### An exact change of coordinates, with its information cost

The Esau reconstruction identifies two complete ways to encode the same nine source counts. One uses selected clean and female cumulative measurements with the grand total. The other uses the complete clean and unclean walks. Their measurement matrices have determinants +1 and −1, respectively. Consequently, each encoding has an integer inverse: every count is recovered by exact additions and subtractions, without fractional corrections.

This result explains why a family can look different while retaining all its numerical information. Counts, successive landings and selected cumulative registers can serve as alternative coordinates for one labelled object. The calculation removes redundant reporting, but nine independent measurements still carry nine independent numeric inputs. It does not derive those source values from fewer free choices. Nor does an arbitrary integer measurement vector automatically describe admissible animal counts: nonnegativity and the source’s category meanings remain additional requirements.

Using every clean, female and unclean prefix gives fourteen measurements of rank nine. Five dependencies therefore follow from the measurement structure itself. Other equalities have a different status. The clean and unclean walks both reach 20 and 50, but through different animal rows; those agreements depend on the particular printed counts. Distinguishing structural dependencies from count-specific matches keeps the common grammar explanatory without multiplying the evidence supplied by equivalent descriptions.

*Sources:* File_58 §13.1, §13.2, §13.3, §13.4.

*Research record:* C1011, C1012, C1013.