{ "formula": "p_i = (A*c_i + B*c_(i-1) + C*c_(i-2) + K) mod 29; A,C nonzero", "normalization": "a=B/A, b=C/A modulo 29; q_i=c_i+a*c_(i-1)+b*c_(i-2). A*q_i+K only permutes histogram bins.", "assumptions": [ "Natural Gematria index order 0..28; one output rune per input rune.", "Coefficients fixed across all eight sections for pooled exact extrema; separate per-section extrema also reported.", "Continuous across punctuation, words and pages inside each section; first two runes of every section excluded.", "No nulls, F skips, counter, additional key stream or arbitrary reordering before arithmetic.", "Benchmarks measure solved rune text, not a necessary property of every possible plaintext." ], "scope": "Eight substantial sections in LP2 pages 0..55; nine-rune section 3 excluded. Previously studied data, not a fresh holdout.", "pooled": { "runes": 12931, "normalized_rules": 812, "maximum_single_rune_count": 541, "maximum_single_rune_frequency": 0.04183744489985307, "peak_example": { "a": 1, "b": 25, "peak_count": 541, "entropy_bits": 4.855211116973677, "histogram": [ 440, 483, 451, 422, 477, 393, 476, 486, 431, 415, 418, 461, 461, 442, 448, 434, 437, 418, 428, 459, 435, 456, 436, 418, 446, 445, 429, 541, 445 ] }, "minimum_entropy_bits": 4.854648899025486, "entropy_example": { "a": 8, "b": 8, "peak_count": 511, "entropy_bits": 4.854648899025486, "histogram": [ 457, 438, 407, 419, 450, 441, 453, 478, 470, 449, 454, 441, 499, 408, 466, 438, 390, 449, 511, 406, 464, 421, 439, 483, 436, 401, 413, 503, 447 ] }, "all_normalized_rules": [ { "a": 0, "b": 1, "peak_count": 488, "entropy_bits": 4.855896320865082, "histogram": [ 404, 437, 404, 437, 480, 475, 431, 432, 469, 475, 451, 488, 445, 467, 443, 453, 468, 411, 396, 424, 430, 457, 470, 448, 435, 454, 474, 437, 436 ] 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411, 448, 468, 467, 459, 466, 412, 431, 424, 435, 441, 477, 419, 433, 439, 470, 434, 473, 473, 414 ] }, { "a": 28, "b": 21, "peak_count": 487, "entropy_bits": 4.856814944551871, "histogram": [ 457, 412, 432, 453, 441, 446, 458, 467, 442, 469, 460, 408, 419, 429, 437, 465, 457, 422, 446, 451, 424, 461, 464, 487, 438, 447, 444, 449, 446 ] }, { "a": 28, "b": 22, "peak_count": 483, "entropy_bits": 4.857042510943305, "histogram": [ 440, 448, 473, 466, 442, 455, 432, 429, 483, 418, 459, 437, 439, 437, 432, 425, 439, 440, 434, 445, 446, 424, 458, 464, 468, 452, 473, 439, 434 ] }, { "a": 28, "b": 23, "peak_count": 477, "entropy_bits": 4.856790069792444, "histogram": [ 456, 409, 420, 457, 460, 473, 447, 463, 446, 440, 448, 422, 454, 458, 420, 467, 450, 441, 456, 450, 410, 423, 462, 440, 448, 422, 458, 477, 454 ] }, { "a": 28, "b": 24, "peak_count": 483, "entropy_bits": 4.8561735774629575, "histogram": [ 446, 469, 423, 483, 391, 466, 428, 405, 460, 470, 441, 415, 450, 465, 481, 432, 428, 470, 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477, 429 ] } ] }, "per_section": [ { "section": 0, "runes": 727, "normalized_rules": 812, "maximum_single_rune_count": 46, "maximum_single_rune_frequency": 