<!--
/**
 * ============================================================================
 * TAXONOMY AND UNIVERSAL CONVERGENCE
 * ============================================================================
 * 
 * A comprehensive web-based tool for exploring the complete classification 
 * and convergence structure of Collatz sequences through the lens of 
 * parameter repetitions and pseudocycles. This tool demonstrates the 
 * exhaustive partition of all positive integers into structural classes 
 * S(m_r) and visualizes the three taxonomic types (A, B or C) that determine 
 * convergence behavior.
 * 
 * MATHEMATICAL FOUNDATION:
 * Based on "On the Convergence of Collatz Sequences: A New Algebraic Framework"
 * by Javier Hernandez (2026).
 * https://doi.org/10.5281/zenodo.15546925
 *
 * CORE CONCEPTS:
 * - First Repeated Parameter (Definition 6.4): Each n belongs to exactly 
 *   one class S(m_r) determined by its first repeated parameter value
 * - Invariant Properties (Theorems 6.6, 6.7 and 6.9): Distance d(m_r), 
 *   local maximum m* and wormhole structure are constant for all n in S(m_r)
 * - Taxonomic Types (Definition 7.3): Position of global maximum M* relative 
 *   to pseudocycle determines Type A, B or C
 * - Regular and Irregular Elements (Definition 8.1): n is regular if 
 *   n = e(m_r)·2^k for some k ≥ 0, irregular otherwise
 * - Universal Convergence (Theorem 9.4): Every positive integer converges 
 *   to the cycle (4, 2, 1) — every n is regular or irregular, both converge
 * 
 * VISUALIZATION FEATURES:
 * 1. Classification Display: Shows n ∈ S(m_r) with taxonomy type
 * 2. Defragmentation View: Horizontal bars showing complete m-sequences
 * 3. Color Coding System:
 *    - Orange (#f39c12): m = 0 (trivial cycle reached)
 *    - Red (#e74c3c): m_r occurrences (pseudocycle markers)
 *    - Green (#2ecc71): m* (local maximum of pseudocycle)
 *    - Blue (#3498db): M* (global maximum of sequence)
 *    - Purple (#8e44ad): m* = M* (Type B indicator)
 *    - Gray (#555): Regular intranscendent values
 * 4. Interactive Tooltips: Detailed information on hover
 * 5. Class Comparison: View multiple sequences from same S(m_r)
 * 6. Badge System: "YOU" for user input, "LATEST m_r^type" for newest found
 *    (LATEST badge suppressed for S(0): class is structurally obvious and unbounded)
 * 
 * CLASSIFICATION METRICS:
 * - Class: S(m_r) - The structural equivalence class
 * - Taxonomy: Type A, B or C based on M* position
 * - m_r: First repeated parameter value
 * - m*: Local maximum within pseudocycle (invariant); null for S(0) where
 *   the trivial repetition 2→1→4 (m=0, m=0) has nothing between the two zeros
 * - M*: Global maximum of this specific sequence
 * - Distance d(m_r): Steps between m_r repetitions (invariant)
 * 
 * TAXONOMY DETERMINATION (Definition 7.3):
 * Type A: M* > m* and occurs before first m_r (pre-pseudocycle maximum)
 * Type B: M* = m* and occurs within pseudocycle (between first and second m_r)
 * Type C: M* > m* and occurs after second m_r (post-pseudocycle maximum)
 * 
 * KNOWN S(m_r) CLASSES:
 * Computational verification for n < 2^40 identifies exactly 42 distinct 
 * parameter values exhibiting repetitions (|M_rep| = 42), creating 42 
 * structural classes. Each class has:
 * - Fixed distance d(m_r)
 * - Fixed local maximum m* (null for S(0))
 * - Fixed entry point e(m_r)
 * - Invariant wormhole W(m_r) to m = 0
 * 
 * INTERACTIVE DEMONSTRATION:
 * 1. Enter positive integer n
 * 2. Click "Process" to compute classification
 * 3. View class S(m_r) and taxonomy type
 * 4. Explore defragmentation view showing:
 *    - Your sequence (highlighted row)
 *    - Other examples from same class
 *    - Color-coded special values
 * 5. Hover over cells for detailed information
 * 6. Observe that all sequences in S(m_r) share same m* and d(m_r)
 *    (m* is null for S(0))
 * 
 * CONVERGENCE PROOF STRUCTURE:
 * Universal: every n is regular or irregular, so both converge (Theorem 9.4)
 * 
 * AUTHOR: Javier Hernandez
 * EMAIL: 271314@pm.me
 * LICENSE: https://github.com/hhvvjj/a-new-algebraic-framework-for-the-collatz-conjecture/blob/main/LICENSE
 * REPOSITORY: https://github.com/hhvvjj/a-new-algebraic-framework-for-the-collatz-conjecture/tree/main/step04-taxonomy-and-universal-convergence
 * 
 * ============================================================================
 */
-->

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    <div class="header">
        <div>
            <h1>Taxonomy and Universal Convergence</h1>
        </div>
        
        <div class="input-section">
            <input type="number" id="numberInput" min="1" placeholder="Enter n > 0">
            <button onclick="processNumber()">Process</button>
        </div>
    </div>

    <div id="error" style="display: none;"></div>

    <div id="content" class="content">
        <div class="info-section">
            <div class="section-title" id="classificationTitle">Classification Results</div>
            <div class="info-grid" id="infoGrid">
            </div>
        </div>

        <div class="defrag-section">
            <div class="section-title">Defragmentation View for Class S(<span id="classLabel">-</span>)</div>
            
            <div class="legend">
                <div class="legend-item">
                    <div class="legend-box" style="background: #555;"></div>
                    <span class="legend-text">Intranscendent values</span>
                </div>
                <div class="legend-item">
                    <div class="legend-box" style="background: #e74c3c;"></div>
                    <span class="legend-text">m<sub>r</sub></span>
                </div>
                 <div class="legend-item">
                    <div class="legend-box" style="background: #2ecc71;"></div>
                    <span class="legend-text">m*</span>
                </div>
                <div class="legend-item">
                    <div class="legend-box" style="background: #3498db;"></div>
                    <span class="legend-text">M*</span>
                </div>
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                    <span class="legend-text">m* = M*</span>
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                <div class="legend-item">
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                    <span class="legend-text">m = 0 (terminal)</span>
                </div>
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            </div>
        </div>

    </div>

    <div id="tooltip" class="tooltip"></div>

    <div class="author-info">
        <p><strong>Author:</strong> Javier Hernandez | <strong>Email:</strong> 271314@pm.me</p>
        <a href="https://github.com/hhvvjj/a-new-algebraic-framework-for-the-collatz-conjecture/tree/main/step04-taxonomy-and-universal-convergence" target="_blank">Go To GitHub Repository</a> |
        <a href="https://github.com/hhvvjj/a-new-algebraic-framework-for-the-collatz-conjecture/blob/main/LICENSE" target="_blank">License</a> | 
        <a href="https://doi.org/10.5281/zenodo.15546925" target="_blank">Research Reference</a>
    </div>

