In the first two papers of this series, we explored a daring idea: that the universe itself behaves like a geometric manifold shaped by entropy. In The Universe as a Geometric Manifold, structure emerged from balance — geometry was not a backdrop but the outcome of thermodynamic optimization. Then, in Algebraic Cycles from Entropy Minimization, we pushed that insight into the heart of algebraic geometry, showing that the stable energy configurations on Kähler manifolds correspond to the very algebraic cycles that define Hodge theory. Reality, we discovered, could be described as geometry seeking equilibrium.
This new paper, An Information-Theoretic Approach to the Birch–Swinnerton-Dyer Conjecture, takes that same principle — geometry = thermodynamics = reality — and extends it into the arithmetic world of numbers and algebra. It asks a bold question: what if the same laws of entropy and information that govern physical structure also shape the hidden symmetries of arithmetic itself?
Arithmetic as an Information System
In this work, elliptic curves — equations like y2=x3+ax+by^2 = x^3 + ax + by2=x3+ax+b — are treated not just as algebraic objects but as information systems. Each rational solution corresponds to a point that can be “distinguished” only up to a certain number of bits of precision. The central theorem proves that the number of distinguishable solutions grows like Br/2B^{r/2}Br/2, where rrr is the rank of the curve and BBB is the number of bits available.
This result means something profound: the rank of an elliptic curve is not just an abstract algebraic invariant — it is an information-scaling exponent. Doubling your information capacity doesn’t double the number of rational points you can resolve; it grows like the square root of that capacity. This bridges pure number theory with physical and computational principles, uniting the abstract and the empirical under a single law of informational geometry.
Entropy, Geometry, and the Algebraic Heart of the Universe
Where the earlier VERSF papers showed that geometric structure and physical reality emerge from entropy minimization, this work reveals that the same principle extends all the way down to algebra — to the equations that define number itself.
In the VERSF vision, entropy and information are two faces of the same creative process: one organizes space, the other organizes meaning. The Birch–Swinnerton-Dyer conjecture, long seen as a purely analytic problem about LLL-functions, now appears as the algebraic counterpart of the same universal rule of balance. The equation’s rank measures how the universe stores and differentiates information — a hidden thermodynamic symmetry encoded within arithmetic.
In short, this paper brings the trilogy full circle. Geometry gave us the stage. Entropy gave it motion. Now algebra gives it language — the grammar through which the universe counts, connects, and communicates its own structure.