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A cut divides the variables into two groups, such as A B on one side and C on the other.

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A proposal score predicts how useful that split will be for the declared task.

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Supervision supplies training examples with target cuts or rankings.

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The learner uses them to predict a proposal.

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C3, C4 and C5 change examples, target feedback and variable-conditioned scoring; none replaces exact verification.

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ANF means algebraic normal form, an exact exclusive-or of products representation with constants allowed.

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It serves as an exact control in this comparison.

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The next lesson explains that representation in detail.

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This overview explains how C3 through C5 changed
examples, cut supervision, and variable-conditioned scoring.

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Better fit on familiar inputs is different from passing
held-out transfer and full charged-cost gates.

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Use one function as a target.

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A learner may propose a candidate from examples, a decomposition routine may inspect a

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declared matrix partition, and a restriction routine may fix an input.

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These are different questions about the same function,

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not interchangeable success criteria.

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Fixing C to zero leaves A and B.

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Fixing it to one makes the entire function true.

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Both restricted functions are exact, but they describe different slices.

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A method must report which input values were fixed.

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Rename the variables while preserving their meanings.

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The original row group A B becomes Z X, and the column C becomes Y.

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Sorting the new names alphabetically and taking X Y as rows would

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define a different partition.

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A renaming-invariant claim must carry the semantic split along.

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A renaming-invariant claim must carry the semantic split along.

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C3, C4 and C5 change different parts of a learned cut proposal.

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C3 strengthens natural examples and targets.

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C4 supervises cuts and ranking more directly.

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C5 conditions scores on variables so the readout
can respect a changed variable order.

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Take a conceptual partition with A and B together, C separate.

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Renaming A to Z, B to X, and C to Y carries
the partition to Z and X together, with Y separate.

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A readout that keeps old positions instead of the
corresponding variables predicts a different partition.

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Test that distinction before exact verification.

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The toy partition demonstrates what equivariance
asks; it is not a recovered held-out circuit.

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The retained learned arms improved some slices but still failed
required held-out transfer and charged promotion criteria.

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Better fitting, correct accepted outputs and low
total cost are independent achievements.

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Keep the exact ANF control visible as the
alternative that already solves the admitted task.

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A candidate can fit observed examples while remaining wrong elsewhere.

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And matches or at zero zero and one one.

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If those are the only training examples shown, the fit does not reveal their difference.

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The input zero one exposes the disagreement.

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Exact checking rejects this candidate independently of its training fit.

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Passing a training score cannot substitute for preserving the complete function.

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A different candidate may reconstruct perfectly yet take too long to propose and verify.

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Keep fit, exactness, and complete-request cost as three reports.

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The retained transfer results and exact algebraic controls remain attached

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to their own experimental scopes.

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Retain the strong exact control in the comparison.

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Retain the strong exact control in the comparison.

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Require prospectively frozen transfer and cost
evidence before promoting learned advice.