Search the frontier of
elliptic curves.

Generate a broad candidate set and rank it with Rank Hunter's historical Nagao-style finite-field score at a modest prime bound.

Rescore only the survivors at a larger prime bound with the same Rank Hunter Nagao-style scorer, preserving scorer/prime provenance, then keep a smaller shortlist.

Annotate/filter the final family shortlist against plugin-declared Research Libraries without importing historical library curves.

Fan one exact curve out into squarefree quadratic twists E^(d).

Generate exact quadratic twists, optionally filter by exact root-number sign, rank survivors with Rank Hunter's audited Nagao-style log-cardinality heuristic, and keep only the strongest children under explicit stage/per-twist hard budgets.

Invoke an exact family-plugin transform hook that constructs published or researcher-supplied rank-jump/base-change children.

Apply exact nondegenerate Möbius maps to the current family parameter and continue on the resulting rational specializations.

Enumerate rational prime-degree isogeny neighbors and continue the search on alternate models.

Run several independent classical 2-descent/Selmer/covering branches on the same curve, persist rigorous uppers, collect exact points, then exact-certify any rank growth.

Ask the active family plugin for generic-section construction artifacts in an explicitly named search-height notion.

Ask the family plugin for trace-section artifacts bound to an exact source section and explicit extension/Galois provenance.

Ask the family plugin for typed divisibility conditions such as nP=Q, bound to an exact source trace and an explicit supported division/slope mode.

Search plugin-supplied exact division conditions for rational specializations behind a core-owned hard timeout.

Run successive section-height shells through trace construction, forced division, and exact square-condition specialization, durably checkpointing each completed shell so resume starts at the first unfinished height without repeating verified symbolic work.

Try several bounded rigorous rank-upper routes in sequence: quick PARI, exact Q-isogenous PARI retries, optional short 2-Selmer, known-basis covering descents, and true higher-descent plugin hooks.

Adaptive generator-recovery strategy: try structured descent/covering hooks, then search MW feedback geometry and higher denominator bands before spending time on late saturation/LLL rescue.

Escalate only curves whose rigorous lower bound has reached a configured record-hunt threshold.

Ask the active family plugin for an exact alternative elliptic fibration/surface model and continue the pipeline on the derived family or curve with full lineage.

Cache reusable exact Frobenius data for good primes below a bound.

Compute Rank Hunter's historical log-cardinality Nagao-style score cumulatively and by prime band from cached #E(F_p).

Combine three RH prime signals (log-cardinality, sign-reversed a_p log(p)/p, and normalized trace) into one experimental consensus scheduling score.

Rank Hunter experimental scheduling heuristic: score how consistently the RH Nagao-style signal survives across disjoint prime bands, using the weakest normalized band.

Compute a smoothed good-prime/prime-power explicit-formula proxy for central-zero pressure.

Run Sage's Mestre-Bober zero-sum analytic-rank upper computation in an isolated worker.

For curves with E(Q)[2]=0, compute the Brumer-Kramer class-group upper bound dim Sel_2(E/Q) <= g(E)+u(E)+n(E) from the cubic 2-division field and exact bad-reduction/splitting data.

Run PARI's unconditional 2-descent and 2-part Cassels pairing in an isolated worker.

For curves with a rational 2-isogeny, run eclib/mwrank descent through each rational 2-isogeny and its dual, record the exact phi- and dual-phi Selmer dimensions, verify every route against Sage's exact maps, and promote only the resulting rigorous Mordell-Weil upper.

Compute the exact bad-prime local root factors and global root number; optionally use an exact sign filter for scheduling.

Record bad primes, Kodaira symbols, discriminant/conductor valuations, Tamagawa numbers and local root factors; optional exact scheduling filters.

Use good-reduction injections into E(F_p) to rigorously reject impossible rational torsion targets by order divisibility and exact finite-group structure when available; passing is not proof.

Apply rigorous mod-p obstruction tests to stored degree-4 coverings and pre-sieve downstream point-centered quartics before ratpoints.

Compute Sage E(Q)_tors exactly in an isolated bounded worker.

Ask the active family plugin for preserved exact specialization point bundles and independently certify them before using any historical rank metadata as rigorous evidence.

Specialize plugin-provided generic sections and exact-certify their independence on the stored curve.

Certify the Gusic-Tadic specialization criterion for eligible FormulaFamily models and persist generic-family rank evidence separately from fiber rank evidence.

Compute the exact global root number through the shared bounded prime/local arithmetic service and publish it through Curve Arithmetic authority.

Try a short rigorous PARI rank upper bound with a screening-budget timeout.

Run bounded eclib/mwrank 2-descent and use mwrank rank_bound() as a rigorous Mordell-Weil rank upper bound.

Rank unresolved fibers by the gap between the rigorous mwrank rank_bound() upper and current rigorous lower bound.

Rank Hunter experimental search-yield statistic: prepare one fixed exact search model for the stage, then run shallow denominator-1 searches at increasing heights and score fresh exact-point yield only from completed comparable rounds.

Search the exact stored/family model directly with bounded ratpoints, avoiding global-minimal-model preprocessing by default.

Legacy/alternative Denis Simon two-descent search.

Run bounded standard eclib/mwrank full 2-descent/covering search.

Run rigorous built-in 2-descent first, then escalate through true higher-descent levels supplied by a compatible research plugin.

Replace active rank42.covering.v1 square quartics by smaller exactly equivalent search models.

For each active square-quartic covering, test every completion of Q relevant to local solubility and attach a ranked point-search budget.

Import exact mapped covering objects from the active plugin and/or existing Rank Hunter covering store, search each covering independently, map exact rational hits back to E(Q), and certify growth.

Delegate a true p-adic covering-point search to an installed research engine/plugin through the core-owned hard-timeout worker.

Run the selected Family adapter search for one stored-pool Pipeline candidate inside a temporary database copy.

Invoke the selected family plugin's native target geometry behind a core-owned scientific-write boundary.

Build exact quartic search geometry around already-known points and search mapped rational points.

Repeatedly rebuild point-centered quartic geometry from the enlarged exact point set; each successful round feeds its new points into the next round.

Search adaptive affine charts inside the configured chart-coordinate denominator interval, then exact-certify any new independent points.

Score continued fresh exact-point discovery across completed comparable, progressively deeper chart-denominator searches.

Run exact bounded ratpoints searches over adaptive affine charts.

Certify exact ledger points against the rigorous witness basis.

Saturate the complete rigorous witness subgroup through a bounded prime.

Run an exact bounded saturation prime ladder on the complete rigorous witness basis, ingest any recovered saturated basis points, and re-certify them.

Compute the Néron–Tate height-pairing matrix on the rigorous witness basis, LLL-reduce it, persist the lattice, and exact-certify the reduced point basis.

Spend a larger PARI budget after point discovery to try to close the rigorous rank interval.

Keep only the strongest fibers for later stages.

Progressively narrow the surviving population while moving bounded exact point-search screens through higher chart-denominator bands.

Stop the run as soon as a candidate reaches the configured rigorous lower-bound goal.