Research · Pedagogy
How this site teaches
Calyx runs on its own learning theory. UTML models learning as a move from entropy (surprise) toward coherence (durable, connected understanding), described by four factors — ΔP, ΔK, ΔΩ, ΔΞ. The predict→reveal→explain-back→mastery loop you move through is that theory running as code, not a metaphor bolted on after.
The loop you move through
Every concept walks the same path. Each step is an instance of a UTML factor, implemented by a real feature on this site (docs/04 §2).
- ΔP
Predict
You commit an answer before the reveal, so the surprise between guess and truth is measured.
Predict-then-reveal on every concept - ΔP
Reveal
The grounded answer is shown only after you commit — feedback lands on a real prediction error.
The grounded, cited reveal - ΔK
Explain back
You say the idea in your own words; the tutor scores it for coherence (self-explanation / teach-back).
Explain-back, then the grounded tutor - ΔΞ
Master
Progress is gated behind a transfer check — fluent talk with a failed transfer banks nothing.
The mastery gate (practice quiz + Ward) - ΔΩ
Stay in flow
Difficulty is tuned toward ~80% success so you stay challenged but not stalled.
Adaptive difficulty + affect-aware help - ΔK
Space it out
Retrieval is scheduled over time so the coherence you built does not decay.
Spaced review (FSRS)
The four factors, mapped to features
The factors are not a metaphor here — they are the control signals the engine reads (docs/04 §1).
| Factor | What it is | How the site runs it |
|---|---|---|
| ΔP — prediction error | The surprise that drives an update: the raw signal before grounding. | Forces a prediction before every reveal; sizes items so you are genuinely uncertain. |
| ΔK — coherence | Knowledge becoming connected, consistent, and durable — the kernel forming. | Drives explain-back and elaboration, then schedules spaced retrieval so it persists. |
| ΔΩ — optimizer state | The health of the learning process itself — keeping you in the flow channel. | Holds ~80% success with adaptive difficulty; detects affect and reduces friction. |
| ΔΞ — non-collapse | The guard against illusory coherence — fluent pattern-matching that is not understanding. | Gates mastery behind a high-confidence transfer check (a Ward); refuses to certify it unearned. |
Why this is honest, not aspirational
These are the same mechanisms the engine actually uses: a prediction is forced before every reveal, mastery is gated behind a transfer check (a fluent explanation alone banks nothing), and review is scheduled so coherence does not decay. The theory is the rigorous source; this page is the map from it to the buttons you click.