Constraint Language at the Arithmetic Layer B: Height as Information—Counting, Northcott, and Covering Entropy
Year: 2026
Type: preprint
Abstract: This paper establishes the two-reading structure of the “information” vertex inthe arithmetic layer. The height function is an arithmetic degree when summingover each place, and it is a description length when counting points up to a givenbound. The two readings are welded into the same object by a conservation identity, with conversion factor log2. This paper uses Schanuel’s theorem to give theexact counting asymptotic for projective space in the description-length reading,and completes an independent recomputation and reconciliation on the projectiveline. The Northcott finiteness theorem gives the finiteness of the set of pointsof bounded height, but does not give a computable a priori bound; the effectivity problem is stratified by class of varieties, the curve case is covered by Baker’smethod and the Chabauty–Coleman method, and practical effective height boundsfor general varieties remain open. Counting dimension is defined as the growthexponent of the counting function; the counting dimension of projective space isthe projective dimension plus one; the Batyrev–Manin program predicts that theheight-bound counting of Fano varieties has the form of a main power multipliedby a logarithmic power, with Picard number information hidden in the logarithmicexponent. The covering entropy dictionary maps resolution to a description-lengthbound and covering number to a counting function, and registers two hard boundaries: opposite directions and difference in regularity. On the network side, theproved anchor is that topological entropy does not increase under factor maps; thearithmetic-side counterpart is the functoriality of height behavior under compression maps, in which the O(1) error term has not yet been eliminated. Open problemB-1 asks which counting structures admit a primitive height function; the SFT endand the arithmetic variety end provide two verifiable extreme samples, and thenetwork class between the two ends remains open. The whole paper distinguishesfive classes of statements: proved theorems, literature anchors, machine-checkedassertions, framework analyses, and open problems.
Authors: zhou, changzheng, zhou, ziqing
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221435699