{"id": "card_solve_refine", "kind": "reference", "title": "Refine — tighten the bound, each step sealed", "body": "A monotone loop: one more term of the series, one more Newton step, one more zero counted — each tightening sealed, the window shrinking. Success guides the direction, failure narrows by proving the bound cannot tighten here. Runs only if the decision needs more than the first estimate gave.", "source": {"label": "Narrow Highway — a step of the solve path", "url": "", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "solve_step", "bands": ["solve path", "method", "refine"], "subject": "Refine — tighten the bound, each step sealed", "connections": [{"to_card_id": "card_spine_solve_path", "relationship": "member_of", "evidence": "a step of the solve path"}, {"to_card_id": "card_solve_get_close", "relationship": "builds_on", "evidence": "refine follows get_close on the solve path"}, {"to_card_id": "card_solve_change_domain", "relationship": "builds_on", "evidence": "refine follows change_domain on the solve path"}, {"to_card_id": "card_solve_answer", "relationship": "enables", "evidence": "answer follows refine on the solve path"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.solve.step", "facets": {"subject": ["method", "refine", "solve path"]}, "presentation": {"glyph": "•", "kind_label": "solve step", "by": "Narrow Highway — a step of the solve path", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_spine_solve_path", "title": "The solve path — a spine", "relationship": "on the shelf of", "why": "a step of the solve path", "href": "/card/card_spine_solve_path", "resolved": true}, {"id": "card_solve_get_close", "title": "Get close — a bounded estimate (the approximate flip)", "relationship": "builds on", "why": "refine follows get_close on the solve path", "href": "/card/card_solve_get_close", "resolved": true}, {"id": "card_solve_change_domain", "title": "Change of domain — solve in the eigenbasis, map back (the exact flip)", "relationship": "builds on", "why": "refine follows change_domain on the solve path", "href": "/card/card_solve_change_domain", "resolved": true}, {"id": "card_solve_answer", "title": "The answer — the surviving window closes", "relationship": "enables", "why": "answer follows refine on the solve path", "href": "/card/card_solve_answer", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_solve_refine", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_solve_refine"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}