    <script>

        /**
         * Dictionary mapping m_r values to their properties (m*, distance, latest, examples).
         * Based on computational verification for n < 2^40.
         * @constant {Object.<number, {emr: number, mstar: number|null, distance: number, latest: number, examples: number[]}>}
         */
        const MR_MSTAR_DICT = {
            0: { emr: 1, mstar: null, distance: 1, latest: 1, examples: [1, 2, 4, 5, 8, 10, 11, 13, 16, 17, 20, 21, 22, 23, 26, 29, 32, 34, 35, 40, 42, 44, 45, 46, 52, 53, 58, 61, 64, 68, 69, 70, 75, 77, 80, 81, 84, 85, 88, 90, 92, 93, 103, 104, 106, 111, 113, 116, 117, 122, 127, 128, 136, 137, 138, 140, 141, 150, 151, 154, 155, 160, 162, 163, 167, 168, 169, 170, 175, 176, 180, 181, 184, 186, 191, 201, 205, 206, 207, 208, 212, 213, 217, 222, 223, 225, 226, 227, 232, 233, 234, 239, 241, 244, 245, 251, 254, 255, 256, 263, 267, 272, 273, 274, 275, 276, 277, 280, 282, 283, 287, 289, 297, 300, 301, 302, 303, 308, 309, 310, 311, 319, 320, 321, 324, 325, 326, 331, 334, 335, 336, 338, 339, 340, 341, 350, 351, 352, 359, 360, 362, 365, 368, 369, 372, 373, 377, 382, 383, 385, 395, 401, 402, 403, 410, 412, 413, 414, 416, 423, 424, 425, 426, 431, 433, 434, 435, 441, 444, 445, 446, 450, 451, 452, 453, 454, 455, 464, 466, 467, 468, 469, 475, 478, 479, 482, 483, 488, 490, 495, 497, 502, 503, 507, 508, 509, 510, 512, 513, 526, 527, 534, 535, 537, 539, 544, 546, 547, 548, 549, 550, 552, 554, 555, 560, 564, 565, 566, 567, 571, 574, 575, 577, 578, 579, 583, 593, 594, 595, 600, 601, 602, 604, 605, 606, 607, 616, 618, 620, 621, 622, 625, 633, 635, 637, 638, 639, 640, 642, 643, 647, 648, 650, 651, 652, 653, 662, 663, 668, 669, 670, 672, 676, 677, 678, 680, 682, 683, 687, 700, 701, 702, 704, 711, 713, 715, 718, 719, 720, 724, 725, 729, 730, 733, 736, 738, 739, 743, 744, 746, 753, 754, 755, 761, 764, 765, 766, 769, 770, 771, 777, 790, 791, 793, 801, 802, 803, 804, 805, 806, 809, 820, 821, 824, 826, 828, 829, 832, 833, 846, 847, 848, 849, 850, 851, 852, 853, 857, 862, 863, 866, 867, 868, 869, 870, 875, 882, 883, 888, 890, 891, 892, 893, 900, 901, 902, 904, 906, 907, 908, 909, 910, 911, 927, 928, 932, 933, 934, 936, 938, 939, 950, 951, 953, 955, 956, 957, 958, 959, 964, 965, 966, 971, 973, 976, 977, 980, 981, 985, 987, 990, 991, 994, 995] },
            1: { emr: 3, mstar: 2, distance: 5, latest: 3, examples: [3] },
            2: { emr: 6, mstar: 4, distance: 3, latest: 12, examples: [6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368] },
            3: { emr: 7, mstar: 25, distance: 13, latest: 7, examples: [7] },
            6: { emr: 14, mstar: 25, distance: 8, latest: 28, examples: [9, 14, 28, 37, 43, 49, 56, 57, 65, 74, 86, 87, 98, 99, 112, 114, 115, 130, 131, 135, 148, 149, 153, 172, 173, 174, 196, 197, 198, 203, 224, 228, 229, 230, 260, 261, 262, 270, 271, 296, 298, 305, 306, 307, 344, 346, 348, 349, 361, 363, 367, 379, 392, 394, 396, 397, 406, 407, 409, 427, 447, 448, 456, 458, 460, 461, 465, 481, 489, 505, 511, 520, 522, 524, 525, 529, 540, 541, 542, 545, 551, 559, 569, 592, 596, 597, 603, 610, 611, 612, 613, 614, 641, 671, 673, 679, 681, 688, 692, 693, 696, 698, 705, 721, 722, 723, 726, 727, 734, 735, 745, 758, 759, 767, 784, 788, 789, 792, 794, 795, 799, 807, 812, 813, 814, 817, 818, 819, 827, 839, 854, 855, 894, 895, 896, 897, 905, 912, 916, 917, 920, 922, 930, 931, 943, 961, 962, 963, 969, 978, 979, 993] },
            7: { emr: 15, mstar: 79, distance: 13, latest: 120, examples: [15, 30, 60, 120, 240, 480, 960, 1920, 3840, 7680, 15360, 30720, 61440, 122880, 245760, 491520, 983040, 1966080, 3932160, 7864320, 15728640, 31457280, 62914560, 125829120, 251658240, 503316480, 1006632960, 2013265920] },
            8: { emr: 18, mstar: 16, distance: 8, latest: 36, examples: [18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216, 18432, 36864, 73728, 147456, 294912, 589824, 1179648, 2359296, 4718592, 9437184, 18874368, 37748736, 75497472, 150994944, 301989888, 603979776]},
            9: { emr: 19, mstar: 43, distance: 13, latest: 76, examples: [19, 38, 76, 59, 67, 79, 89, 101, 105, 118, 119, 123, 134, 139, 152, 157, 158, 178, 179, 185, 187, 202, 209, 210, 211, 219, 236, 237, 238, 246, 247, 249, 268, 269, 278, 279, 281, 295, 304, 314, 316, 317, 329, 356, 357, 358, 370, 371, 374, 375, 393, 404, 405, 418, 419, 420, 421, 422, 438, 439, 443, 472, 474, 476, 477, 492, 493, 494, 498, 499, 519, 536, 538, 556, 557, 558, 561, 562, 563, 585, 590, 591, 608, 615, 628, 629, 632, 634, 657, 658, 659, 665, 699, 703, 712, 714, 716, 717, 740, 741, 742, 748, 749, 750, 779, 786, 787, 808, 810, 836, 837, 838, 840, 842, 843, 844, 845, 876, 877, 878, 886, 887, 923, 937, 944, 948, 949, 952, 954, 984, 986, 988, 989, 996, 997, 998, 999]},
            12: { emr: 25, mstar: 43, distance: 13, latest: 50, examples: [25, 50 ,100, 133, 177, 200, 266, 315, 354, 355, 400, 473, 532, 533, 630, 631, 708, 709, 710, 747, 800, 841, 945, 946, 947]},
            16: { emr: 33, mstar: 49, distance: 13, latest: 66, examples: [33, 66, 132, 264, 528, 1056, 2112, 4224, 8448, 16896, 33792, 67584, 135168, 270336, 540672, 1081344, 2162688, 4325376, 8650752, 17301504, 34603008, 69206016, 138412032, 276824064, 553648128 ]},
            19: { emr: 39, mstar: 151, distance: 26, latest: 156, examples: [39, 78, 156, 312, 624, 1248, 2496, 4992, 9984, 19968, 39936, 79872, 159744, 319488, 638976, 1277952, 2555904, 5111808, 10223616, 20447232, 40894464, 81788928, 163577856, 327155712, 654311424] },
            25: { emr: 51, mstar: 115, distance: 13, latest: 204, examples: [51, 102, 204, 408, 816, 1632, 3264, 6528, 13056, 26112, 52224, 104448, 208896, 417792, 835584, 1671168, 3342336, 6684672, 13369344, 26738688, 53477376, 106954752, 213909504, 427819008, 855638016] },
