Milestone 5: controller-routed four-prime M6 re-attestation/search

[mdheller]

Purpose

Start the first real research move after proof-apparatus validation: controller-routed M6 re-attestation / fresh search for the four-prime cohort left without certified second independent points in the v0.5 baseline.

Precondition

Completed:

• State A import PR State A import BSD v0.5 executable baseline payload #8 merged.
• Candidate 2 PR Candidate 2: positive replay against BSD v0.5 baseline #9 merged.
• SocioSphere #320 closed as completed.

Target cohort

{257, 313, 353, 457}

Baseline status from Candidate 2:

n	claim_tier	alg_rank_method	exact rank known	exact rank	second independent point certified
257	H-E1-alg	explicit_point_HB_selmer4	false	null	false
313	H-E1-alg	explicit_point_HB_selmer4	false	null	false
353	H-E1-alg	explicit_point_HB_selmer4	false	null	false
457	H-E1-alg	explicit_point_HB_selmer4	false	null	false

Split by failure mode

257 and 457

Search returned no certified second independent point in the inherited baseline.

Milestone 5 route:

• extended-bound second-generator search;
• or four-descent certificate;
• or Heegner/Gross-Zagier/Kolyvagin route if explicitly scoped.

313 and 353

Inherited search had false positives: descent-image difference was not enough because candidates were in the torsion orbit of P1.

Milestone 5 route:

• orbit-membership-aware search from the start;
• record descent image and torsion-orbit membership for every candidate;
• reject torsion translates before any independence claim.

Required controller-witnessed evidence

Any claim-state movement requires a State B PR with:

• controller-routed gate execution;
• exact gate input digest(s), including inherited P1 / row data / search parameters;
• exact gate output digest(s);
• on-curve verification;
• torsion-orbit membership check;
• independence certificate or typed failure;
• updated non-claims for unresolved primes;
• no repo-local promoted state.

Acceptance

One of the following for each target prime:

• certified second independent point with controller-witnessed evidence;
• certified no-result within declared bounds, with bound and search transcript digests;
• typed obstruction wall explaining what method failed and what bridge remains.

Non-claims

This milestone does not by itself prove BSD-I, BSD-II, Tate-Shafarevich finiteness, or rank exactness for primes where second-generator / four-descent evidence remains unresolved.

First implementation step

Create an orbit-membership-aware candidate schema and runner before any new search is interpreted as independence evidence.

[mdheller]

Milestone 5 scaffold step completed.

Merged PR #11: Milestone 5: orbit-membership-aware scaffold.

Observed validation:

Milestone 5 orbit scaffold: success
Proof apparatus continuous validation: success
milestone5_orbit_scaffold: PASS
target_count=4
event_count=4
distinct_input_digest_count=4
distinct_output_digest_count=4
all_p1_on_curve=True

Merged result recorded at:

docs/status/MILESTONE_5_SCAFFOLD_RESULT.md

The scaffold is fail-closed:

• no rank exactness claim;
• no second independent point certification;
• no four-descent execution;
• no BSD-I/BSD-II/Sha claim;
• no adapter gate marked pass;
• no claim promotion.

Target classification is now durable:

257: no_second_generator_found
313: historical_false_positive_requires_orbit_filter
353: historical_false_positive_requires_orbit_filter
457: no_second_generator_found

Next step: implement the first controller-routed candidate runner that consumes records conforming to schemas/milestone5_orbit_candidate.schema.json and enforces candidate on-curve verification, descent-image recording, torsion-orbit membership checking, digest-addressed events, and typed no-result outputs.