Saturation¶

Saturation studies the index of the subgroup generated by known Mordell–Weil points.

It does not normally create a new rank direction.

If a rank-\(r\) subgroup is not saturated, saturation can replace it by a larger subgroup of the same rank with smaller/more primitive generators.

Why saturation matters¶

An unsaturated basis can make later work harder:

Saturation is therefore often a conditioning step.

sage -python -m rank42.cli saturate \
  --db rank42.db \
  --curve-id 123 \
  --min-prime 2 \
  --max-prime 100 \
  --timeout 900

This runs one bounded saturation request over the configured prime range.

sage -python -m rank42.cli saturate \
  --db rank42.db \
  --curve-id 123 \
  --mode ladder \
  --min-prime 2 \
  --max-prime 100 \
  --timeout 120

In ladder mode, the timeout is per-prime step.

This can be easier to diagnose because progress/failure is associated with a specific prime.

Saturation should operate on the rigorous witness basis, not an arbitrary collection of unclassified point rows.

If the basis is incomplete, fix independence/evidence first.

A nontrivial saturation index means the original subgroup was not primitive over the searched range.

It does not mean rank increased.

If saturation returns replacement points, Rank Hunter can persist the improved basis/evidence for later reuse.

A timeout means saturation over the requested range was not completed.

Do not record “saturated” for the full range unless the engine completed that claim.

Saturation can help resolve hard independence problems by replacing a bad basis.

It is not a substitute for the final exact independence certificate.