0.06327372764786796, "peak_example": { "a": 24, "b": 12, "peak_count": 46, "entropy_bits": 4.8206291439566336, "histogram": [ 26, 27, 30, 28, 27, 24, 22, 17, 17, 21, 20, 26, 30, 18, 20, 25, 22, 22, 21, 20, 26, 25, 27, 27, 46, 34, 34, 20, 25 ] }, "minimum_entropy_bits": 4.79684605481209, "entropy_example": { "a": 11, "b": 27, "peak_count": 38, "entropy_bits": 4.79684605481209, "histogram": [ 31, 10, 31, 27, 20, 38, 23, 36, 22, 21, 21, 23, 17, 32, 20, 31, 11, 23, 19, 22, 25, 17, 32, 35, 21, 36, 29, 30, 24 ] } }, { "section": 1, "runes": 1143, "normalized_rules": 812, "maximum_single_rune_count": 68, "maximum_single_rune_frequency": 0.0594925634295713, "peak_example": { "a": 5, "b": 2, "peak_count": 68, "entropy_bits": 4.833753926464031, "histogram": [ 53, 46, 30, 33, 37, 38, 48, 39, 33, 34, 42, 37, 34, 39, 40, 33, 36, 37, 39, 41, 39, 30, 35, 45, 38, 47, 38, 68, 34 ] }, "minimum_entropy_bits": 4.82301810563961, "entropy_example": { "a": 24, "b": 4, "peak_count": 56, "entropy_bits": 4.82301810563961, "histogram": [ 33, 38, 47, 50, 27, 47, 31, 32, 48, 39, 37, 56, 30, 45, 30, 46, 43, 43, 37, 48, 39, 47, 31, 22, 48, 53, 29, 31, 36 ] } }, { "section": 2, "runes": 1727, "normalized_rules": 812, "maximum_single_rune_count": 93, "maximum_single_rune_frequency": 0.05385060799073538, "peak_example": { "a": 20, "b": 8, "peak_count": 93, "entropy_bits": 4.84125866820564, "histogram": [ 61, 49, 56, 56, 57, 55, 66, 43, 66, 70, 67, 93, 56, 65, 53, 54, 60, 68, 59, 60, 59, 67, 50, 66, 49, 52, 56, 49, 65 ] }, "minimum_entropy_bits": 4.835446716177928, "entropy_example": { "a": 14, "b": 11, "peak_count": 78, "entropy_bits": 4.835446716177928, "histogram": [ 64, 61, 59, 63, 61, 69, 57, 54, 75, 63, 61, 57, 58, 73, 49, 73, 67, 31, 58, 67, 53, 48, 63, 78, 56, 56, 68, 44, 41 ] } }, { "section": 4, "runes": 1892, "normalized_rules": 812, "maximum_single_rune_count": 98, "maximum_single_rune_frequency": 0.05179704016913319, "peak_example": { "a": 17, "b": 15, "peak_count": 98, "entropy_bits": 4.841586250728945, "histogram": [ 74, 57, 65, 63, 66, 59, 51, 67, 77, 79, 70, 61, 67, 75, 60, 59, 45, 98, 70, 52, 64, 66, 61, 65, 59, 65, 78, 60, 59 ] }, "minimum_entropy_bits": 4.833182090739941, "entropy_example": { "a": 15, "b": 3, "peak_count": 97, "entropy_bits": 4.833182090739941, "histogram": [ 79, 59, 80, 64, 62, 57, 81, 59, 85, 75, 63, 63, 65, 43, 70, 57, 54, 63, 58, 49, 56, 97, 67, 73, 62, 80, 42, 67, 62 ] } }, { "section": 5, "runes": 1019, "normalized_rules": 812, "maximum_single_rune_count": 63, "maximum_single_rune_frequency": 0.06182531894013739, "peak_example": { "a": 11, "b": 14, "peak_count": 63, "entropy_bits": 4.829524585986367, "histogram": [ 40, 40, 63, 34, 37, 28, 39, 35, 29, 33, 29, 37, 38, 39, 37, 32, 32, 40, 37, 31, 27, 38, 22, 25, 40, 32, 34, 41, 30 ] }, "minimum_entropy_bits": 4.8167362668528515, "entropy_example": { "a": 4, "b": 1, "peak_count": 55, "entropy_bits": 4.8167362668528515, "histogram": [ 35, 55, 24, 23, 33, 37, 49, 24, 30, 31, 40, 30, 37, 21, 29, 34, 40, 39, 21, 28, 40, 34, 36, 34, 46, 49, 43, 39, 38 ] } }, { "section": 