Rank & Evidence¶

Let \(E/\mathbf Q\) be an elliptic curve.

The Mordell–Weil theorem gives

\[ E(\mathbf Q) \cong E(\mathbf Q)_{\mathrm{tors}} \oplus \mathbf Z^r, \]

where \(r\) is the Mordell–Weil rank.

Rank Hunter separates lower-bound and upper-bound work because they are mathematically different tasks.

Lower bounds from points¶

If Rank Hunter has \(r\) rational points whose images in the free part are independent, then

\[ \operatorname{rank}E(\mathbf Q)\ge r. \]

The difficult word is independent.

Finding many rational points is not enough. They may lie in the subgroup generated by fewer points.

This is why the point ledger and rigorous witness basis are separate.

Upper bounds¶

Descent and related methods can produce rigorous upper bounds

\[ \operatorname{rank}E(\mathbf Q)\le u. \]

The details depend on the engine and assumptions.

Rank Hunter stores the engine/result as evidence and only promotes an upper bound through accepted rigorous paths.

Exact rank¶

When

\[ r=u, \]

the rank is exact.

Rank Hunter should report lower-bound-only results with \(\ge\), not as exact rank.

Torsion¶

Torsion points do not contribute to the free rank.

A point-search result that is torsion can be exact and interesting but does not add a rank direction.

Specializations of families¶

Suppose a Family over \(\mathbf Q(T)\) has rational sections.

Specializing sections at \(T=t\) can give rational points on \(E_t\).

Important distinctions:

generic sections can specialize dependently at exceptional fibers;
a specialization can acquire additional nongeneric points;
generic-rank metadata is not automatically the exact rank of each fiber.

Under an appropriate good specialization, exact independence of specialized section images can be used to prove generic independence: a nontrivial relation among generic sections would specialize to a relation.

That proof step must be explicit.

Heuristic rank signals¶

Many high-rank searches use heuristics based on reductions modulo primes, root numbers, local patterns, or height geometry.

These are useful because exact rank computation on every candidate is too expensive.

They should answer:

Where should we spend exact computation?

not:

What rank have we proved?

External record curves¶

A published or leaderboard curve may come with a known lower bound or exact rank.

Rank Hunter treats external claims as reference data until the relevant exact local evidence is reproduced or imported and checked.

This protects the local evidence chain from accidental trust-by-label.

Evidence conflicts¶

If a rigorous upper is below a rigorous lower, at least one piece of the evidence chain is wrong or mismatched.

Possible causes include:

curve/model mismatch;
bad point transport;
malformed import;
engine/wrapper error;
invalid assumptions/certificate.

The correct response is investigation, not averaging the bounds.

Computational outcome vs theorem¶

A completed bounded search can prove statements about the bounded computation, such as “no point was found in this region.”

It usually does not prove global nonexistence of rational points.

Timeout is even weaker: it proves only that the process exceeded its budget.

Reporting conventions¶

Use:

rank ≥ r for rigorous lower bound only;
rank ≤ u for rigorous upper bound only;
rank = r only when the interval closes.

This convention is enforced throughout the documentation and should also be used in papers/posts derived from Rank Hunter results.