            45: { emr: 91, mstar: 4615, distance: 75, latest: 9172, examples: [63, 91, 95, 126, 143, 182, 190, 199, 215, 235, 253, 265, 286, 299, 313, 323, 337, 353, 364, 380, 381, 398, 399, 417, 430, 449, 470, 471, 485, 506, 530, 531, 572, 573, 598, 599, 626, 627, 646, 674, 675, 706, 707, 728, 760, 762, 796, 797, 798, 834, 835, 860, 861, 898, 899, 903, 940, 941, 942, 970, 1012, 1013, 1060, 1061, 1062, 1113, 1144, 1146, 1196, 1197, 1198, 1252, 1253, 1254, 1292, 1293, 1348, 1349, 1350, 1355, 1412, 1413, 1414, 1419, 1456, 1483, 1520, 1524, 1525, 1592, 1594, 1596, 1597, 1668, 1669, 1670, 1675, 1720, 1722, 1796, 1797, 1798, 1806, 1880, 1882, 1884, 1885, 1940, 1941, 1977, 2024, 2026, 2033, 2120, 2122, 2124, 2125, 2129, 2225, 2226, 2233, 2288, 2292, 2293, 2392, 2394, 2396, 2397, 2504, 2506, 2508, 2509, 2513, 2584, 2586, 2696, 2698, 2700, 2701, 2710, 2824, 2826, 2828, 2829, 2833, 2838, 2912, 2966, 2977, 3040, 3048, 3050, 3057, 3184, 3188, 3192, 3194, 3336, 3338, 3340, 3350, 3440, 3444, 3592, 3594, 3596, 3612, 3760, 3764, 3768, 3770, 3880, 3882, 3954, 4048, 4052, 4066, 4240, 4244, 4248, 4250, 4258, 4450, 4452, 4466, 4576, 4584, 4586, 4784, 4788, 4792, 4794, 5008, 5012, 5016, 5018, 5026, 5168, 5172, 5392, 5396, 5400, 5402, 5420, 5648, 5652, 5656, 5658, 5666, 5676, 5824, 5932, 5954, 6080, 6096, 6100, 6114, 6368, 6376, 6384, 6388, 6672, 6676, 6680, 6700, 6880, 6888, 7184, 7188, 7192, 7224, 7520, 7528, 7536, 7540, 7760, 7764, 7908, 8096, 8104, 8132, 8480, 8488, 8496, 8500, 8516, 8900, 8904, 8932, 9152, 9168, 9172, 751, 871, 879, 1001, 1127, 1161, 1307, 1319, 1471, 1502, 1503, 1563, 1691, 1742, 1743, 1758, 1759, 1807, 1903, 1951, 1961, 1979, 2002, 2003, 2207, 2227, 2239, 2254, 2255, 2322, 2323, 2343, 2345, 2409, 2415, 2523, 2537, 2601, 2614, 2615, 2638, 2639, 2669, 2711, 2779, 2815, 2839, 2855, 2927, 2942, 2943, 2967, 2969, 2985, 3004, 3005, 3006, 3097, 3126, 3127, 3189, 3211, 3311, 3341, 3345, 3351, 3359, 3382, 3383, 3445, 3484, 3485, 3486, 3515, 3516, 3517, 3518, 3597, 3601, 3613, 3614, 3623, 3705, 3753, 3765, 3777, 3783, 3785, 3806, 3807, 3902, 3903, 3922, 3923, 3943, 3955, 3958, 3959, 3969, 4004, 4005, 4006, 4053, 4067, 4129, 4169, 4219, 4223, 4245, 4259, 4267, 4281, 4283, 4391, 4414, 4415, 4451, 4453, 4454, 4467, 4478, 4479, 4508, 4509, 4510, 4593, 4644, 4645, 4646, 4686, 4687, 4689, 4690, 4691, 4735, 4747, 4789, 4801, 4817, 4818, 4819, 4830, 4831, 4935, 4967, 4987, 5013, 5027, 5039, 5046, 5047, 5074, 5075, 5173, 5202, 5203, 5228, 5229, 5230, 5257, 5273, 5276, 5277, 5278, 5295, 5307, 5338, 5341, 5397, 5421, 5422, 5435, 5479, 5505, 5558, 5559, 5611, 5625, 5630, 5631, 5653, 5667, 5675, 5677, 5678, 5689, 5710, 5711, 5799, 5854, 5855, 5884, 5885, 5886, 5915, 5933, 5934, 5937, 5938, 5939, 5947, 5955, 5970, 5971, 6008, 6010, 6012, 6013, 6101, 6115, 6193, 6194, 6195, 6247, 6249, 6252, 6253, 6254, 6303, 6313, 6329, 6335, 6378, 6385, 6389, 6401, 6422, 6423, 6425, 6441, 6523, 6587, 6622, 6623, 6649, 6655, 6677, 6682, 6690, 6691, 6699, 6701, 6702, 6715, 6718, 6719, 6729, 6764, 6765, 6766, 6890, 6897, 6937, 6968, 6970, 6972, 6973, 7009, 7030, 7031, 7032, 7034, 7036, 7037, 7103, 7117, 7121, 7189, 7194, 7202, 7203, 7226, 7228, 7229, 7246, 7247, 7305, 7339, 7403, 7410, 7411, 7451, 7481, 7506, 7507, 7530, 7537, 7541, 7554, 7555, 7559, 7566, 7567, 7569, 7570, 7571, 7585, 7611, 7612, 7613, 7614, 7765, 7804, 7805, 7806, 7844, 7845, 7846, 7886, 7887, 7909, 7910, 7916, 7917, 7918, 7929, 7938, 7939, 7943, 7961, 7975, 8008, 8010, 8012, 8013, 8017, 8106, 8133, 8134, 8153, 8219, 8223, 8257, 8258, 8259, 8329, 8331, 8337, 8338, 8339, 8367, 8383, 8417, 8438, 8439, 8446, 8447, 8490, 8501, 8513, 8517, 8518, 8523, 8534, 8535, 8539, 8562, 8563, 8566, 8567, 8587, 8697, 8699, 8782, 8783, 8828, 8829, 8830, 8865, 8873, 8901, 8902, 8906, 8908, 8909, 8921, 8933, 8934, 8953, 8956, 8957, 8958, 8967, 8971, 8991, 9016, 9018, 9020, 9021, 9173, 9186, 9187, 9247, 9249, 9288, 9290, 9292, 9293, 9297, 9345, 9371, 9372, 9373, 9374, 9378, 9379, 9380, 9381, 9382, 9435, 9447, 9451, 9455, 9470, 9471, 9489, 9494, 9495, 9503, 9568, 9576, 9578, 9584, 9585, 9588, 9589, 9602, 9603, 9607, 9634, 9635, 9636, 9637, 9638, 9660, 9661, 9662, 9785, 9787, 9870, 9871, 9881, 9883, 9934, 9935, 9974, 9975, 9983] },
            53: { emr: 108, mstar: 106, distance: 13, latest: 6912, examples: [108, 216, 432, 864, 1728, 3456, 6912, 13824, 27648, 55296, 110592, 221184, 442368, 884736, 1769472, 3538944, 7077888, 14155776, 28311552, 56623104, 113246208, 226492416, 452984832, 905969664] },
            60: { emr: 121, mstar: 4615, distance: 75, latest: 9160, examples: [27, 31, 41, 47, 54, 55, 62, 71, 73, 82, 83, 94, 97, 107, 109, 110, 121, 124, 125, 129, 142, 145, 146, 147, 161, 164, 165, 166, 171, 188, 193, 194, 195, 214, 218, 220, 221, 231, 242, 248, 250, 257, 258, 259, 285, 290, 291, 292, 293, 294, 322, 327, 328, 330, 332, 342, 343, 345, 347, 376, 386, 387, 388, 389, 390, 391, 415, 428, 429, 436, 437, 440, 442, 457, 459, 462, 463, 484, 491, 496, 500, 514, 515, 516, 517, 518, 521, 523, 543, 553, 570, 580, 581, 582, 584, 586, 587, 588, 589, 609, 617, 619, 623, 644, 645, 654, 655, 656, 660, 661, 664, 684, 685, 686, 689, 690, 691, 694, 695, 697, 731, 737, 752, 763, 772, 773, 774, 775, 776, 778, 780, 781, 782, 783, 785, 811, 815, 823, 825, 830, 856, 858, 859, 872, 873, 874, 880, 881, 884, 885, 913, 914, 915, 918, 919, 921, 924, 925, 926, 929, 935, 967, 968, 982, 983, 992, 1000, 1017, 1028, 1029, 1030, 1032, 1033, 1034, 1035, 1036, 1037, 1041, 1042, 1043, 1046, 1047, 1081, 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9921, 9927, 9929, 9947, 9968, 9972, 9973, 9976, 9978] },
            79: { emr: 159, mstar: 4615, distance: 44, latest: 5088, examples: [159, 318, 636, 1272, 2544, 5088, 10176, 20352, 40704, 81408, 162816, 325632, 651264, 1302528, 2605056, 5210112, 10420224, 20840448, 41680896, 83361792, 166723584, 333447168, 666894336] },
            91: { emr: 183, mstar: 4615, distance: 75, latest: 5856, examples: [183, 366, 732, 1464, 2928, 5856, 11712, 23424, 46848, 93696, 187392, 374784, 749568, 1499136, 2998272, 5996544, 11993088, 23986176, 47972352, 95944704, 191889408, 383778816, 767557632] },
            121: { emr: 243,mstar: 4615, distance: 75, latest: 7776, examples: [243, 486, 972, 1944, 3888, 7776, 15552, 31104, 62208, 124416, 248832, 497664, 995328, 1990656, 3981312, 7962624, 15925248, 31850496, 63700992, 127401984, 254803968, 509607936, 1019215872] },
            125: { emr: 252, mstar: 889, distance: 44, latest: 8064, examples: [252, 504, 1008, 2016, 4032, 8064, 16128, 32256, 64512, 129024, 258048, 516096, 1032192, 2064384, 4128768, 8257536, 16515072, 33030144, 66060288, 132120576, 264241152, 528482304, 1056964608] },
            141: { emr: 284, mstar: 889, distance: 44, latest: 9088, examples: [189, 284, 568, 757, 1009, 1136, 1195, 1345, 1514, 1593, 1793, 2018, 2019, 2272, 2390, 2690, 3028, 3029, 3186, 3586, 4036, 4038, 4544, 4780, 5380, 6056, 6058, 6372, 7172, 8072, 8076, 9088, 1767, 2391, 2651, 2691, 2983, 3187, 3534, 3535, 3587, 3977, 4037, 4249, 4475, 4713, 4781, 4782, 5035, 5302, 5303, 5381, 5382, 5665, 5966, 5967, 6283, 6373, 6374, 6713, 7068, 7069, 7070, 7173, 7174, 7179, 7553, 7954, 7955, 8074, 8077, 8377, 8379, 8497, 8498, 8499, 8950, 8951, 9425, 9426, 9427, 9560, 9562, 9564, 9565, 10070, 10071, 10604, 10605, 10606, 10760, 10762, 10764, 10765, 10769] },
