{"query": "Measuring information: one bit, and a distance", "count": 20, "results": [{"id": "card_works_information", "title": "Measuring information: one bit, and a distance", "shelf": "the-works", "surface": "secular", "snippet": "A single fair coin carries exactly one bit of Shannon entropy; the Hamming distance counts how many positions two codewords differ in — the ground of error correction.  Worked & sealed by the engine —", "authority_tier": "verified", "source": "The Works — worked & sealed", "generated": false}, {"id": "card_theory_shannon_information_theory", "title": "Shannon information theory (entropy, channel capacity)", "shelf": "theories", "surface": "secular", "snippet": "Shannon information theory (entropy, channel capacity) — an engine domain that can touch it: information_theory. Calibration: seals. H = -Σ p log p. Information is measured by how much UNCERTAINTY a m", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_quantum_information___entanglement__von_neumann_entr", "title": "Quantum information & entanglement (von Neumann entropy)", "shelf": "theories", "surface": "secular", "snippet": "Quantum information & entanglement (von Neumann entropy) — an engine domain that can touch it: information_theory. Calibration: seals. S = -Tr(ρ log ρ), the entropy of a density matrix, which reduces ", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_landauer", "title": "Landauer's principle (erasing information costs heat)", "shelf": "theories", "surface": "secular", "snippet": "Landauer's principle (erasing information costs heat) — an engine domain that can touch it: information_theory. Calibration: seals — the energy bound computes exactly. ERASING one bit must dissipate a", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_network_theory", "title": "Network theory (routing, addressing / CIDR)", "shelf": "theories", "surface": "secular", "snippet": "Network theory (routing, addressing / CIDR) — an engine domain that can touch it: networking. Calibration: seals — subnet masks, address ranges and prefix matches compute exactly. An IP address is spl", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_digital_analog", "title": "Digital versus analog (why discreteness survives copying)", "shelf": "theories", "surface": "secular", "snippet": "Digital versus analog (why discreteness survives copying) — an engine domain that can touch it: computer_science. Calibration: map-only — sampling rates and bit depth are ordinary arithmetic, and THIS", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_discrete_vs_continuous", "title": "Discrete versus continuous (the oldest dividing line)", "shelf": "theories", "surface": "secular", "snippet": "Discrete versus continuous (the oldest dividing line) — an engine domain that can touch it: number_theory. Calibration: map-only — a foundational distinction; specific cardinality results compute. Som", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_theory_double_entry_accounting_identity", "title": "Double-entry accounting identity (A = L + E)", "shelf": "theories", "surface": "secular", "snippet": "Double-entry accounting identity (A = L + E) — an engine domain that can touch it: finance. Calibration: seals — the identity is checkable to the penny. Assets equal Liabilities plus Equity. Every tra", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "generated": false}, {"id": "card_domchk_information_theory_claimed_entropy_bits", "title": "Information Theory: entropy bits", "shelf": "information_theory", "surface": "secular", "snippet": "A worked check in information theory: entropy bits.\n\nGIVEN\n  bsc_error_rate = 0.1\n  probabilities = [0.5, 0.5]\n  string_a = 10101010\n  string_b = 10100100\n\nCLAIMED\n  claimed_entropy_bits = 1\n\nTHE ENGI", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "generated": false}, {"id": "card_domchk_cybersecurity_claimed_entropy_bits", "title": "Cybersecurity: entropy bits", "shelf": "cybersecurity", "surface": "secular", "snippet": "A worked check in cybersecurity: entropy bits.