0fb7d98a56aa961b03afc081c4017273 algorithms_a2026v19n1art7.pdf dfb5ed431c6644c9611640cb248fb46f156cb09e algorithms_a2026v19n1art7.pdf 102887d09a5e1e40467f7ce1bdc95361a9184be4eed1b8fe5513a551cb292123 algorithms_a2026v19n1art7.pdf Title: RAMA: A Meta-Algorithmic Framework for Ramanujan-Style Heuristic Discovery Using Large Language Models Subject: This work introduces RAMA (Recursive Aesthetic Modular Approximation), a metaheuristic framework that models a restricted form of mathematical intuition inspired by the notebooks of Srinivasa Ramanujan. While Ramanujan often produced deep results without formal proofs, the heuristic processes guiding such discoveries remain poorly understood. RAMA treats large language models (LLMs) as proposal mechanisms within an iterative search that generates, evaluates, and refines candidate conjectures under an explicit energy functional balancing fit, description length, and aesthetic structure. A small set of Ramanujan-inspired heuristics—modular symmetries, integrality cues, aesthetic compression, and near-invariance detection—is formalized as micro-operators acting on symbolic states. We instantiate RAMA in two domains: (i) inverse engineering eta-quotients from partial q-series data and (ii) designing cyclotomic fingerprints with shadow gadgets for quantum circuits. In both settings, RAMA recovers compact structures from limited information and improves separation from classical baselines, illustrating how intuitive heuristic patterns can be rendered as explicit, reproducible computational procedures. Keywords: heuristic discovery; Ramanujan; mock modular forms; meta-algorithm; large language models; aesthetic compression; modular symmetries Author: Jordi Vallverdú Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25; modified using OpenPDF 1.4.2 CreationDate: Tue Dec 30 02:14:56 2025 CET ModDate: Tue Dec 30 02:19:46 2025 CET Custom Metadata: yes Metadata Stream: no Tagged: no UserProperties: no Suspects: no Form: none JavaScript: no Pages: 18 Encrypted: no Page size: 595.276 x 841.89 pts (A4) Page rot: 0 File size: 486480 bytes Optimized: no PDF version: 1.7 name type encoding emb sub uni object ID ------------------------------------ ----------------- ---------------- --- --- --- --------- JRINBR+URWPalladioL-Roma Type 1 Custom yes yes yes 158 0 LCRHSP+URWPalladioL-Ital Type 1 Custom yes yes yes 159 0 WJCSVL+URWPalladioL-Bold Type 1 Custom yes yes yes 160 0 WCYBKT+TimesNewRomanPS-BoldItalicMT Type 1C WinAnsi yes yes no 148 0 NYLFJB+PazoMath-Italic Type 1 Builtin yes yes yes 194 0 CDGXOS+CMSY10 Type 1 Builtin yes yes yes 195 0 HMZIWF+SFRM0900 Type 1 Custom yes yes yes 196 0 SOSTRQ+CMR10 Type 1 Builtin yes yes yes 197 0 ARIMYE+URWPalladioL-BoldItal Type 1 Custom yes yes yes 255 0 GJRTEW+CMMI10 Type 1 Builtin yes yes yes 256 0 IKCRZM+CMEX10 Type 1 Builtin yes yes yes 295 0 HVVMYW+MSBM10 Type 1 Builtin yes yes yes 296 0 MLQDBH+PazoMath Type 1 Builtin yes yes yes 297 0 EWAJSR+SFTT1000 Type 1 Custom yes yes yes 326 0 WOTBNN+CMSS10 Type 1 Builtin yes yes yes 358 0 BMQQDV+DejaVuSans Type 3 Custom yes yes no 379 0 BMQQDV+DejaVuSans Type 3 Custom yes yes no 385 0 BMQQDV+DejaVuSans Type 3 Custom yes yes no 483 0 AAAAAA+DejaVuSans TrueType WinAnsi yes yes yes 475 0 Helvetica Type 1 WinAnsi no no no 476 0 BMQQDV+DejaVuSans Type 3 Custom yes yes no 480 0 SFDDRB+SFTT0900 Type 1 Custom yes yes yes 614 0 Jhove (Rel. 1.28.0, 2023-05-18) Date: 2026-02-05 04:17:39 CET RepresentationInformation: algorithms_a2026v19n1art7.pdf