{"id": "card_solve_change_domain", "kind": "reference", "title": "Change of domain — solve in the eigenbasis, map back (the exact flip)", "body": "Put the problem in the space where it diagonalizes (its eigenbasis, the frequency basis), solve it as independent scalars, map back — exact when the basis is known and cheap (Fourier, diagonalization; Fibonacci becomes Binet). The door is src/concordance/verifiers/spectral.py; the engine checks the flip (A v = λ v) before trusting it. Where no cheap basis exists, fall through to get close.", "source": {"label": "Narrow Highway — a step of the solve path", "url": "", "authority_tier": "engine_derived", "ref": "stick_change_of_domain_solve_in_the_eigenbasis_map_back"}, "shelf": "codex", "box": "solve_step", "bands": ["solve path", "method", "change_domain"], "subject": "Change of domain — solve in the eigenbasis, map back (the exact flip)", "connections": [{"to_card_id": "card_spine_solve_path", "relationship": "member_of", "evidence": "a step of the solve path"}, {"to_card_id": "card_instr_spectral", "relationship": "connects_at", "evidence": "Change of domain: uses card_instr_spectral"}, {"to_card_id": "card_solve_converge", "relationship": "builds_on", "evidence": "change_domain follows converge on the solve path"}, {"to_card_id": "card_solve_refine", "relationship": "enables", "evidence": "refine follows change_domain 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": ["change_domain", "method", "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_instr_spectral", "title": "Change of domain - the eigenbasis flip", "relationship": "connects at", "why": "Change of domain: uses card_instr_spectral", "href": "/card/card_instr_spectral", "resolved": true}, {"id": "card_solve_converge", "title": "Converge from inside — anchor on what you know exactly", "relationship": "builds on", "why": "change_domain follows converge on the solve path", "href": "/card/card_solve_converge", "resolved": true}, {"id": "card_solve_refine", "title": "Refine — tighten the bound, each step sealed", "relationship": "enables", "why": "refine follows change_domain on the solve path", "href": "/card/card_solve_refine", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_solve_change_domain", "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_change_domain"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}