INTERACTIVE FIGURE 5Where scale debt goes to die
chain row sum s 40
press Propagate ▸ — the ×s chip will walk the chain and each block will rule on it.
The fold in one animation. A trained row's positive weights sum to s ≫ 1; dividing the row by s makes it legal but leaves a ×s debt owed to whatever consumes the output. Homogeneous blocks pass it (ReLU), linear blocks fold it into their columns and re-emit it, normalizers kill it exactly (BatchNorm, RMSNorm — recorded statistics recalibrate), and non-homogeneous gates stop it, where it parks as a recorded temperature — still exact, just named. The forbidden move is the mirror image: "inflating" a row whose sum is below 1 emits amplification debt that normalizer statistics were never calibrated for, and it compounds — flip the toggle and watch ten blocks turn ×2.5 into the KL ≈ 10²³ disaster from the footnote. Chain choice = legality fraction: that is the transplant law in miniature.