            166: { emr: 333, mstar: 889, distance: 44, latest: 5328, examples: [333, 666, 1332, 2664, 5328, 10656, 21312, 42624, 85248, 170496, 340992, 681984, 1363968, 2727936, 5455872, 10911744, 21823488, 43646976, 87293952, 174587904, 349175808, 698351616]},
            188: { emr: 378, mstar: 889, distance: 44, latest: 6048, examples: [378, 756, 1512, 3024, 6048, 12096, 24192, 48384, 96768, 193536, 387072, 774144, 1548288, 3096576, 6193152, 12386304, 24772608, 49545216, 99090432, 198180864, 396361728, 792723456] },
            205: { emr: 411, mstar: 2347, distance: 44, latest: 6576, examples: [411, 822, 1644, 3288, 6576, 13152, 26304, 52608, 105216, 210432, 420864, 841728, 1683456, 3366912, 6733824, 13467648, 26935296, 53870592, 107741184, 215482368, 430964736, 861929472] },
            243: { emr: 487, mstar: 4615, distance: 119, latest: 7792, examples: [487, 974, 1948, 2597, 3896, 5194, 7792, 1215, 1823, 2431, 2735, 3241, 3646, 3647, 4103, 4321, 4551, 4861, 4862, 5470, 5471, 5761, 6155, 6481, 6482, 6483, 6827, 6925, 7292, 7293, 7294, 7681, 8091, 8206, 8207, 8641, 8642, 8643, 9102, 9103, 9233, 9722, 9724, 9725] },
            250: { emr: 501, mstar: 889, distance: 44, latest: 8016, examples: [501, 1002, 2004, 4008, 8016, 16032, 32064, 64128, 128256, 256512, 513024, 1026048, 2052096, 4104192, 8208384, 16416768, 32833536, 65667072, 131334144, 262668288, 525336576, 1050673152] },
            324: { emr: 649, mstar: 4615, distance: 119, latest: 5192, examples: [649, 1298, 2596, 5192, 3461, 6922, 7291, 8203, 9229, 9721, 10384, 10937, 12305, 12961, 13844, 13845, 14383, 14582, 14583] },
            333: { emr: 667, mstar: 10843, distance: 75, latest: 21344, examples: [667, 1334, 2107, 2371, 2668, 2809, 3161, 3557, 3745, 4214, 4742, 4993, 5336, 5618, 6322, 6657, 7114, 7490, 8428, 9484, 9986, 10672, 11236, 12644, 13314, 14228, 14980, 16856, 18968, 19972, 21344, 4215, 5619, 5919, 6235, 6323, 7015, 7491, 8313, 8429, 8430, 8879, 9353, 9485, 9987, 9991, 10239, 10523, 11237, 11238, 11835, 11838, 11839, 12135, 12470, 12471, 12645, 12646, 13315, 13319, 13321, 14030, 14031, 14229, 14779, 14981, 14982, 14987, 15359, 15783, 15785, 15787, 16626, 16627, 16858, 16860, 16861, 17515, 17753, 17758, 17759, 17761, 18203, 18706, 18707, 18970, 19705, 19707, 19973, 19974, 19979, 19982, 19983, 20478, 20479, 21046, 21047, 21049, 22169, 22171, 22472, 22474, 22476, 22477, 22481, 23039, 23353, 23670, 23671, 23675, 23676, 23677, 23678, 23681, 24270, 24271, 24940, 24941, 24942, 24943, 25288, 25290, 25292, 25293] },
            432: { emr: 865, mstar: 4615, distance: 119, latest: 6920, examples: [865, 1730, 3460, 6920, 3075, 4613, 6150, 9226, 12300, 12301, 12783, 13840, 16401, 18452, 18453, 19175, 24600, 24602, 25566, 25567, 27303, 27680, 28763, 32359, 32802, 32803, 33627, 34089, 35911] },
            444: { emr: 889, mstar: 10843, distance: 75, latest: 18964, examples: [889, 1778, 3556, 4741, 6321, 7112, 9482, 12642, 14224, 18964, 12643, 14983, 16857, 18965, 19977, 22475, 24607, 25284, 25285, 25286, 28063, 28448, 29163, 29563, 29966, 29967, 32809, 33259, 33713, 33714, 33715, 35515, 36911, 37417, 37928, 37930, 39417, 39954, 39955] },
            487: { emr: 975, mstar: 4939, distance: 119, latest: 7800, examples: [975, 1950, 3900, 7800, 15600, 31200, 62400, 124800, 249600, 499200, 998400, 1996800, 3993600, 7987200, 15974400, 31948800, 63897600, 127795200, 255590400, 511180800, 1022361600] },
            576: { emr: 1153, mstar: 4615, distance: 119, latest: 9224, examples: [1153, 1537, 2049, 2306, 3074, 4098, 4612, 6148, 8196, 9224, 4099, 5465, 6149, 7675, 8197, 8198, 8635, 9715, 10233, 10929, 10930, 10931, 11513, 12296, 12298, 12953, 14573, 15350, 15351, 16392, 16394, 16396, 16397, 17270, 17271, 18448, 19430, 20466, 20467, 21858, 21859, 21860, 21861, 21862, 23026, 23027] },
            592: { emr: 1185, mstar: 10843, distance: 75, latest: 18960, examples: [1185, 2370, 4740, 9480, 18960,37920, 75840, 151680, 303360, 606720, 1213440, 2426880, 4853760, 9707520, 19415040, 38830080, 77660160, 155320320, 310640640, 621281280] },
            649: { emr: 1299, mstar: 4615, distance: 119, latest: 5196, examples: [1299, 2598, 5196, 10392, 20784, 41568, 83136, 166272, 332544, 665088, 1330176, 2660352, 5320704, 10641408, 21282816, 42565632, 85131264, 170262528, 340525056, 681050112] },
            667: { emr: 1335, mstar: 10843, distance: 75, latest: 21360, examples: [1335, 2670, 5340, 10680, 21360, 42720, 85440, 170880, 341760, 683520, 1367040, 2734080, 5468160, 10936320, 21872640, 43745280, 87490560, 174981120, 349962240, 699924480] },
            683: { emr: 1368, mstar: 3643, distance: 88, latest: 5472, examples: [1368, 2736, 5472, 10944, 21888, 43776, 87552, 175104, 350208, 700416, 1400832, 2801664, 5603328, 11206656, 22413312, 44826624, 89653248, 179306496, 358612992, 717225984] },
            865: { emr: 1731,  mstar: 4615, distance: 119, latest: 6924, examples: [1731, 3462, 6924, 13848, 27696, 55392, 110784, 221568, 443136, 886272, 1772544, 3545088, 7090176, 14180352, 28360704, 56721408, 113442816, 226885632, 453771264, 907542528 ] },
            889: { emr: 1779, mstar: 10843, distance: 75, latest: 14232, examples: [1779, 3558, 7116, 14232, 28464, 56928, 113856, 227712, 455424, 910848, 1821696, 3643392, 7286784, 14573568, 29147136, 58294272, 116588544 ] },
            1153: { emr: 2307, mstar: 5191, distance: 119, latest: 9228, examples: [2307, 4614, 9228, 18456, 36912, 73824, 147648, 295296, 590592, 1181184, 2362368, 4724736, 9449472, 18898944, 37797888, 75595776, 151191552, 302383104, 604766208] },
            1214: { emr: 2430, mstar: 13849, distance: 119, latest: 19440, examples: [2430, 4860, 9720, 19440, 38880, 77760, 155520, 311040, 622080, 1244160, 2488320, 4976640, 9953280, 19906560, 39813120, 79626240, 159252480, 318504960, 637009920] },
            1821: { emr: 3643, mstar: 9223, distance: 119, latest: 14572, examples: [3643, 7286, 14572, 19429, 25905, 29144, 38858, 51111, 51810, 51811, 58288, 69081, 72775, 76667, 77716, 77717, 86251, 92107, 97033, 102222, 102223, 103620, 103621, 103622, 109163, 115001, 116576, 122809, 122811, 129377, 136297, 138161, 138162, 138163, 145550, 145551] },
            2428: { emr: 4857, mstar: 9223, distance: 119, latest: 9714, examples: [4857, 9714, 19428, 38856, 77712, 155424, 310848, 621696, 1243392, 2486784, 4973568, 9947136, 19894272, 39788544, 79577088, 159154176, 318308352, 636616704] },
            3643: { emr: 7287, mstar: 24595, distance: 119, latest: 29148, examples: [7287, 14574, 29148, 58296, 116592, 233184, 466368, 932736, 1865472, 3730944, 7461888, 14923776, 29847552, 59695104, 119390208, 238780416, 477560832, 955121664] }
        };

