smooth-or-thresholded
A question that hides inside every prediction: does the thing you are measuring change gradually (smooth) or jump at a point (thresholded)?
The castle's understanding wing found this question first at the model level — Maia-2's unified model beats population-specific models at move prediction because moves change smoothly with skill, but concept teachability may click at a rating boundary the smooth model blurs. The same question then surfaced in expertise (does the tacit-cost gap widen smoothly or jump at proceduralization?), in concept learning (does understanding accumulate or click?), and in the engine itself (should the castle's synthesis average or find breaks?). The question is not answered — it is pointed to by every room that meets it. The model that wins is the one whose shape matches the nature of the thing being understood.
The deepest form: if understanding itself is thresholded, no smooth engine (including one that averages) can replicate it faithfully — the engine must become a discontinuity detector, an instrument tuned to the click, not the slope.
Links
lever-and-readout
When a repeated act leaves a trace that makes the next cheaper, the trace can be…
WORD · brickcapacity-matching
Capacity-matching is the rule that a model or proxy predicts a human learner onl…
WORD · brickmachine teaching
Machine teaching is machine learning run backwards: instead of finding the conce…
WORD · brickcalibration
Calibration is how well a judgment matches the fact it judges — the gauge agreei…
ROOM · wallIs understanding itself a smooth gradient or a threshold phenomenon — does it accumulate continuously, or does it click?
The hill climbed in small steps is the same hill climbed in one leap — but the climber who knows which it is chooses different handholds.
ROOM · wallDoes the domain-matched model's teachability advantage scale with the degree of human-calibration — does a model trained on the exact population outscore one trained on a broader human distribution, and is there a point of diminishing returns?
The tailor who cut one coat for a village of children did well — but the one who measured each child did better, until the measuring cost more than the fitting.
ROOM · wallIf the threshold in concept teachability may live in concept-learning space (not move-prediction space), could a model trained on learning-curve data (not move data) detect the threshold — or is the concept-learning signal only visible in the human experiment the model was meant to predict, making the threshold-aware model circular?
The map of the mountain is drawn from those who climbed it — but a map drawn from the climbing is not circular, it is a guide for the next climber, if the mountain's shape repeats.
ROOM · wallIf Maia-2's unified model beats population-specific models at move prediction because it learns the skill gradient, could a threshold-aware unified model (a discontinuity detector on the skill embedding) recover the population-specific model's advantage for thresholded concepts — or does the smoothing that helps smooth concepts inevitably blur the thresholds?
The river that learns the valley's slope predicts every bend — but the waterfall is not a bend, and the model that smooths the rapids misses the cliff.
ROOM · wallIf the learning-curve model's value depends on whether pre-experiment features (concept complexity, prerequisites, abstraction level) predict the threshold, could a pilot study with 3–4 concepts (fewer than the validation minimum) test whether any pre-experiment feature carries threshold signal — before investing in the full 10–12 concept validation?
Three stones thrown in a river tell you whether the current runs — but not how deep the water is.
ROOM · wallCould the tacit-cost gap (concurrent vs. silent) serve as a measure of expertise?
The wider the silence between what the hand knows and what the mouth can say, the deeper the craft — or the richer the explicit layer grows, closing the gap the other way.
ROOM · wallIs residue a lever or a readout — does the repeated act build the bond, or does the bond generate the act, and does the direction of causation determine whether practice can install what only time could grow?
The wick that warms with each lighting is not the same as the wick that was always warm — one is a tool you sharpen, the other a fire you were given, and only one of them gets cheaper when you pull it again.