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from north","zero_display":675,"alpha_circle_positions":1800,"beta_circle_positions":675,"node_rule":"Native residues divisible by 3, including zero; preserves the approved M6/M9/M12 reference selection. Additional regular formations of all native values are catalogued separately.","m1_convention":"Modulo 1 has period 1 and residue 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Extended catalogue, Chevron and native/cycle-based Epsilon are Python-checked.","evidence":"m1001-verification.json"},"reduction_to_m9":{"version":"m9-reduction-v1","target_modulus":9,"status":"Exact match","applicable":true,"research_question":"Does the shared M9 pattern have shared biological relevance, or does native-modulus complexity change it? Unassessed: requires peer-reviewed evidence, in silico modelling (including proposed CUDA/CUDA-Q work), and appropriate experimental validation. 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and M9 share the same 24-position Gamma geometry. 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Human myometrial-cell experiments and records from 2,367 patients add distinct translational observations.","relevance":"a direct OXTR connection to the Alpha theme.","limit":"genetic perturbation and promoter experiments strengthen mechanism. Animal, cell and retrospective components need separate appraisal; the human observations can be confounded. Uterine findings do not establish effects in brain tissue.","access":"final publication abstract and selected full-text results. The 2024 preprint is superseded for this register and is not counted as another study.","scoreParts":{"design":3,"validation":2,"review":2,"provenance":1},"score":8,"scoreReason":"Genetic and promoter experiments support mechanism; the human observational component has separate confounding limits.","jadad":"Not applicable","citationCount":3,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Interaction between time-of-day and oxytocin efficacy in mice and humans with and without gestational diabetes"},{"id":"E04","title":"Cryptochromes mediate rhythmic repression of the glucocorticoid receptor","author":"Lamia et al.","year":"2011","journal":"Nature","doi":"10.1038/nature10700","theme":"Alpha · receptor regulation","hero":null,"methods":"Mouse cells and mice; genetic and biochemical experiments","finding":"Cry1/Cry2 interacted with the glucocorticoid receptor and altered glucocorticoid-responsive transcription in mouse cells, with related metabolic phenotypes in mice.","relevance":"direct circadian–receptor evidence provides a natural NR3C1 example beyond oxytocin.","limit":"molecular interactions, genetic perturbations and physiological observations complement one another. Multiple pathways change in knockout animals; the results do not isolate a universal geometric explanation or establish human clinical effects.","access":"primary abstract and selected experimental/results text. NR3C1 identifies the receptor target; the principal genetic perturbations here involved cryptochromes.","scoreParts":{"design":3,"validation":2,"review":2,"provenance":1},"score":8,"scoreReason":"Genetic perturbations, molecular interactions and physiological measurements support the reported mechanism.","jadad":"Not applicable","citationCount":550,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Cryptochromes mediate rhythmic repression of the glucocorticoid receptor"},{"id":"E05","title":"Experimental and theoretical study of mitotic spindle orientation","author":"Théry et al.","year":"2007","journal":"Nature","doi":"10.1038/nature05786","theme":"Beta · mitotic geometry","hero":null,"methods":"Human HeLa cells; controlled adhesion geometry and modelling","finding":"controlled adhesion patterns in human HeLa cells, combined with a physical model, accounted for spindle-orientation distributions across different microenvironment geometries.","relevance":"direct evidence that geometry is a meaningful biological variable; supports interest in the Beta/division theme and related shape families.","limit":"manipulation of the cell environment offers more than an observational resemblance. The model uses specific force assumptions and a transformed cell line. Division orientation is distinct from division rate, population doubling or a Fibonacci sequence.","access":"publisher abstract and figure descriptions. Full methods and sample accounting remain to be appraised.","scoreParts":{"design":3,"validation":1,"review":1,"provenance":1},"score":6,"scoreReason":"Controlled adhesion geometry tests a quantitative model; full methods and sample accounting remain open for appraisal.","jadad":"Not applicable","citationCount":424,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Experimental and theoretical study of mitotic spindle orientation"},{"id":"E06","title":"Oxytocin promotes epicardial cell activation and heart regeneration after cardiac injury","author":"Wasserman et al.","year":"2022","journal":"Frontiers in Cell and Developmental Biology","doi":"10.3389/fcell.2022.985298","theme":"Beta · proliferation","hero":"beta","methods":"Human stem-cell-derived epicardial cells and zebrafish","finding":"OXT promoted activation and proliferation-related outcomes in human stem-cell-derived epicardial cells; OXTR knockdown removed the response. Zebrafish experiments supplied a separate regeneration context.","relevance":"a direct OXTR-related example for the Beta research theme.","limit":"receptor knockdown strengthens causal interpretation within the model. Check independent cell lines and differentiation batches; images are subsamples, not donors. Cultured epicardial cells and fish regeneration do not establish regeneration of adult human hearts.","access":"primary full-text methods/results and relevant figure captions, including OXTR knockdown and proliferation measurements.","scoreParts":{"design":3,"validation":2,"review":3,"provenance":1},"score":9,"scoreReason":"Receptor knockdown provides a mechanistic test alongside cell and zebrafish experiments; methods and results appraised.","jadad":"Not applicable","citationCount":25,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Oxytocin promotes epicardial cell activation and heart regeneration after cardiac injury"},{"id":"E07","title":"Oxytocin stimulates hippocampal neurogenesis via oxytocin receptor expressed in CA3 pyramidal neurons","author":"Lin et al.","year":"2017","journal":"Nature Communications","doi":"10.1038/s41467-017-00675-5","theme":"Beta · survival / proliferation","hero":null,"methods":"Mice; conditional Oxtr deletion","finding":"conditional Oxtr deletion