{
  "schema_version": "public-current-state-v3",
  "as_of": "2026-10-07",
  "research_completed": "2026-10-06",
  "rh_status": "OPEN",
  "CMP": "OPEN",
  "ES": "OPEN",
  "combined_case": "CASE D — CMP OPEN + ES OPEN",
  "previous_state": "current-state-2026-10-05.json",
  "historical_state": "research-state.json",
  "public_summaries": {
    "ja": "../docs/ja/updates/2026-10-07-core-s.md",
    "en": "../docs/en/updates/2026-10-07-core-s.md"
  },
  "core_s": {
    "scope": "Every canonical actual finite Weil matrix at a>0, integer N>=1",
    "statement": "e_+<e_- implies dim ker(Q^+-e_+ I)=1",
    "finite_ES_equivalence": "ES(a,N) iff e_-(a,N)>e_+(a,N)",
    "derived_from": "CCM, arXiv:2511.22755v1, Section 5.1-5.2, Lemmas 5.1-5.2",
    "primary_source": "https://arxiv.org/html/2511.22755v1#S5",
    "independent_even_simplicity_obligation_removed": true,
    "uniform_gap_bound_proved": false,
    "bibliographic_novelty_claimed": false,
    "internal_independent_audit": "PASS",
    "audit_is_external_peer_review": false,
    "numerical_observations_used_as_proof": false
  },
  "joint_hypotheses": [
    "On one and the same sequence: a_j -> infinity, a_j/(N_j+1) ->0",
    "C_j=||(I-G_{+,j})P_{N_j}R_{a_j}k||_2 ->0, with G_+ the full even ground eigenspace projector",
    "e_-(a_j,N_j)>e_+(a_j,N_j) eventually"
  ],
  "shortest_sufficient_route": "Same-sequence capture + strict parity -> ES -> adopted transfer -> RH; actual antecedents OPEN",
  "conditional_joint_transfer": "ACCEPTED_CONDITIONAL_ONLY",
  "ordinary_L2_alone_suffices": false,
  "lambda_power_rate_required_in_joint_branch": false,
  "old_ES_independent_CMP_R_preserved": true,
  "actual_strong_L2_capture_proved": false,
  "actual_eventual_strict_parity_proved": false,
  "actual_eventual_ES_proved": false,
  "same_cofinal_sequence_established": false,
  "cofinal_compatibility_proved": false,
  "cofinal_incompatibility_proved": false,
  "full_even_ground_capture_proved": false,
  "rh_closed": false,
  "prior_import_audit": {
    "items": 19,
    "generic_or_fixed_scope_delegated": 12
  },
  "new_actual_asymptotic_obligations_discharged": 0,
  "private_source_directory_published": false,
  "source_bundle_published": false,
  "diagnostics_or_internal_agent_logs_published": false,
  "new_research_automatically_started": false,
  "next_research_question": "Capture and strict parity on the same resolving cofinal sequence; publication ends without automatic restart."
}