        /**
         * Decomposes a Collatz sequence into algebraic tuples (p, m).
         * Each element c_i is written as c_i = 2·m_i + p_i, where p_i ∈ {1, 2}
         * is the parity component and m_i is the scale parameter (Definition 2.6).
         * Valid tuples satisfy conditions V1–V4 (Lemma 2.12).
         * This implementation uses the fixed transform φ_1 (q = 1).
         * Only the m component of each tuple is used downstream (see getMSequence).
         * 
         * @param {number} n - Starting number for the Collatz sequence
         * @returns {Array<{p: number, m: number}>} Array of tuples from the algebraic decomposition
         * @throws {Error} If the Collatz sequence exceeds 10,000 iterations
         * 
         * @example
         * // For n = 7: Collatz sequence [7, 22, 11, 34, 17, ...]
         * // Returns [{p: 1, m: 3}, {p: 2, m: 10}, {p: 1, m: 5}, ...]
         */
        function collatzDecompose(n) {
            const collatzSeq = [];
            let current = n;
            
            // Generate standard Collatz sequence
            while (current !== 1) {
                collatzSeq.push(current);
                current = (current % 2 === 0) ? current / 2 : 3 * current + 1;
                if (collatzSeq.length > 10000) {
                    throw new Error(`n = ${n} exceeds the safety limit of 10,000 iterations. Try a smaller value.`);
                }
            }
            collatzSeq.push(1, 4, 2, 1)

