{"query": "H(½,½) = 1 bit · Hamming(1011, 1001) = 1", "count": 20, "results": [{"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", "readable": false, "generated": false}, {"id": "card_theory_error_correcting_codes", "title": "Error-correcting codes (Hamming, Reed–Solomon)", "shelf": "theories", "surface": "secular", "snippet": "Error-correcting codes (Hamming, Reed–Solomon) — an engine domain that can touch it: information_theory. Calibration: seals — code distances, correctable error counts and parity all compute exactly. A", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"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", "readable": false, "generated": false}, {"id": "card_calc_v_info_theory_hamming_distance", "title": "Hamming distance", "shelf": "calculations", "surface": "secular", "snippet": "Hamming distance — information_theory. Formula: count of differing positions. Canonical FORM: combinatorial (C(n,k) = n! / (k!(n-k)!)) — engine verifier info_theory.hamming_distance — the deterministi", "authority_tier": "reference", "source": "The Calculation Map — every calculation, mapped by form", "readable": false, "generated": false}, {"id": "card_bridge_theory_central_dogma_of_molecular_biology__error_correcting_codes", "title": "Bridge: Central dogma of molecular biology (DNA→RNA→protein)  ↔  Error-correcting codes (Hamming, Reed–Solomon)", "shelf": "bridges", "surface": "secular", "snippet": "Central dogma of molecular biology (DNA→RNA→protein) and Error-correcting codes (Hamming, Reed–Solomon) are the same form in different domains. the genetic code's REDUNDANCY -- several codons per amin", "authority_tier": "reference", "source": "The Bridges — cross-domain isomorphisms", "readable": false, "generated": false}, {"id": "card_src_word_hamming", "title": "hamming", "shelf": "dictionary", "surface": "secular", "snippet": "hamming: (noun) poor acting by a ham actor — syn: overacting", "authority_tier": "reference", "source": "WordNet 3.0, Princeton University (WordNet License)", "readable": false, "generated": false}, {"id": "card_seal_ca78855e99416219f6c304f52d74315f3164bef8f8f033c4ca441761d319f693", "title": "Receipt ca78855e9941… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"information_theory\",\n  \"dimensions\": [\n   \"conservation_balance\",\n   \"discreteness\",\n   \"encoding\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"det", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_96b05ebc77392d3af3a44888429b46df46475a310023944ec34f2a635f9024c6", "title": "Receipt 96b05ebc7739… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_chart_self_healing_matrix", "title": "One path, many entries, a self-healing matrix - not many paths", "shelf": "codex", "surface": "secular", "snippet": "One path, many entries, a self-healing matrix - not many paths. A chart on the one map: this is the tick stick stick_one_path_many_entries_a_self_healing_matrix_not_many_paths, placed at the node it c", "authority_tier": "engine_derived", "source": "Narrow Highway - a chart: a tick stick placed on the one map", "readable": false, "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)", "readable": false, "generated": false}, {"id": "card_theory_casimir_effect", "title": "The Casimir effect (structured energy of the quantum vacuum)", "shelf": "theories", "surface": "secular", "snippet": "The Casimir effect (structured energy of the quantum vacuum) — an engine domain that can touch it: physics. Calibration: seals — a deterministic verifier can CONFIRM or BREAK checkable claims drawn fr", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "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)", "readable": false, "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)", "readable": false, "generated": false}, {"id": "card_theory_neuroscience", "title": "Neurons & the action potential (how a nerve carries a signal)", "shelf": "theories", "surface": "secular", "snippet": "Neurons & the action potential (how a nerve carries a signal) — an engine domain that can touch it: medicine. Calibration: seals — conduction velocities, membrane potentials and Nernst equilibria comp", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_vsepr_theory", "title": "VSEPR theory (molecular geometry)", "shelf": "theories", "surface": "secular", "snippet": "VSEPR theory (molecular geometry) — an engine domain that can touch it: molecular_geometry. Calibration: seals — shapes and bond angles predict directly. Valence Shell Electron Pair Repulsion: electro", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_electromagnetic_spectrum", "title": "The electromagnetic spectrum & modulation (radio to gamma, one axis)", "shelf": "theories", "surface": "secular", "snippet": "The electromagnetic spectrum & modulation (radio to gamma, one axis) — an engine domain that can touch it: physics. Calibration: seals — c = fλ, photon energy E = hf, and bandwidth relations all recom", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_exercise_physiology", "title": "Exercise physiology (VO₂max, HR zones)", "shelf": "theories", "surface": "secular", "snippet": "Exercise physiology (VO₂max, HR zones) — an engine domain that can touch it: exercise_science. Calibration: seals — max HR estimates, zones and work rates compute. VO₂max is the maximum rate of oxygen", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_noncommutative_geometry", "title": "Noncommutative geometry & the spectral triple (Connes)", "shelf": "theories", "surface": "secular", "snippet": "Noncommutative geometry & the spectral triple (Connes) — an engine domain that can touch it: mathematics. Calibration: map-only — out of scope for sealing — foundational, empirical or interpretive (RE", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (lone-domain seeding)", "readable": false, "generated": false}, {"id": "card_theory_honeycomb_conjecture", "title": "Optimal packing & the honeycomb conjecture (Hales)", "shelf": "theories", "surface": "secular", "snippet": "The hexagonal grid is the most efficient way to divide a surface into regions of equal area with the LEAST TOTAL PERIMETER. Stated by Pappus of Alexandria around 300 AD as an observation about bees, i", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "generated": false}, {"id": "card_theory_dna_structure", "title": "The double helix (structure that explains its own copying)", "shelf": "theories", "surface": "secular", "snippet": "The double helix (structure that explains its own copying) — an engine domain that can touch it: genetics. Calibration: seals — base-pair ratios, melting temperatures and complement sequences compute.", "authority_tier": "reference", "source": "The Theory Assay — calibrated, not judged (docs/THEORY_CATALOG.md)", "readable": false, "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_v_a2c57d5d0b2a"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}