ReportingModule: PDF-hul, Rel. 1.12.4 (2023-03-16) LastModified: 2026-02-04 14:29:24 CET Size: 486480 Format: PDF Version: 1.7 Status: Well-Formed and valid SignatureMatches: PDF-hul MIMEtype: application/pdf PDFMetadata: Objects: 701 FreeObjects: 1 IncrementalUpdates: 0 DocumentCatalog: ViewerPreferences: HideToolbar: false HideMenubar: false HideWindowUI: false FitWindow: true CenterWindow: false DisplayDocTitle: false NonFullScreenPageMode: UseNone Direction: L2R ViewArea: CropBox ViewClip: CropBox PrintArea: CropBox PageClip: CropBox PageLayout: SinglePage PageMode: UseNone Outlines: Item: Title: Introduction Item: Title: Historical and Cognitive Foundations Item: Title: The RAMA Meta-Algorithm: A Metacognitive Framework Children: Item: Title: Motivation and Problem Statement Item: Title: Metacognitive Architecture Item: Title: Operational Description Item: Title: Philosophical Interpretation Item: Title: Definition (Formal Summary) Item: Title: Scope and Philosophical Position Item: Title: Meta-Algorithmic Rationale Item: Title: Meta-Algorithmic Framework and Final Output Children: Item: Title: Energy Functional: Track-Specific Forms Item: Title: Objects, Characters, and Cyclotomic Moments Item: Title: Mock Symmetries and Shadow Corrections Item: Title: Algorithmic Skeleton (Meta-Algorithm) Item: Title: Pseudocode Realization of RAMA Item: Title: Instantiation for Quantum Circuits: Modular Fingerprints Item: Title: What the Meta-algorithm Outputs in Practice Item: Title: Prompt Chains as Algorithms: A Precise Equivalence Item: Title: Empirical Validation Protocols Item: Title: RAMA Compliance and Interpretative Justification Item: Title: Discussion Children: Item: Title: Potential Use Cases Beyond the Present Experiments Item: Title: Comparison with Computer Algebra Systems Item: Title: Conclusions Item: Title: References Info: Title: RAMA: A Meta-Algorithmic Framework for Ramanujan-Style Heuristic Discovery Using Large Language Models Author: Jordi Vallverdú Subject: This work introduces RAMA (Recursive Aesthetic Modular Approximation), a metaheuristic framework that models a restricted form of mathematical intuition inspired by the notebooks of Srinivasa Ramanujan. While Ramanujan often produced deep results without formal proofs, the heuristic processes guiding such discoveries remain poorly understood. RAMA treats large language models (LLMs) as proposal mechanisms within an iterative search that generates, evaluates, and refines candidate conjectures under an explicit energy functional balancing fit, description length, and aesthetic structure. A small set of Ramanujan-inspired heuristics—modular symmetries, integrality cues, aesthetic compression, and near-invariance detection—is formalized as micro-operators acting on symbolic states. We instantiate RAMA in two domains: (i) inverse engineering eta-quotients from partial q-series data and (ii) designing cyclotomic fingerprints with shadow gadgets for quantum circuits. In both settings, RAMA recovers compact structures from limited information and improves separation from classical baselines, illustrating how intuitive heuristic patterns can be rendered as explicit, reproducible computational procedures. Keywords: heuristic discovery; Ramanujan; mock modular forms; meta-algorithm; large language models; aesthetic compression; modular symmetries Creator: LaTeX with hyperref Producer: pdfTeX-1.40.25; modified using OpenPDF 1.4.2 CreationDate: Tue Dec 30 02:14:56 CET 2025 ModDate: