | How much card difficulty penalizes recovery stability after a lapse. | 0.06 |
| w₁₃ | Prior Stability Preservation | Fraction of pre-lapse stability preserved after a forgotten review. | 0.33 |
| w₁₄ | Lapse Retrievability Damping | Impact of retrievability at lapse time on post-lapse stability. | 0.28 |
| w₁₅ | Hard Grade Penalty | Stability growth multiplier when user selects 'Hard' (G=2). | 0.22 |
| w₁₆ | Easy Grade Bonus | Stability growth multiplier when user selects 'Easy' (G=4). | 2.94 |
Evolution of Spaced Repetition (1972 → 2024)
A head-to-head comparison across 5 generations of cognitive scheduling algorithms.
| Algorithm | Memory Model | Scheduling Formula | 90-Day Retention | Workload Efficiency |
|---|---|---|---|---|
| Leitner System (1972) | 1D (Box index) | Fixed interval doubling | ~65% | High (Rigid buckets) |
| SuperMemo SM-2 (1987) | 1D (Ease Factor) | I(n) = I(n-1) * EF | Baseline | Fixed intervals |
| Duolingo HLR (2016) | 2D (Half-Life) | Logistic regression on features | Cloud-reported | Adaptive |
| FSRS-4.5 (2023) | 3D (DSR Model) | 17-parameter power forgetting | Open benchmark | Fewer than SM-2 (design goal) |
| FSRS-5 (2024, open source) | 3D (DSR + Mean Reversion) | Refined power law with lapse decay | Open benchmark | Fewer than SM-2 (design goal) |
Scientific Literature & Empirical Studies
Frequently Asked Scientific Questions
Q: Why is 90% retention the recommended default?
Targeting 95%+ retention requires an exponential surge in daily reviews (roughly doubling review load for just a 5% gain). Targeting 85% causes too many lapses. 90% sits at the exact mathematical sweet spot of knowledge acquisition efficiency.
Q: Can I customize my target retention in ClozeForge?
Yes! In Web App 2.0 settings, you can tune your target retention between 80% (light maintenance) and 97% (high-stakes exam prep) and watch review intervals adjust instantly.
Experience FSRS-5 in Live Practice
Stop wasting 30% of your time on premature reviews. Train with the 2024 memory engine in Web App 2.0.