{"id": "card_floor_the_hamiltonian", "kind": "note", "title": "The Hamiltonian — the law of the flow", "body": "The energy operator and the generator of time evolution: it governs how the many potentials develop between the one source and the one end. Four pillars rest on the floors already seeded - the generator (unitary, conserves the total, on the capstone), the eigenbasis (the change-of-domain door, on the solve path), the conserved invariant (Noether's energy), and the no-cut bridge (one generator classical to quantum, the reply to Copenhagen). Its one open end is the measurement gap: H never selects a single outcome. The seals and the four fits are on the stick (stick_the_hamiltonian_the_law_of_the_flow); this card places it on the map (tools/seed_the_hamiltonian.py).", "source": {"label": "Narrow Highway — the Hamiltonian", "url": "", "authority_tier": "engine_derived", "ref": "stick_the_hamiltonian_the_law_of_the_flow"}, "shelf": "codex", "box": "hamiltonian", "bands": ["floor", "hamiltonian", "energy", "time evolution", "quantum", "one map"], "subject": "The Hamiltonian — the law of the flow", "connections": [{"to_card_id": "card_ham_generator", "relationship": "has_part", "evidence": "a pillar of the Hamiltonian (The generator of time evolution)"}, {"to_card_id": "card_ham_eigenbasis", "relationship": "has_part", "evidence": "a pillar of the Hamiltonian (The energy eigenbasis)"}, {"to_card_id": "card_ham_noether", "relationship": "has_part", "evidence": "a pillar of the Hamiltonian (The conserved invariant)"}, {"to_card_id": "card_ham_no_cut", "relationship": "has_part", "evidence": "a pillar of the Hamiltonian (The no-cut bridge)"}, {"to_card_id": "card_question_measurement_gap", "relationship": "has_open_end", "evidence": "the measurement gap - the one outcome H does not select"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the law of the flow on the one map"}, {"to_card_id": "card_lag_legendre_dual", "relationship": "connects_at", "evidence": "the two faces of one dynamics: L is T - V, H is T + V, joined by the Legendre transform"}, {"to_card_id": "card_noether_time_gives_energy", "relationship": "connects_at", "evidence": "time-translation symmetry -> conserved energy = the Hamiltonian; the seam between L and H"}, {"to_card_id": "card_thermo_statistical_mechanics", "relationship": "connects_at", "evidence": "statistical mechanics bridges the microscopic Hamiltonian flow to the one macrostate"}, {"to_card_id": "card_rel_mass_energy", "relationship": "connects_at", "evidence": "E = m c^2: mass is rest energy, part of the Hamiltonian"}, {"to_card_id": "card_qm_quantization", "relationship": "connects_at", "evidence": "the discrete energy spectrum - the Hamiltonian's eigenvalues come in quanta"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.hamiltonian", "facets": {"subject": ["energy", "floor", "one map", "quantum", "time evolution"]}, "presentation": {"glyph": "•", "kind_label": "hamiltonian", "by": "Narrow Highway — the Hamiltonian", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_ham_generator", "title": "The generator of time evolution — unitary, conserves the total", "relationship": "has part", "why": "a pillar of the Hamiltonian (The generator of time evolution)", "href": "/card/card_ham_generator", "resolved": true}, {"id": "card_ham_eigenbasis", "title": "The energy eigenbasis — the stationary states", "relationship": "has part", "why": "a pillar of the Hamiltonian (The energy eigenbasis)", "href": "/card/card_ham_eigenbasis", "resolved": true}, {"id": "card_ham_noether", "title": "The conserved invariant — energy, the same end through the flow", "relationship": "has part", "why": "a pillar of the Hamiltonian (The conserved invariant)", "href": "/card/card_ham_noether", "resolved": true}, {"id": "card_ham_no_cut", "title": "The no-cut bridge — one generator, classical to quantum", "relationship": "has part", "why": "a pillar of the Hamiltonian (The no-cut bridge)", "href": "/card/card_ham_no_cut", "resolved": true}, {"id": "card_question_measurement_gap", "title": "The measurement gap — what the Hamiltonian does not determine", "relationship": "has open end", "why": "the measurement gap - the one outcome H does not select", "href": "/card/card_question_measurement_gap", "resolved": true}, {"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the law of the flow on the one map", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_lag_legendre_dual", "title": "The Legendre dual of the Hamiltonian — H = p q' − L", "relationship": "connects at", "why": "the two faces of one dynamics: L is T - V, H is T + V, joined by the Legendre transform", "href": "/card/card_lag_legendre_dual", "resolved": true}, {"id": "card_noether_time_gives_energy", "title": "Time-translation symmetry gives energy — the Hamiltonian", "relationship": "connects at", "why": "time-translation symmetry -> conserved energy = the Hamiltonian; the seam between L and H", "href": "/card/card_noether_time_gives_energy", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_the_hamiltonian", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_the_hamiltonian"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}