            // q = 1 fixes the transform φ_1; declared once outside the loop.
            const q = 1;
            const tuples = [];
            // Find (p, m) for each consecutive pair
            for (let i = 0; i < collatzSeq.length - 1; i++) {
                const ci = collatzSeq[i];
                const ci_plus_1 = collatzSeq[i + 1];
                
                let foundP = null, foundM = null;
                for (let p = 1; p <= 2 * q; p++) {
                    // p must have same parity as c_i
                    if ((p % 2) !== (ci % 2)) continue;
                    // Check if (c_i - p) is divisible by 2q
                    if ((ci - p) % (2 * q) !== 0) continue;
                    const m = (ci - p) / (2 * q);
                    
                    if (m < 0) continue;
                    
                    // Verify the relation with c_{i+1}
                    const f_p = (p % 2 === 0) ? p / 2 : 3 * p + 1;
                    const expected = (p % 2 === 0) ? q * m + f_p : 6 * q * m + f_p;
                    
                    if (expected === ci_plus_1) {
                        foundP = p;
                        foundM = m;
                        break;
                    }
                }
                
                if (foundP !== null) {
                    tuples.push({ p: foundP, m: foundM });
                }
            }
            
            return tuples;
        }

       /**
         * Extracts the m-sequence from the algebraic decomposition.
         * The m-sequence {m_i} is the parameter sequence fundamental for 
         * classification into S(m_r) classes (Definition 6.1).
         * 
         * @param {number} n - The number to analyze
         * @returns {number[]} Array of m values from the decomposition
         * 
         * @example
         * // For n = 7
         * // Returns [3, 10, 5, 16, 8, 25, 12, 6, 3, 10, 5, ...]
         */
        function getMSequence(n) {
            const tuples = collatzDecompose(n);
            return tuples.map(t => t.m);
        }

        /**
         * Finds the first repeated value in the m-sequence.
         * m_r is the value whose second occurrence appears earliest in the
         * sequence, not necessarily the first value to appear chronologically
         * (Definition 6.3). By Theorems 5.5, 5.6 and 5.7, every sequence must
         * contain at least one repeated parameter, so m_r is always well-defined.
         * When the sequence reaches the trivial cycle without a prior repetition,
         * m_r = 0, placing n in class S(0) (Lemma 7.5).
         * 
         * @param {number[]} sequence - The m-sequence to analyze
         * @returns {number} The first repeated value m_r; 0 if the only
         *   repetition is the trivial m = 0 of the cycle (4, 2, 1)
         * 
         * @example
         * // For n = 27: m=30 appears at positions 5 and 93, m=60 at positions 17 
         * // and 92. Although m=30 appears first, m=60 closes its pair earlier.
         * // Returns 60 (m_r = 60, per Example 6.2)
         */
        function findFirstRepeated(sequence) {
            const seen = new Set();
            for (let m of sequence) {
                if (seen.has(m)) {
                    return m;
                }
                seen.add(m);
            }
            // The loop never reaches here in practice: collatzDecompose always appends
            // [..., 1, 4, 2, 1] to the sequence, producing m = 0 twice at the end.
            // For n ∈ S(0) that carry no earlier repetition, the loop returns 0 when
            // it encounters that second m = 0 — correctly identifying m_r = 0 (Lemma 7.5).
            // For all other n, an earlier repetition is found and returned before this point.
            return 0; // m_r = 0 → S(0)
        }

        /**
         * Finds all positions where a specific value appears in the sequence.
         * Used to locate where m_r occurs, which is required for taxonomy
         * classification (Definition 7.3).
         * 
         * @param {number[]} sequence - The sequence to search
         * @param {number} value - The value to locate
         * @returns {number[]} Array of 0-based indices where the value appears
         * 
         * @example
         * // For sequence [3, 10, 5, 16, 8, 3, 12, ...]
         * // findPositions(sequence, 3) returns [0, 5]
         */
        function findPositions(sequence, value) {
            const positions = [];
            for (let i = 0; i < sequence.length; i++) {
                if (sequence[i] === value) {
                    positions.push(i);
                }
            }
            return positions;
        }

        /**
         * Main processing function that analyzes a number and displays results.
         * Orchestrates the complete analysis pipeline:
         * 1. Validates input: rejects empty strings, floats, non-numeric input,
         *    scientific notation, and any value that is not a strict positive integer (n ≥ 1)
         * 2. Computes the m-sequence
         * 3. Identifies m_r (first repeated parameter)
         * 4. Looks up m* and distance from MR_MSTAR_DICT
         * 5. Calculates M* (global maximum of the sequence)
         * 6. Delegates to displayResults for classification and visualization
         * 
         * @returns {void}
         */
        function processNumber() {
            const errorDiv = document.getElementById('error');
            errorDiv.style.display = 'none';
            errorDiv.innerHTML = '';

            const input = document.getElementById('numberInput');
            const raw = input.value.trim();
            const n = Number(raw);
            if (!raw || !Number.isInteger(n) || n < 1) {
                errorDiv.innerHTML = '<div class="error-message">Please enter a valid positive integer for n</div>';
                errorDiv.style.display = 'block';
                return;
            }

            const content = document.getElementById('content');
            const infoGrid = document.getElementById('infoGrid');
            
            infoGrid.innerHTML = '<div style="grid-column: 1 / -1; text-align: center; padding: 20px;"><div class="loading">Processing</div></div>';
            content.classList.add('active');

            // Async processing to keep UI responsive
            setTimeout(() => {
                try {
                    const mSequence = getMSequence(n);
                    const mr = findFirstRepeated(mSequence);
                    if (!Object.prototype.hasOwnProperty.call(MR_MSTAR_DICT, mr)) {
                        throw new Error(`n = ${n} produced an unrecognized first repeated parameter m_r = ${mr}. This value is not present in the known 42 classes (n < 2^40).`);
                    }
                    const mrPositions = findPositions(mSequence, mr);
                    const mstar = MR_MSTAR_DICT[mr].mstar;
                    const Mstar = mSequence.reduce((a, b) => a > b ? a : b, 0);
                    const distance = MR_MSTAR_DICT[mr].distance;
                    const examples = MR_MSTAR_DICT[mr].examples;
                    
                    displayResults(n, mr, mstar, Mstar, distance, mSequence, mrPositions, examples);
                } catch (error) {
                    infoGrid.innerHTML = `<div style="grid-column: 1 / -1; text-align: center; color: #ff6b6b; padding: 20px;">Error: ${error.message}</div>`;
                }
            }, 100);
        }