Tue Dec 30 02:19:46 CET 2025 ID: 0xde55ce33c64618c43c397475ad13bc2d, 0x0396e3e6d07cf69334b089614d3b24f2 Filters: FilterPipeline: FlateDecode Fonts: Type1: Font: BaseFont: WJCSVL+URWPalladioL-Bold FontSubset: true FirstChar: 2 LastChar: 250 FontDescriptor: FontName: WJCSVL+URWPalladioL-Bold Flags: Symbolic FontBBox: -152, -301, 1000, 935 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: GJRTEW+CMMI10 FontSubset: true FirstChar: 60 LastChar: 96 FontDescriptor: FontName: GJRTEW+CMMI10 Flags: Symbolic FontBBox: -32, -250, 1048, 750 FontFile: true ToUnicode: true Font: BaseFont: NYLFJB+PazoMath-Italic FontSubset: true FirstChar: 97 LastChar: 113 FontDescriptor: FontName: NYLFJB+PazoMath-Italic Flags: Symbolic FontBBox: -70, -277, 902, 733 FontFile: true ToUnicode: true Font: BaseFont: CDGXOS+CMSY10 FontSubset: true FirstChar: 0 LastChar: 106 FontDescriptor: FontName: CDGXOS+CMSY10 Flags: Symbolic FontBBox: -29, -960, 1116, 775 FontFile: true ToUnicode: true Font: BaseFont: HMZIWF+SFRM0900 FontSubset: true FirstChar: 25 LastChar: 25 FontDescriptor: FontName: HMZIWF+SFRM0900 Flags: Symbolic FontBBox: -196, -322, 1502, 937 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: SOSTRQ+CMR10 FontSubset: true FirstChar: 40 LastChar: 93 FontDescriptor: FontName: SOSTRQ+CMR10 Flags: Symbolic FontBBox: -40, -250, 1009, 750 FontFile: true ToUnicode: true Font: BaseFont: EWAJSR+SFTT1000 FontSubset: true FirstChar: 40 LastChar: 121 FontDescriptor: FontName: EWAJSR+SFTT1000 Flags: Symbolic FontBBox: -208, -360, 1374, 838 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: WOTBNN+CMSS10 FontSubset: true FirstChar: 66 LastChar: 80 FontDescriptor: FontName: WOTBNN+CMSS10 Flags: Symbolic FontBBox: -61, -250, 999, 759 FontFile: true ToUnicode: true Font: BaseFont: SFDDRB+SFTT0900 FontSubset: true FirstChar: 46 LastChar: 121 FontDescriptor: FontName: SFDDRB+SFTT0900 Flags: Symbolic FontBBox: -210, -359, 1376, 844 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: IKCRZM+CMEX10 FontSubset: true FirstChar: 0 LastChar: 12 FontDescriptor: FontName: IKCRZM+CMEX10 Flags: Symbolic FontBBox: -24, -2960, 1454, 772 FontFile: true ToUnicode: true Font: BaseFont: HVVMYW+MSBM10 FontSubset: true FirstChar: 67 LastChar: 90 FontDescriptor: FontName: HVVMYW+MSBM10 Flags: Symbolic FontBBox: -55, -420, 2343, 920 FontFile: true ToUnicode: true Font: BaseFont: MLQDBH+PazoMath FontSubset: true FirstChar: 68 LastChar: 229 FontDescriptor: FontName: MLQDBH+PazoMath Flags: Symbolic FontBBox: -40, -283, 878, 946 FontFile: true ToUnicode: true Font: BaseFont: JRINBR+URWPalladioL-Roma FontSubset: true FirstChar: 2 LastChar: 242 FontDescriptor: FontName: JRINBR+URWPalladioL-Roma Flags: Symbolic FontBBox: -166, -283, 1021, 943 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: LCRHSP+URWPalladioL-Ital FontSubset: true FirstChar: 2 LastChar: 233 FontDescriptor: FontName: LCRHSP+URWPalladioL-Ital Flags: Symbolic FontBBox: -170, -305, 1010, 941 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Font: BaseFont: ARIMYE+URWPalladioL-BoldItal FontSubset: true FirstChar: 45 LastChar: 116 FontDescriptor: FontName: ARIMYE+URWPalladioL-BoldItal Flags: Symbolic FontBBox: -170, -300, 1073, 935 FontFile: true EncodingDictionary: Differences: true ToUnicode: true Pages: Page: Sequence: 1 Annotations: Annotation: Subtype: Link Rect: 34, 216, 124, 235 Annotation: Subtype: Link Rect: 89, 77, 152, 84 Annotation: Subtype: Link Rect: 34, 64, 120, 72 Annotation: Subtype: Link Rect: 34, 773, 154, 809 