\n\nGIVEN\n  charset_size = 94\n  cidr_prefix = 24\n  cvss_base_score = 9.1\n  password_length = 16\n  port_number = 443\n  tls_version = 1.2\n\nCLAIMED\n  claimed_e", "authority_tier": "reference", "source": "The verifier's own documented relation, and a run this engine performed against it (deterministic; re-runnable with tools/domain_goldens.py)", "generated": false}, {"id": "card_src_word_check_bit", "title": "check bit", "shelf": "dictionary", "surface": "secular", "snippet": "check bit: (noun) (computer science) a bit that is used in an error detection procedure in which a 0 or 1 is added to each group of bits so that it will have either an odd number of 1's or an even num", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_src_word_parity_bit", "title": "parity bit", "shelf": "dictionary", "surface": "secular", "snippet": "parity bit: (noun) (computer science) a bit that is used in an error detection procedure in which a 0 or 1 is added to each group of bits so that it will have either an odd number of 1's or an even nu", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "generated": false}, {"id": "card_alm_connection_entropy_is_thermodynamic", "title": "Almanac: Shannon entropy and thermodynamic entropy are the same quantity", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  The entropy of a message and the entropy of a gas are not analogies -- they are one formula. information_theory confirmed Shannon entropy numerically: a fair coin carries 1 bit, four equal", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_alm_connection_independence_multiplies_information_adds", "title": "Almanac: Independence multiplies probability but adds information", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  Flip two fair coins. The probability verifier confirmed that independent events MULTIPLY: P(A and B) = 0.5 x 0.5 = 0.25. The information_theory verifier confirmed that one fair coin carrie", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_n_49d1ebe5ba6a", "title": "The genome as information — and the algebra of a population", "shelf": "science", "surface": "secular", "snippet": "Two bits per base, sealed: log(4)/log(2) = 2 — so the ~3.2 billion base pairs of the human\ngenome hold ~6.4 billion bits, about 800 MB, a large app's worth of code in every cell (https://narrowhighway", "authority_tier": "engine_derived", "source": "Concordance assay — 2026-07-09", "generated": false}, {"id": "card_v_a2c57d5d0b2a", "title": "H(½,½) = 1 bit · Hamming(1011, 1001) = 1", "shelf": "information_theory", "surface": "secular", "snippet": "Verdict: HOLDS.\n• info_theory.shannon_entropy: H = 1.000000 bits (matches claim 1.0)\n• info_theory.hamming_distance: Hamming('1011', '1001') = 1 (matches claim)\nSealed and independently re-checkable: ", "authority_tier": "engine", "source": "Verified by the engine — information_theory", "generated": false}, {"id": "card_src_rfc_3190", "title": "RFC3190 — RTP Payload Format for 12-bit DAT Audio and 20- and 24-bit Linear Sampled Audio", "shelf": "rfcs", "surface": "secular", "snippet": "RFC3190: RTP Payload Format for 12-bit DAT Audio and 20- and 24-bit Linear Sampled Audio (January 2002). Status: PROPOSED STANDARD.", "authority_tier": "reference", "source": "The RFC Index (RFC Editor / IETF) — public domain", "generated": false}, {"id": "card_alm_connection_birthday_bound_halves_hash_security", "title": "Almanac: Pigeonhole and the birthday bound: why a 256-bit hash gives only 128-bit collision resistance", "shelf": "almanac", "surface": "secular", "snippet": "SITUATION\n  A hash maps unlimited inputs to a fixed set of outputs, so collisions are not just possible -- they are guaranteed, and they come sooner than intuition says. mathematics confirmed the pige", "authority_tier": "reference", "source": "The Almanac — verified-only practical wisdom (sealed)", "generated": false}, {"id": "card_n_26706d587f17", "title": "Easton: Bit", "shelf": "dictionary", "surface": "secular", "snippet": "The curb put into the mouths of horses to restrain them. The Hebrew word (metheg) so rendered in Ps. 32:9 is elsewhere translated “bridle” (2 Kings 19:28; Prov. 26:3; Isa. 37:29). Bits were generally ", "authority_tier": "external_aligned", "source": "Matthew Easton, Illustrated Bible Dictionary (1897)", "generated": false}, {"id": "card_src_food_169893", "title": "Bacon Bits, Meatless", "shelf": "nutrition", "surface": "secular", "snippet": "Bacon bits, meatless — a food in the USDA FoodData Central reference. Per 100 g: 476 kcal, 32 g protein, 25.9 g fat, 28.6 g carbohydrate.", "authority_tier": "reference", "source": "USDA FoodData Central (public domain)", "generated": false}]}