        /**
         * Displays comprehensive results including classification and taxonomy.
         * 
         * @param {number} userN - The analyzed number
         * @param {number} mr - First repeated parameter value (determines class S(m_r))
         * @param {number|null} mstar - Local maximum m* (invariant for the class, Definition 6.9);
         *   null for S(0) where m* is undefined (the two consecutive m=0 values of the
         *   trivial cycle 2→1→4 have nothing between them)
         * @param {number} Mstar - Global maximum M* of this specific sequence
         * @param {number} distance - d(m_r), the invariant distance metric for the class (Theorem 6.7)
         * @param {number[]} userSequence - Complete m-sequence for userN
         * @param {number[]} userMrPositions - 0-based positions where m_r occurs in userSequence
         * @param {number[]} examples - Other known members of S(m_r) from MR_MSTAR_DICT
         * 
         * @description
         * Entry point e(m_r) is read from MR_MSTAR_DICT[m_r].emr (Definition 6.10).
         * Special case: e(0) = 1, consistent with the trivial cycle (4, 2, 1).
         * 
         * Regularity (Definition 8.1):
         *   n is regular if n = e(m_r)·2^k for some k ≥ 0, i.e. n/e(m_r) is a
         *   positive integer power of 2. Irregular otherwise.
         * 
         * Taxonomy (Definition 7.3):
         *   All elements of S(0) are Type A (Lemma 7.5).
         *   Type A: M* > m* and M* occurs before the first m_r.
         *   Type B: M* = m* and M* occurs between the first and second m_r.
         *   Type C: M* > m* and M* occurs after the second m_r.
         * 
         * Each sequence object in allSequences stores type and isRegular to avoid
         * redundant recalculation during rendering.
         * 
         * @returns {void}
         */
        function displayResults(userN, mr, mstar, Mstar, distance, userSequence, userMrPositions, examples) {

            document.getElementById('classificationTitle').textContent = `Classification of n = ${userN}`;
            document.getElementById('classLabel').textContent = mr;
            
            const infoGrid = document.getElementById('infoGrid');

            const userFirstMrPos = userMrPositions[0];
            const userSecondMrPos = userMrPositions[1];
            const userMstarPos = userSequence.indexOf(Mstar);

            const latest = MR_MSTAR_DICT[mr]?.latest;

            // Entry point e(m_r) from Definition 6.10, precomputed in MR_MSTAR_DICT
            const entryPoint = MR_MSTAR_DICT[mr].emr;

            // Regularity check (Definition 8.1): n is regular if n = e(m_r)·2^k
            const ratio = userN / entryPoint;
            const isRegular = ratio >= 1 && Number.isInteger(ratio) && (ratio & (ratio - 1)) === 0;

            // Taxonomy (Definition 7.3); S(0) is always Type A (Lemma 7.5)
            let userType;
            if (mr === 0) {
                userType = 'A';
            } else if (userMstarPos < userFirstMrPos && Mstar > mstar) {
                userType = 'A';
            } else if (userMstarPos >= userFirstMrPos && userMstarPos <= userSecondMrPos && Mstar === mstar) {
                userType = 'B';
            } else {
                userType = 'C';
            }

            // Display classification metrics
            infoGrid.innerHTML = `
            <div class="info-card">
                <div class="info-label">Class</div>
                <div class="info-value">S(${mr})</div>
                </div>
                <div class="info-card">
                    <div class="info-label">Taxonomy</div>
                    <div class="info-value">Type ${userType}</div>
                </div>
                <div class="info-card">
                    <div class="info-label">m<sub>r</sub> (First Repetition of m)</div>
                    <div class="info-value">${mr}</div>
                </div>
                <div class="info-card">
                    <div class="info-label">m* (Local Max)</div>
                    <div class="info-value">${mstar !== null ? mstar : '-'}</div>
                </div>
                <div class="info-card">
                    <div class="info-label">M* (Global Max)</div>
                    <div class="info-value">${Mstar}</div>
                </div>
                <div class="info-card">
                    <div class="info-label">Distance d(m<sub>r</sub>)</div>
                    <div class="info-value">${distance}</div>
                </div>
                <div class="info-card">
                    <div class="info-label">Regularity</div>
                    <div class="info-value">${isRegular ? 'R' : 'I'}</div>
                </div>
            `;

            // Prepare all sequences for defragmentation view.
            // type and isRegular are computed once here per sequence to avoid
            // redundant recalculation during rendering.
            const allSequences = [];
            
            allSequences.push({
                n: userN,
                mSequence: userSequence,
                mrPositions: userMrPositions,
                isUser: true,
                Mstar: Mstar,
                isRegular: isRegular,
                type: userType
            });
            
            // Add example sequences from the same class
            examples.forEach(n => {
                if (n !== userN) {
                    const seq = getMSequence(n);
                    const mrPos = findPositions(seq, mr);
                    const seqMstar = seq.reduce((a, b) => a > b ? a : b, 0);

                    // Regularity check for this example (Definition 8.1)
                    const seqRatio = n / entryPoint;
                    const seqIsRegular = seqRatio >= 1 && Number.isInteger(seqRatio) && (seqRatio & (seqRatio - 1)) === 0;

                    // Taxonomy for this example (Definition 7.3)
                    const seqFirstMrPos = mrPos[0];
                    const seqSecondMrPos = mrPos[1];
                    const seqMstarPos = seq.indexOf(seqMstar);

                    let seqType;
                    if (mr === 0) {
                        seqType = 'A';
                    } else if (seqMstarPos < seqFirstMrPos && seqMstar > mstar) {
                        seqType = 'A';
                    } else if (seqMstarPos >= seqFirstMrPos && seqMstarPos <= seqSecondMrPos && seqMstar === mstar) {
                        seqType = 'B';
                    } else {
                        seqType = 'C';
                    }

                    allSequences.push({
                        n: n,
                        mSequence: seq,
                        mrPositions: mrPos,
                        isUser: false,
                        Mstar: seqMstar,
                        isRegular: seqIsRegular,
                        type: seqType
                    });
                }
            });
            
            // Sort by n value
            allSequences.sort((a, b) => a.n - b.n);

            // Render defragmentation visualization
            const container = document.getElementById('defragContainer');
            container.innerHTML = '';
            allSequences.forEach(seq => {
                renderDefragRow(container, seq, mr, mstar, seq.type, latest, seq.isRegular);
            });
        }

        /**
         * Renders a single row in the defragmentation view.
         * Each cell represents one m value in the sequence, color-coded by role.
         * 
         * @param {HTMLElement} container - Parent container for the visualization
         * @param {Object} seqData - Data for this sequence
         * @param {number} seqData.n - Starting value
         * @param {number[]} seqData.mSequence - The m-sequence (m_i values)
         * @param {number[]} seqData.mrPositions - 0-based positions of m_r occurrences
         * @param {boolean} seqData.isUser - Whether this is the user-provided sequence
         * @param {number} seqData.Mstar - Global maximum M* for this sequence
         * @param {number} mr - The m_r value for this class
         * @param {number|null} mstar - The invariant local maximum m* for this class; null for S(0)
         * @param {string} type - Taxonomy type for this sequence: 'A', 'B', or 'C'
         * @param {number} latest - The largest known example value in this class;
         *   not displayed for S(0) where the class is structurally obvious and unbounded
         * @param {boolean} isRegular - Whether this sequence is regular (Definition 8.1)
         * 
         * @description
         * Each row displays badges (YOU / LATEST), the n value with taxonomy type and
         * regularity label (R/I), followed by the color-coded m-sequence bar. The bar is
         * rendered right-to-left over the m-sequence array: the leftmost cell corresponds
         * to the last m value (m = 0, terminal), so the sequence reads chronologically
         * from right to left with the trivial cycle anchor on the left.
         * 
         * Color coding:
         *   Orange (#f39c12): m = 0 (trivial cycle reached)
         *   Red (#e74c3c):    m_r occurrences (pseudocycle markers)
         *   Green (#2ecc71):  m* (local maximum of pseudocycle)
         *   Blue (#3498db):   M* (global maximum of sequence)
         *   Purple (#8e44ad): m* = M* (Type B: global max coincides with local max)
         *   Gradient:         position is both m_r and m*
         *   Gray (#555):      all other intranscendent values
         * 
         * @returns {void}
         */
        function renderDefragRow(container, seqData, mr, mstar, type, latest, isRegular) {
            const { n, mSequence, mrPositions, isUser, Mstar } = seqData;
            
            const row = document.createElement('div');
            row.className = 'defrag-row';

            const label = document.createElement('div');
            label.className = 'n-label';