Annotation: Subtype: Link Rect: 515, 773, 560, 809 Annotation: Subtype: Link Rect: 107, 664, 118, 677 Annotation: Subtype: Link Rect: 413, 260, 420, 269 ActionDest: 120 Annotation: Subtype: Link Rect: 402, 218, 409, 227 ActionDest: 120 Annotation: Subtype: Link Rect: 412, 218, 419, 227 ActionDest: 120 Annotation: Subtype: Link Rect: 462, 204, 469, 213 ActionDest: 120 Annotation: Subtype: Link Rect: 472, 204, 484, 213 ActionDest: 120 Annotation: Subtype: Link Rect: 423, 190, 435, 199 ActionDest: 120 Annotation: Subtype: Link Rect: 438, 190, 450, 199 ActionDest: 120 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 2 Annotations: Annotation: Subtype: Link Rect: 409, 520, 416, 532 ActionDest: 81 Annotation: Subtype: Link Rect: 282, 506, 289, 518 ActionDest: 173 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 3 Annotations: Annotation: Subtype: Link Rect: 221, 646, 228, 658 ActionDest: 13 Annotation: Subtype: Link Rect: 486, 646, 493, 658 ActionDest: 18 Annotation: Subtype: Link Rect: 354, 632, 361, 644 ActionDest: 45 Annotation: Subtype: Link Rect: 384, 632, 391, 644 ActionDest: 53 Annotation: Subtype: Link Rect: 395, 618, 402, 630 ActionDest: 81 Annotation: Subtype: Link Rect: 388, 604, 395, 616 ActionDest: 81 Annotation: Subtype: Link Rect: 298, 590, 305, 602 ActionDest: 96 Annotation: Subtype: Link Rect: 203, 579, 210, 588 ActionDest: 101 Annotation: Subtype: Link Rect: 230, 579, 242, 588 ActionDest: 115 Annotation: Subtype: Link Rect: 247, 548, 254, 560 ActionDest: 13 Annotation: Subtype: Link Rect: 472, 548, 479, 560 ActionDest: 18 Annotation: Subtype: Link Rect: 245, 480, 252, 489 ActionDest: 120 Annotation: Subtype: Link Rect: 384, 452, 391, 461 ActionDest: 120 Annotation: Subtype: Link Rect: 394, 452, 401, 461 ActionDest: 120 Annotation: Subtype: Link Rect: 251, 411, 258, 420 ActionDest: 120 Annotation: Subtype: Link Rect: 505, 410, 512, 420 ActionDest: 120 Annotation: Subtype: Link Rect: 515, 411, 527, 420 ActionDest: 120 Annotation: Subtype: Link Rect: 528, 397, 540, 406 ActionDest: 120 Annotation: Subtype: Link Rect: 543, 397, 555, 406 ActionDest: 120 Annotation: Subtype: Link Rect: 210, 244, 216, 256 ActionDest: 18 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 4 Annotations: Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 5 Annotations: Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 6 Annotations: Annotation: Subtype: Link Rect: 286, 434, 298, 443 ActionDest: 120 Annotation: Subtype: Link Rect: 300, 434, 312, 443 ActionDest: 120 Annotation: Subtype: Link Rect: 477, 66, 489, 75 ActionDest: 289 Annotation: Subtype: Link Rect: 492, 66, 504, 75 ActionDest: 289 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 7 Annotations: Annotation: Subtype: Link Rect: 419, 723, 426, 735 ActionDest: 18 Annotation: Subtype: Link Rect: 315, 335, 322, 346 ActionDest: 53 Annotation: Subtype: Link Rect: 212, 66, 224, 75 ActionDest: 289 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 8 Annotations: Annotation: Subtype: Link Rect: 436, 597, 448, 606 ActionDest: 120 Annotation: Subtype: Link Rect: 451, 597, 463, 606 ActionDest: 120 Annotation: Subtype: Link Rect: 213, 370, 220, 382 ActionDest: 68 Annotation: Subtype: Link Rect: 277, 330, 291, 340 ActionDest: 53 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 9 Annotations: Annotation: Subtype: Link Rect: 214, 726, 221, 737 ActionDest: 53 Annotation: Subtype: Link Rect: 280, 686, 292, 695 ActionDest: 289 Annotation: Subtype: Link Rect: 292, 686, 304, 695 ActionDest: 