{"id": "card_floor_logarithm", "kind": "note", "title": "The logarithm - the instrument the joints share", "body": "Two trees: Napier's table (1614) and Saint-Vincent's hyperbola (1647) - a number's logarithm as a tabulated count, and as an area. They become one function in Euler's Introductio (1748): e, and exp and log as inverses. From there the logarithm enters each of the seven questions as the instrument that turns a product into a sum (log zeta, Euler 1737 and Riemann 1859), a size into a count (the height of a point, Weil 1929 and Néron 1965), a scale into a step (the running coupling, Gell-Mann-Low 1954), a profile into a line (the law of the wall, von Kármán 1930), a number into its length (Cobham 1965), a pole into a form (Deligne 1971), a count of states into an entropy (Boltzmann 1877, Shannon 1948, Perelman 2002). Found, cited; sealed where it is arithmetic (tools/tick.py logarithm_chain).", "source": {"label": "Narrow Highway - a chain on the one map (operator seed)", "url": "", "ref": "docs/MILLENNIUM_PREPAREDNESS.md", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "floor", "bands": ["floor", "chain", "two trees", "logarithm", "napier", "euler", "entropy", "instrument"], "subject": "The logarithm - the instrument the joints share", "connections": [{"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the logarithm chain rests on the one Floor of Discovery"}, {"to_card_id": "card_chain_napier_1614", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_saint_vincent_1647", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_question_riemann", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_bsd", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_navier_stokes", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_yang_mills", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_p_vs_np", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_hodge", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_question_poincare", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chart_e", "relationship": "charted_by", "evidence": "the stick seals e = sum 1/k!; e is the base of the natural logarithm this floor's two trees become one function in"}, {"to_card_id": "card_chart_entropy", "relationship": "charted_by", "evidence": "the stick seals the Maxwell-Boltzmann speeds; entropy is S = k log W (Boltzmann), the logarithm as the count of states - a root of this floor"}, {"to_card_id": "card_instr_mathematics", "relationship": "connects_at", "evidence": "instruments the map: mathematics' numeric mode evaluates every expression the whole map seals - including li(1000), log 2, the law of the wall - the universal instrument"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.floor", "facets": {"subject": ["chain", "entropy", "euler", "instrument", "logarithm", "napier", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "floor", "by": "Narrow Highway - a chain on the one map (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"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 logarithm chain rests on the one Floor of Discovery", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_chain_napier_1614", "title": "Napier 1614 — Mirifici logarithmorum canonis descriptio", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_napier_1614", "resolved": true}, {"id": "card_chain_saint_vincent_1647", "title": "Saint-Vincent 1647 — Opus geometricum quadraturae circuli et sectionum coni", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_saint_vincent_1647", "resolved": true}, {"id": "card_question_riemann", "title": "The Riemann hypothesis", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_riemann", "resolved": true}, {"id": "card_question_bsd", "title": "The Birch and Swinnerton-Dyer conjecture", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_bsd", "resolved": true}, {"id": "card_question_navier_stokes", "title": "Navier-Stokes existence and smoothness", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_navier_stokes", "resolved": true}, {"id": "card_question_yang_mills", "title": "Yang-Mills existence and the mass gap", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_yang_mills", "resolved": true}, {"id": "card_question_p_vs_np", "title": "P versus NP", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_p_vs_np", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_logarithm", "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_logarithm"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}