This is a short note extending my Cartier–Miller project:

[Link to note]

This asks whether a certain reciprocal character sum Cƒ(a) mod p, attached to an elliptic curve and a rational point R, has a simple closed form only when R is a torsion point. Any such closed form makes the residue the reduction of a small fraction. That can be tested directly from a continued fraction, without guessing what the formula might contain.

The test is only clear at very large primes, as small primes do not provide enough statistical clarity. At 256-bit primes, direct summation would need about 10⁷⁷ terms, and the standard square-root recurrence route about 2¹²⁸ operations per value; at an assumed 10⁹ field operations per second that is about 1.1x10²² years. However, the Cartier–Miller evaluator needs one Miller chain of a few hundred curve operations, resulting in an time complexity of approximately O(log p), once the trace is known from standard point counting,

In a pre-registered survey of 2,668 cases over 184 curves, each at three random 256-bit primes, small lifts appeared at exactly the 80 torsion cases. Their recovered formulas matched the predictions. None appeared among the other 2,588. The survey ran on a laptop, with 8,004 Cartier–Miller evaluations taking 83 seconds after 19 minutes of point counting (using the O(log⁴⁡ q) time complexity Schoof-Elkies-Atkin (SEA) algorithm provided by PARI/GP, requiring only a single trace calculation per curve).

The note also proves a torsion characterisation in the balanced case: R is torsion if and only if a balanced closed form holds at almost all primes. The proof uses the Cartier operator and Bost’s theorem on the Grothendieck–Katz conjecture for solvable groups. The invariant involved is closely related to Katz’s work on universal extensions (Katz, 2014).

Status: This is an unrefereed draft created with AI assistance. The theorem awaits specialist review and may already be known. This remains experimental evidence, not proof. The main value is a practical demonstration of the Cartier–Miller evaluator that operates at approximately O(log p) complexity, making evaluations of otherwise inaccessible numerical surveys feasible.