289 Annotation: Subtype: Link Rect: 305, 686, 317, 695 ActionDest: 289 Annotation: Subtype: Link Rect: 261, 352, 273, 361 ActionDest: 289 Annotation: Subtype: Link Rect: 276, 352, 288, 361 ActionDest: 289 Annotation: Subtype: Link Rect: 430, 230, 442, 239 ActionDest: 289 Annotation: Subtype: Link Rect: 479, 216, 491, 225 ActionDest: 289 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 10 Annotations: Annotation: Subtype: Link Rect: 481, 99, 488, 111 ActionDest: 314 Annotation: Subtype: Link Rect: 508, 99, 515, 111 ActionDest: 314 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 11 Annotations: Annotation: Subtype: Link Rect: 284, 489, 297, 501 ActionDest: 81 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 12 Annotations: Annotation: Subtype: Link Rect: 218, 617, 225, 629 ActionDest: 315 Annotation: Subtype: Link Rect: 196, 603, 203, 615 ActionDest: 315 Annotation: Subtype: Link Rect: 212, 166, 219, 178 ActionDest: 315 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 13 Annotations: Annotation: Subtype: Link Rect: 387, 627, 394, 639 ActionDest: 314 Annotation: Subtype: Link Rect: 414, 627, 421, 639 ActionDest: 314 Annotation: Subtype: Link Rect: 277, 600, 284, 612 ActionDest: 314 Annotation: Subtype: Link Rect: 553, 600, 560, 612 ActionDest: 314 Annotation: Subtype: Link Rect: 217, 502, 223, 514 ActionDest: 315 Annotation: Subtype: Link Rect: 209, 474, 216, 486 ActionDest: 315 Annotation: Subtype: Link Rect: 218, 446, 225, 458 ActionDest: 96 Annotation: Subtype: Link Rect: 262, 171, 269, 183 ActionDest: 314 Annotation: Subtype: Link Rect: 376, 117, 383, 129 ActionDest: 18 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 14 Annotations: Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 15 Annotations: Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 16 Annotations: Annotation: Subtype: Link Rect: 355, 476, 362, 488 ActionDest: 13 Annotation: Subtype: Link Rect: 230, 324, 237, 333 ActionDest: 81 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 17 Annotations: Annotation: Subtype: Link Rect: 423, 689, 560, 700 Annotation: Subtype: Link Rect: 165, 676, 258, 687 Annotation: Subtype: Link Rect: 307, 561, 560, 572 Annotation: Subtype: Link Rect: 165, 548, 253, 559 Annotation: Subtype: Link Rect: 300, 535, 560, 546 Annotation: Subtype: Link Rect: 165, 522, 233, 533 Annotation: Subtype: Link Rect: 530, 485, 536, 493 ActionDest: 81 Annotation: Subtype: Link Rect: 519, 444, 526, 455 ActionDest: 314 Annotation: Subtype: Link Rect: 544, 444, 551, 455 ActionDest: 314 Annotation: Subtype: Link Rect: 519, 108, 557, 119 Annotation: Subtype: Link Rect: 55, 96, 91, 106 Annotation: Subtype: Link Rect: 519, 82, 557, 93 Annotation: Subtype: Link Rect: 55, 70, 91, 81 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Page: Sequence: 18 Annotations: Annotation: Subtype: Link Rect: 409, 731, 446, 742 Annotation: Subtype: Link Rect: 453, 731, 489, 742 Annotation: Subtype: Link Rect: 339, 705, 377, 716 Annotation: Subtype: Link Rect: 55, 642, 92, 652 Annotation: Subtype: Link Rect: 485, 616, 523, 627 Annotation: Subtype: Link Rect: 406, 603, 444, 614 Annotation: Subtype: Link Rect: 520, 590, 557, 601 Annotation: Subtype: Link Rect: 391, 450, 428, 461 Annotation: Subtype: Link Rect: 431, 27, 560, 37 Checksum: cdb0d032 Type: CRC32 Checksum: 0fb7d98a56aa961b03afc081c4017273 Type: MD5 Checksum: dfb5ed431c6644c9611640cb248fb46f156cb09e Type: SHA-1 Checksum: 102887d09a5e1e40467f7ce1bdc95361a9184be4eed1b8fe5513a551cb292123 Type: SHA-256