6, "runes": 1522, "normalized_rules": 812, "maximum_single_rune_count": 89, "maximum_single_rune_frequency": 0.05847568988173456, "peak_example": { "a": 25, "b": 20, "peak_count": 89, "entropy_bits": 4.834274874649735, "histogram": [ 56, 53, 54, 42, 53, 42, 59, 54, 46, 45, 89, 52, 52, 46, 36, 52, 49, 45, 50, 43, 52, 64, 73, 54, 58, 48, 54, 46, 55 ] }, "minimum_entropy_bits": 4.824085450414543, "entropy_example": { "a": 27, "b": 16, "peak_count": 83, "entropy_bits": 4.824085450414543, "histogram": [ 44, 57, 52, 51, 48, 44, 59, 41, 52, 44, 57, 45, 63, 32, 60, 49, 27, 78, 58, 60, 45, 45, 50, 58, 53, 46, 58, 83, 63 ] } }, { "section": 7, "runes": 1587, "normalized_rules": 812, "maximum_single_rune_count": 86, "maximum_single_rune_frequency": 0.05419029615626969, "peak_example": { "a": 21, "b": 1, "peak_count": 86, "entropy_bits": 4.834715445937616, "histogram": [ 48, 50, 86, 59, 55, 51, 58, 46, 59, 49, 48, 56, 69, 58, 54, 61, 48, 48, 31, 52, 58, 56, 55, 46, 69, 41, 64, 63, 49 ] }, "minimum_entropy_bits": 4.8319922326687195, "entropy_example": { "a": 0, "b": 18, "peak_count": 76, "entropy_bits": 4.8319922326687195, "histogram": [ 56, 61, 44, 56, 76, 66, 57, 40, 51, 38, 41, 63, 73, 57, 45, 42, 70, 65, 48, 64, 39, 61, 61, 54, 45, 49, 62, 49, 54 ] } }, { "section": 8, "runes": 3314, "normalized_rules": 812, "maximum_single_rune_count": 155, "maximum_single_rune_frequency": 0.04677127338563669, "peak_example": { "a": 20, "b": 2, "peak_count": 155, "entropy_bits": 4.84962372264329, "histogram": [ 113, 120, 121, 102, 119, 115, 91, 116, 106, 110, 108, 117, 120, 122, 128, 130, 155, 119, 118, 120, 121, 101, 116, 91, 104, 103, 113, 117, 98 ] }, "minimum_entropy_bits": 4.846413656620303, "entropy_example": { "a": 23, "b": 6, "peak_count": 138, "entropy_bits": 4.846413656620303, "histogram": [ 102, 118, 113, 94, 111, 132, 125, 116, 122, 136, 93, 113, 95, 100, 108, 120, 126, 100, 138, 113, 134, 98, 121, 134, 117, 124, 124, 105, 82 ] } } ], "independently_keyed_sections_maximum_single_rune_frequency": 0.053978810610161625, "lp1_benchmark": { "runes": 1953, "histogram": [ 31, 95, 73, 189, 135, 62, 23, 73, 66, 111, 97, 0, 0, 27, 2, 132, 128, 35, 247, 55, 62, 24, 0, 72, 141, 2, 46, 14, 11 ], "entropy_bits": 4.261170010714537, "maximum_frequency": 0.1264720942140297 }, "lp2_page57_benchmark": { "runes": 93, "histogram": [ 3, 6, 4, 3, 7, 5, 1, 3, 1, 9, 9, 0, 0, 0, 0, 5, 5, 1, 15, 3, 3, 1, 0, 4, 4, 0, 1, 0, 0 ], "entropy_bits": 4.029195076130102, "maximum_frequency": 0.16129032258064516 }, "repeated_phrase": { "runes": "\u16de\u16c4\u16a2\u16d2\u16d6\u16c1", "labels": [ "D", "J", "U", "B", "E", "I" ], "occurrences": [ { "page": 27, "page_rune_offset": 28, "prior_two_values": [ 17, 0 ], "prior_two_labels": [ "B", "F" ] }, { "page": 55, "page_rune_offset": 70, "prior_two_values": [ 5, 14 ], "prior_two_labels": [ "K", "X" ] } ], "predecessor_differences_earlier_minus_later": [ 12, 15 ], "BC_pairs_checked": 812, "difference_formula": [ "15*B+12*C mod 29", "15*C mod 29", 0, 0, 0, 0 ], "note": "Under a shared key, last four