Supervision, ranking, and transfer

A cut divides the variables into two groups, such as A B on one side and C on the other. A proposal score predicts how useful that split will be for the declared task. Supervision supplies training examples with target cuts or rankings. The learner uses them to predict a proposal. C3, C4 and C5 change examples, target feedback and variable-conditioned scoring; none replaces exact verification. ANF means algebraic normal form, an exact exclusive-or of products representation with constants allowed. It serves as an exact control in this comparison. The next lesson explains that representation in detail. This overview explains how C3 through C5 changed examples, cut supervision, and variable-conditioned scoring. Better fit on familiar inputs is different from passing held-out transfer and full charged-cost gates. Use one function as a target. A learner may propose a candidate from examples, a decomposition routine may inspect a declared matrix partition, and a restriction routine may fix an input. These are different questions about the same function, not interchangeable success criteria. Fixing C to zero leaves A and B. Fixing it to one makes the entire function true. Both restricted functions are exact, but they describe different slices. A method must report which input values were fixed. Rename the variables while preserving their meanings. The original row group A B becomes Z X, and the column C becomes Y. Sorting the new names alphabetically and taking X Y as rows would define a different partition. A renaming-invariant claim must carry the semantic split along. C3, C4 and C5 change different parts of a learned cut proposal. C3 strengthens natural examples and targets. C4 supervises cuts and ranking more directly. C5 conditions scores on variables so the readout can respect a changed variable order. Take a conceptual partition with A and B together, C separate. Renaming A to Z, B to X, and C to Y carries the partition to Z and X together, with Y separate. A readout that keeps old positions instead of the corresponding variables predicts a different partition. Test that distinction before exact verification. The toy partition demonstrates what equivariance asks; it is not a recovered held-out circuit. The retained learned arms improved some slices but still failed required held-out transfer and charged promotion criteria. Better fitting, correct accepted outputs and low total cost are independent achievements. Keep the exact ANF control visible as the alternative that already solves the admitted task. A candidate can fit observed examples while remaining wrong elsewhere. And matches or at zero zero and one one. If those are the only training examples shown, the fit does not reveal their difference. The input zero one exposes the disagreement. Exact checking rejects this candidate independently of its training fit. Passing a training score cannot substitute for preserving the complete function. A different candidate may reconstruct perfectly yet take too long to propose and verify. Keep fit, exactness, and complete-request cost as three reports. The retained transfer results and exact algebraic controls remain attached to their own experimental scopes. Retain the strong exact control in the comparison. Require prospectively frozen transfer and cost evidence before promoting learned advice.