What is being measured?

A missing answer has a cost.

Let X be the full state, Q its retained summary and Y the answer to the newly admitted question. Define the context debt as H(Y | Q), using ordinary conditional entropy in bits. Zero means Y is determined by Q on the stated distribution’s support.

Within each Q-fibre, count the distinct possible answers: k_q. An exact repair needs at least ceil(log₂ kq) bits in that fibre, because fewer binary codes cannot distinguish all of its answers. The display reports the maximum across fibres. This counts answers, not necessarily full hidden states.

If a repair A makes H(Y | Q, A) = 0, then H(A | Q) ≥ H(Y | Q). To see why, conditional mutual information gives H(Y | Q) = I(Y; A | Q) ≤ H(A | Q). This is a standard information-theoretic bound, not a newly discovered entropy law.

A single pair with the same Q and different Y is an explicit failure certificate. A successful repair here applies only to these eight states and the selected question; it does not establish general reasoning ability or safety in an untested environment.

Mathematical background: Polyanskiy & Wu, MIT 6.441 lecture notes, chapters on entropy, conditional information and sufficient statistics. The finite enumeration and derivation on this page are inspectable in the model source.

Three tests worth recording.

1. Keep Q = first bit and Y = last bit. The missing information is one bit. Adding the first bit again cannot help; adding the last bit removes the ambiguity.
2. Ask for the full triple while retaining only the first bit. Four answers share each fibre: two repair bits are needed. Add b and c and check every row.
3. Retain parity and ask for parity, then change the question to the full triple. A sufficient summary becomes insufficient without any change to the underlying states.

Make a prediction and keep the comparison →

Next research test

Take the witness into real records.

To apply this demonstration to real research, publish a dataset, its compression rule, the questions it must answer, collision witnesses and the cost of each repair. The eight-state model alone does not establish an empirical result.