Independence¶

Exact rational points do not automatically give the same number of Mordell–Weil generators.

The Independence tools determine whether candidate exact points add directions beyond the current rigorous witness basis.

Standard command¶

sage -python -m rank42.independence_check \
  --db rank42.db \
  --curve-id 123 \
  --timeout 120

Select specific ledger rows with repeated --point-id options.

Use:

sage -python -m rank42.independence_check --help

for the release's full option set.

Core options¶

The current parser includes:

per-candidate certificate timeout;
maximum halvings;
maximum prime;
maximum matrix columns;
strategy: standard, saturation, or trial-saturation;
saturation timeout/prime controls;
option to disable full trial-saturation fallback.

These are proof-search budgets, not rank knobs.

Increasing them may resolve an inconclusive candidate; it cannot make a dependent point independent.

Authoritative basis¶

Independence begins from the authoritative rigorous witness basis for the curve.

It does not simply take every exact point row and assume those points form a basis.

This distinction prevents previously unresolved/dependent discoveries from contaminating later certification.

Basis-first hard-case strategy¶

For difficult candidates, Rank Hunter can:

prepare or reuse a 2-saturated rigorous basis;
retry the exact certificate against that prepared basis;
if still unresolved and allowed, perform full-trial 2-saturation on basis + candidate;
retry exact certification.

Completed saturated bases can be persisted and reused for remaining hard cases.

Outcomes¶

A candidate can end as:

independent — accepted into the working basis;
dependent — exact relation/no new rank direction;
inconclusive — certificate budget/path did not resolve it;
timeout — bounded computation expired;
error — worker/certificate failure.

Only the first outcome can increase the rigorous lower bound.

Why numerical height rank is not enough¶

A floating-point height Gram matrix can be useful for screening and ordering candidate points.

It is not by itself the exact certificate used to promote rank.

Numerical near-singularity is especially dangerous in high-rank, large-height groups.

Saturation vs independence¶

Saturation and independence answer different questions.

Independence:

Does this point add a new \(\mathbf Z\)-rank direction?

Saturation:

Is the subgroup generated by the current basis missing divisible points of the same rank?

A saturated subgroup can still have the same rank.

Hard-case escalator¶

For unresolved cases:

sage -python -m rank42.hard_case_escalator \
  --db rank42.db \
  --curve-id 123

Use this when the standard certificate path is genuinely stuck, not as a replacement for the normal workflow.

Relation attack¶

When lattice geometry suggests a relation:

sage -python -m rank42.mw_relation_attack \
  --db rank42.db \
  --curve-id 123

A relation attack is useful for understanding dependence; it should not be confused with finding a new generator.

Generic sections¶

For a Family with rational sections over \(\mathbf Q(T)\), one rigorous route to generic independence is to specialize at a good fiber and exact-certify the specialized images.

If a nontrivial generic relation existed, it would specialize. Therefore an exact independent specialization can certify the generic packet under the appropriate good-specialization assumptions.

This argument must be documented explicitly; a metadata field saying “generic rank 17” is not the proof.

Reporting¶

Prefer:

17 exact specialized points were certified independent, proving rank ≥ 17 on this fiber.

Avoid:

The search found 17 points, therefore rank is 17.