The difference between the product exponent and the Haar exponent is exactly one – one whole power of the observer's size per level of structural regularity in the bulk marginal. The gap is stable under every deformation tested, and it survived the entire Paper II → Paper III rewrite unchanged. Integer gaps in scaling laws usually mean something is being counted. We still do not know what.
Not approximately one. 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 the structural identity was corrected to the diagonal. 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.
Stability under demolition. A corrected lemma, a new headline result, two separate rewrites of the worst appendix – 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.
What we conjecture and what we don’t. Integer gaps in scaling laws usually mean something is being counted. One power of per level of structural regularity in the bulk marginal is the phrase the paper uses. It is suggestive, but it is a description, not an explanation. The analytic origin of the integer is open. It is Open Problem One of the programme – stated in plain sight as a question we now know how to ask precisely.