            // Add badges for special sequences
            const badgesArray = [];
            if (isUser) {
                badgesArray.push('<span class="you-badge">YOU</span>');
            }
            // S(0) has no meaningful "latest" — the class is structurally obvious and unbounded
            if (n === latest && mr !== 0) {
                badgesArray.push(`<span class="latest-badge">LATEST ${mr}<sub>${type}</sub></span>`);
            }

            const badgesHTML = badgesArray.join('');

            label.innerHTML = `<div style="text-align: left; display: flex; gap: 8px;">${badgesHTML}</div><div style="text-align: right;">n=${n} &nbsp;<strong>${type} ${isRegular ? 'R' : 'I'}</strong></div>`;

            const bar = document.createElement('div');
            bar.className = 'sequence-bar';
           
            // Render sequence right-to-left: leftmost cell = last m value (m = 0 terminal)
            for (let i = mSequence.length - 1; i >= 0; i--) {
                const m = mSequence[i];
                const cell = document.createElement('div');
                cell.className = 'cell';
                
                // Apply color coding based on value significance
                if (m === 0) {
                    cell.classList.add('zero');
                } else if (mrPositions.includes(i) && m === mstar) {
                    cell.classList.add('mr-and-mstar');
                } else if (mrPositions.includes(i)) {
                    cell.classList.add('mr');
                } else if (m === mstar && m === Mstar) {
                    cell.classList.add('mstar-and-Mstar');
                } else if (m === mstar) {
                    cell.classList.add('mstar');
                } else if (m === Mstar) {
                    cell.classList.add('Mstar');
                } else {
                    cell.classList.add('normal');
                }
                
                // Add interactive tooltips
                cell.addEventListener('mouseenter', (e) => showTooltip(e, n, i, m, mr, mstar, Mstar, mrPositions, mSequence));
                cell.addEventListener('mouseleave', hideTooltip);
                
                bar.appendChild(cell);
            }
            
            row.appendChild(label);
            row.appendChild(bar);
            container.appendChild(row);
        }

        /**
         * Shows a tooltip with detailed information about a cell on hover.
         * Positions are displayed as 1-based for readability, though internally
         * all indices are 0-based.
         * 
         * @param {MouseEvent} e - The mouse event that triggered the tooltip
         * @param {number} n - Starting value of the sequence
         * @param {number} position - 0-based index of the cell in the m-sequence
         * @param {number} m - Value of m at this position
         * @param {number} mr - The m_r value for this class
         * @param {number|null} mstar - The invariant local maximum m* (Definition 6.9); null for S(0)
         * @param {number} Mstar - Global maximum M* for this sequence
         * @param {number[]} mrPositions - 0-based positions where m_r occurs
         * @param {number[]} mSequence - Complete m-sequence
         * 
         * @returns {void}
         */
        function showTooltip(e, n, position, m, mr, mstar, Mstar, mrPositions, mSequence) {
            const tooltip = document.getElementById('tooltip');
            const mstarPos = mSequence.indexOf(mstar);
            const MstarPos = mSequence.indexOf(Mstar);
            
            let html = '';
            html += `<div class="tooltip-row"><span class="tooltip-label">n:</span> <span class="tooltip-value">${n}</span></div>`;
            html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
            html += `<div class="tooltip-row"><span class="tooltip-label">Transform length:</span> <span class="tooltip-value">${mSequence.length}</span></div>`;
            html += `<div class="tooltip-row"><span class="tooltip-label">m<sub>${position+1}</sub>:</span> <span class="tooltip-value">${m}</span></div>`;
            html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
            html += `<div class="tooltip-row"><span class="tooltip-label">m*${mstar !== null ? ` (i=${mstarPos+1})` : ''}:</span> <span class="tooltip-value">${mstar !== null ? mstar : '-'}</span></div>`;
            html += `<div class="tooltip-row"><span class="tooltip-label">M* (i=${MstarPos+1}):</span> <span class="tooltip-value">${Mstar}</span></div>`;


            if (m === 0) {
                html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
                html += `<div class="tooltip-row"><span style="color: #f39c12; font-weight: bold;">SPECIAL CASE</span> Terminal</div>`;
                html += `<div class="tooltip-row"><span class="tooltip-label">m<sub>${position + 1}</sub> = 0</span></div>`;
            } else if (m === mstar && m === Mstar) {
                html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
                html += `<div class="tooltip-row"><span style="color: #8e44ad; font-weight: bold;">SPECIAL CASE</span> m* = M*</div>`;
                html += `<div class="tooltip-row"><span></span> m* (local max)</div>`;
                html += `<div class="tooltip-row"><span></span> M* (global max)</div>`;
            } else if (m === mstar) {
                html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
                html += `<div class="tooltip-row"><span style="color: #2ecc71; font-weight: bold;">SPECIAL CASE</span> m* (local max)</div>`;
            } else if (m === Mstar) {
                html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
                html += `<div class="tooltip-row"><span style="color: #3498db; font-weight: bold;">SPECIAL CASE</span> M* (global max)</div>`;
            }

            if (mrPositions.includes(position) && m !== 0) {
                const occurrence = mrPositions.indexOf(position) + 1;
                html += '<hr style="border: 1px solid #444; margin: 8px 0;">';
                html += `<div class="tooltip-row"><span style="color: #e74c3c; font-weight: bold;">SPECIAL CASE</span> m_r (occurrence ${occurrence})</div>`;
            }

            tooltip.innerHTML = html;
            tooltip.style.left = (e.clientX + 10) + 'px';
            tooltip.style.bottom = (window.innerHeight - e.clientY + 10) + 'px';
            tooltip.style.top = 'auto';
            tooltip.classList.add('active');
        }

        /**
         * Hides the tooltip by removing the active class.
         * 
         * @returns {void}
         */
        function hideTooltip() {
            document.getElementById('tooltip').classList.remove('active');
        }

        /**
         * Initializes the application on DOMContentLoaded.
         * Clears the input field and registers the Enter key listener.
         * 
         * @listens DOMContentLoaded
         * @returns {void}
         */
        document.addEventListener('DOMContentLoaded', function() {

            document.getElementById('numberInput').value = '';

            document.getElementById('numberInput').addEventListener('keypress', (e) => {
                if (e.key === 'Enter') {
                    processNumber();
                }
            });
        });

    </script>
</body>
</html>