Across the whole programme, one number never moved.
The two scaling laws have exponents for the product class and for the Haar class. Their difference is exactly one: Not . Not “approximately one within error.” One.
It was one in Paper I, when the foundation was loose and the laws were laws of a model. It was one in Paper II, after we corrected the foundation. It was one in Paper III, after the entropy-replacement theorem made the Haar law a real theorem, after we fixed the grouped-Dirichlet covariance twice, after we threw away the numerics-backed closure of the hardest bound and replaced it with the centered-operator identity. A corrected lemma, a new headline result, two separate rewrites of the worst appendix – and the integer sat there through all of it, unbothered.
When a quantity is that stable under that much demolition, it is not a coincidence of the calculation. It is a feature of the object. In physics, integer-valued gaps in scaling laws almost always mean something is being counted – a dimension, a degree of freedom, a conserved index. One power of per level of structural regularity in the bulk marginal is the phrase we use in the paper, and it is suggestive, but it is a description, not an explanation. We can tell you the integer is robust. We cannot yet tell you what it counts.
This is, deliberately, where Paper III stops. We could have buried the open question in a footnote and let the reader assume we understood it. Instead it is Open Problem number one, stated as a question we now know how to ask precisely, which is itself a kind of progress: at the start of this we did not know enough to even pose it. An integer-valued exponent gap in observer complementarity, stable across state classes and code families, analytic origin unknown.
If you want the one thing to take from three papers, it is not the prefactors and not even the replacement theorem, satisfying as that was to close. It is this: somewhere in the structure of how two observers disagree about a quantum object, something is being counted in whole numbers, and finding out what would be worth more than everything we have proved so far. We left the integer on the page, unexplained, because the honest edge of the work is more useful to the next person than a confident guess would be. That is where the donkey is standing – upright now, in a frock coat, at the edge of the next cliff.