{"query": "Counting: how many ways to choose and to arrange", "count": 20, "results": [{"id": "card_works_counting", "title": "Counting: how many ways to choose and to arrange", "shelf": "the-works", "surface": "secular", "snippet": "From five things, there are 20 ordered pairs but only 10 unordered pairs — order is the whole difference between a permutation and a combination.  Worked & sealed by the engine — P(5,2) = 5!/3! = 20 ·", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_theory_central_place_theory", "title": "Central place theory (Christaller)", "shelf": "theories", "surface": "secular", "snippet": "Central place theory (Christaller) — an engine domain that can touch it: geography. Calibration: partial — specific relations verify; the theory as a whole is not a sealable computation. Settlements a", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_periodic_law", "title": "Periodic law (Mendeleev)", "shelf": "theories", "surface": "secular", "snippet": "Periodic law (Mendeleev) — an engine domain that can touch it: periodic_table. Calibration: seals — element properties and group trends look up and compute. Arrange elements by atomic number and their", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_c_130ff92ca772", "title": "Supersymmetry — pairing as the map's arrange ↔ The periodic table — the map of the elements", "shelf": "connections", "surface": null, "snippet": "the symmetry then PREDICTS the missing partner — a broken pair is a gap that tells us what to look for, the way SUSY predicts a superpartner.  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_n_0be5414e0c37", "title": "Supersymmetry — pairing as the map's arrangement", "shelf": "concepts", "surface": "secular", "snippet": "Arrange the map so each axis/node has a dual; the symmetry then PREDICTS the missing partner — a broken pair is a gap that tells us what to look for, the way SUSY predicts a superpartner.\n\nHONEST STAT", "authority_tier": "matt", "source": "Operator's seed (Matt) — honest grade: resonance", "readable": false, "generated": false}, {"id": "card_bridge_master_binomial_counting", "title": "Master equation: C(n,k) = n! / (k!(n-k)!)", "shelf": "bridges", "surface": "secular", "snippet": "the atom of counting: n! is the number of orderings of n things, and every way to arrange or select is that factorial divided by the orderings that don't matter. Permutations keep order, combinations ", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_c_275d2a32f646", "title": "Supersymmetry — pairing as the map's arrange ↔ Two poles — abstract/formal vs material/embo", "shelf": "connections", "surface": null, "snippet": "Held on for the general method (seek complementary structure) and the refined direction the data points to — NOT the stated pairs.  — a concord the card itself states; mined + verified.", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_c_a9e92568df8b", "title": "Two poles — abstract/formal vs material/embo ↔ Supersymmetry — pairing as the map's arrange", "shelf": "connections", "surface": null, "snippet": "The strongest anti-correlations run ACROSS this divide (reasoning <-> physical_substance, reasoning <-> metabolism), NOT within the proposed supersymmetry duals.  — a concord the card itself states; m", "authority_tier": "engine_derived", "source": "Concordance miner — 2026-07-11", "readable": false, "generated": false}, {"id": "card_src_lex_h6186", "title": "H6186 — עָרַךְ (a.rakh): to arrange", "shelf": "lexicon", "surface": "secular", "snippet": "עָרַךְ (a.rakh), Strong's H6186: to arrange. 1) to arrange, set or put or lay in order, set in array, prepare, order, ordain, handle, furnish, esteem, equal, direct, compare 1a) (Qal) 1a1) to arrange ", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "readable": false, "generated": false}, {"id": "card_src_pron_arranges", "title": "arranges", "shelf": "pronunciation", "surface": "secular", "snippet": "arranges: pronounced (ARPABET) ER0 EY1 N JH IH0 Z. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_src_pron_arrange", "title": "arrange", "shelf": "pronunciation", "surface": "secular", "snippet": "arrange: pronounced (ARPABET) ER0 EY1 N JH. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_src_etym_arrange", "title": "arrange", "shelf": "etymology", "surface": "secular", "snippet": "arrange: etymology (Webster 1913) — v. t.: [OE. arayngen, OF. arengier, F. arranger, fr. a (L. ad) + OF. rengier, rangier, F. ranger. See Range, v. t.]. From Webster's Revised Unabridged Dictionary (1", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_lex_g7262", "title": "G7262 — ἐντάσσω (entassō): to arrange in", "shelf": "lexicon", "surface": "secular", "snippet": "ἐντάσσω (entassō), Strong's G7262: to arrange in. to place or post in :—;Pass., σφενδονᾶν ἐντεταγμένος posted to use the sling, (Xenophon Historicus) (ML)", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "readable": false, "generated": false}, {"id": "card_src_lex_g20234", "title": "G20234 — διαρρυθμίζω (diarruthmizō): to arrange in order", "shelf": "lexicon", "surface": "secular", "snippet": "διαρρυθμίζω (diarruthmizō), Strong's G20234: to arrange in order. arrange in order", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "readable": false, "generated": false}, {"id": "card_src_etym_classify", "title": "classify", "shelf": "etymology", "surface": "secular", "snippet": "classify: etymology (Webster 1913) — v. t.: [L. classis class + Defn: To distribute into classes; to arrange according to a system; to arrnge in sets according to some method founded on common propert", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_lex_g21133", "title": "G21133 — στιχίζω (stichizō): to arrange", "shelf": "lexicon", "surface": "secular", "snippet": "στιχίζω (stichizō), Strong's G21133: to arrange. arranged in a row", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "readable": false, "generated": false}, {"id": "card_src_lex_g4294", "title": "G4294 — προκαταρτίζω (prokatartizō): to arrange", "shelf": "lexicon", "surface": "secular", "snippet": "προκαταρτίζω (prokatartizō), Strong's G4294: to arrange. προ-κατ-αρτίζω to make ready beforehand : 2Co.9:5. † (AS)", "authority_tier": "reference", "source": "STEPBible TBESH/TBESG — Extended Strong's lexicon (CC-BY, Tyndale House)", "readable": false, "generated": false}, {"id": "card_src_etym_assessor", "title": "assessor", "shelf": "etymology", "surface": "secular", "snippet": "assessor: etymology (Webster 1913) — n.: [L., one who sits beside, the assistant of a judge, fr. assid. See Assession. LL., one who arranges of determines the taxes, fr. assid. See Assess, v., and cf.", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_etym_disarrange", "title": "disarrange", "shelf": "etymology", "surface": "secular", "snippet": "disarrange: etymology (Webster 1913) — v. t.: [Pref. dis- + arrange: cf. F. désarranger.]. From Webster's Revised Unabridged Dictionary (1913), public domain.", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}, {"id": "card_src_etym_enrange", "title": "enrange", "shelf": "etymology", "surface": "secular", "snippet": "enrange: etymology (Webster 1913) — v. t.: [Pref. en- + range. Cf. Enrank, Arrange.]. From Webster's Revised Unabridged Dictionary (1913), public domain.", "authority_tier": "reference", "source": "Webster's Revised Unabridged Dictionary (1913), Project Gutenberg eBook #29765 — public domain", "readable": true, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_works_counting"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}