plaintext runes match; second differs; first may match. This does not identify any plaintext. Separate section keys remove this cross-occurrence constraint." }, "controls": [ { "name": "LP1 cycled", "key": [ 7, 11, 23, 3 ], "roundtrip_exact": true, "normalized_decode_exact": true, "plaintext": { "runes": 12931, "histogram": [ 219, 614, 475, 1236, 864, 402, 143, 484, 493, 696, 619, 0, 0, 159, 7, 901, 833, 219, 1661, 380, 397, 141, 0, 492, 1021, 18, 315, 77, 65 ], "entropy_bits": 4.245730842201766, "maximum_frequency": 0.1284510092026912 }, "sweep": { "runes": 12931, "normalized_rules": 812, "maximum_single_rune_count": 1661, "maximum_single_rune_frequency": 0.1284510092026912, "peak_example": { "a": 14, "b": 24, "peak_count": 1661, "entropy_bits": 4.245730842201767, "histogram": [ 1236, 619, 219, 1021, 475, 696, 833, 492, 614, 493, 901, 0, 219, 484, 7, 141, 65, 143, 159, 397, 77, 402, 0, 380, 315, 864, 0, 1661, 18 ] }, "minimum_entropy_bits": 4.245730842201767, "entropy_example": { "a": 14, "b": 24, "peak_count": 1661, "entropy_bits": 4.245730842201767, "histogram": [ 1236, 619, 219, 1021, 475, 696, 833, 492, 614, 493, 901, 0, 219, 484, 7, 141, 65, 143, 159, 397, 77, 402, 0, 380, 315, 864, 0, 1661, 18 ] } } }, { "name": "LP2 page 57 cycled", "key": [ 19, 0, 5, 28 ], "roundtrip_exact": true, "normalized_decode_exact": true, "plaintext": { "runes": 12931, "histogram": [ 407, 817, 544, 408, 955, 680, 137, 409, 136, 1225, 1226, 0, 0, 136, 0, 678, 680, 137, 2038, 411, 409, 136, 0, 544, 681, 0, 137, 0, 0 ], "entropy_bits": 4.075562744164639, "maximum_frequency": 0.15760575361534299 }, "sweep": { "runes": 12931, "normalized_rules": 812, "maximum_single_rune_count": 2038, "maximum_single_rune_frequency": 0.15760575361534299, "peak_example": { "a": 0, "b": 14, "peak_count": 2038, "entropy_bits": 4.075562744164639, "histogram": [ 0, 2038, 136, 0, 137, 409, 137, 680, 137, 0, 678, 680, 681, 0, 955, 544, 136, 408, 0, 0, 544, 136, 0, 817, 409, 1226, 407, 411, 1225 ] }, "minimum_entropy_bits": 4.075562744164639, "entropy_example": { "a": 0, "b": 14, "peak_count": 2038, "entropy_bits": 4.075562744164639, "histogram": [ 0, 2038, 136, 0, 137, 409, 137, 680, 137, 0, 678, 680, 681, 0, 955, 544, 136, 408, 0, 0, 544, 136, 0, 817, 409, 1226, 407, 411, 1225 ] } } } ], "small_alphabet_exhaustive_check": { "alphabet_size": 5, "keys_exhausted": 400, "normalized_rules": 20, "exact_extrema_agreement": true }, "source_sha256": "b0b2e13c3cd35512ee5955a2d9a56c22d7410e283d35dd9c90abbaa7b99279eb", "script_sha256": "f389c821e832d32a822a42f1d405be8c0374e682ab078dac532f48b76d4138c4", "limitations": [ "Exact finite-corpus extrema, not significance tests or a proof of non-English plaintext.", "No unknown-key plaintext recovery attempted. Controls verify inversion, normalization and bound admissibility.", "An arbitrary substitution AFTER this decoder preserves the frequency limits; arbitrary input alphabet ordering is not tested.", "A changing key, nonlinear two-rune rule, longer state or intermediate encoded text remains outside scope." ] }