in mouse hippocampal excitatory neurons affected survival and maturation of newly generated neurons, while measured progenitor proliferation and early differentiation were unchanged.","relevance":"direct evidence that the biological outcome must be named precisely. It bounds generalisations from E06 without contradicting a different tissue-specific finding.","limit":"conditional genetics help localise the process. Appraise cell-type specificity, blinding and precision; an unchanged measured outcome does not prove absence of every possible effect.","access":"primary results/discussion. Retained deliberately to avoid presenting all OXTR growth-related findings as evidence of duplication.","scoreParts":{"design":3,"validation":2,"review":2,"provenance":1},"score":8,"scoreReason":"Conditional genetics and cell-outcome measurements distinguish survival from proliferation in this mouse system.","jadad":"Not applicable","citationCount":149,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Oxytocin stimulates hippocampal neurogenesis via oxytocin receptor expressed in CA3 pyramidal neurons"},{"id":"E08","title":"Structural basis for the activation and ligand recognition of the human oxytocin receptor","author":"Waltenspühl et al.","year":"2022","journal":"Nature Communications","doi":"10.1038/s41467-022-31325-0","theme":"Gamma · molecular recognition","hero":"gamma","methods":"Engineered human receptor complex; cryo-EM and functional assays","finding":"cryo-EM at 3.2 Å and functional experiments characterised OXT recognition and receptor activation. All nine peptide residues participate in binding.","relevance":"a direct structural foundation for Gamma's molecular-recognition theme; useful context, but not confirmation, for the OXT-specific Epsilon proposal.","limit":"structure–function comparisons strengthen interpretation. Receptor stabilisation and an engineered G protein affect generalisation. The six-residue disulfide loop is not itself an aromatic hexagon; the reported contacts do not establish an Epsilon trajectory.","access":"full primary article main text, receptor-engineering description and functional-assay captions. No new molecular analysis performed.","scoreParts":{"design":3,"validation":2,"review":3,"provenance":1},"score":9,"scoreReason":"Structural observations and functional residue tests provide complementary evidence; main text and assay descriptions appraised.","jadad":"Not applicable","citationCount":46,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Structural basis for the activation and ligand recognition of the human oxytocin receptor"},{"id":"E09","title":"X-ray structures and mechanism of the human serotonin transporter","author":"Coleman, Green & Gouaux","year":"2016","journal":"Nature","doi":"10.1038/nature17629","theme":"Gamma · molecular recognition","hero":null,"methods":"Engineered human serotonin transporter; crystallography and assays","finding":"structures of human SERT bound to antidepressants identified central and allosteric binding sites and explained aspects of inhibitor recognition.","relevance":"direct structural evidence broadens Gamma beyond OXTR to SLC6A4.","limit":"structural and binding/transport measurements are complementary. Thermostabilised constructs were used; the ts3 construct had no detectable transport activity. Results describe particular inhibitor-bound conformations rather than every physiological state.","access":"publisher abstract and selected primary methods/results. These data concern the protein product and ligand recognition, not regulation of SLC6A4 gene expression.","scoreParts":{"design":2,"validation":2,"review":2,"provenance":1},"score":7,"scoreReason":"Structures and functional measurements complement one another; engineered constructs constrain interpretation.","jadad":"Not applicable","citationCount":674,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"X-ray structures and mechanism of the human serotonin transporter"},{"id":"E10","title":"Structure of a D2 dopamine receptor–G-protein complex in a lipid membrane","author":"Yin et al.","year":"2020","journal":"Nature","doi":"10.1038/s41586-020-2379-5","theme":"Gamma · molecular recognition","hero":null,"methods":"Engineered human DRD2 complex in a lipid membrane; cryo-EM","finding":"cryo-EM characterised an agonist-bound DRD2–Gi complex within a reconstituted phospholipid membrane, identifying receptor, G-protein and lipid interactions.","relevance":"direct evidence for the structural context of molecular recognition; another Gamma hero example.","limit":"including a lipid environment improves contextual relevance relative to isolated detergent structures. The receptor complex is engineered and ligand-bound; it does not represent every receptor state or establish clinical outcomes.","access":"primary abstract and selected full-text structural results, with deposited structural metadata checked. No geometric correspondence to the Atlas was analysed.","scoreParts":{"design":2,"validation":1,"review":2,"provenance":1},"score":6,"scoreReason":"A membrane-reconstituted structural observation provides a defined molecular state; independent functional validation needs fuller appraisal.","jadad":"Not applicable","citationCount":199,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Structure of a D2 dopamine receptor–G-protein complex in a lipid membrane"},{"id":"E11","title":"Assembly of a π–π stack of ligands in the binding site of an acetylcholine-binding protein","author":"Stornaiuolo et al.","year":"2013","journal":"Nature Communications","doi":"10.1038/ncomms2900","theme":"Epsilon · aromatic stacking","hero":"epsilon","methods":"Molluscan binding protein; crystallography and solution assays","finding":"an Aplysia acetylcholine-binding protein accommodated a stack of three aromatic ligands. Solution binding measurements supported the structural observation.","relevance":"direct experimental support for π-stacking as a molecular-recognition phenomenon; an Epsilon reference that does not depend on OXTR.","limit":"crystallography, mutagenesis and solution assays help address crystallographic-artifact concerns. This soluble molluscan protein is a model for receptor ligand-binding domains, not a complete human receptor. Neither the three-ligand count nor the stacked geometry establishes the Atlas traversal.","access":"full primary main text, including solution validation and scope of the receptor model.","scoreParts":{"design":3,"validation":2,"review":3,"provenance":1},"score":9,"scoreReason":"Structures, mutagenesis and solution assays support the binding interpretation; full main text appraised.","jadad":"Not applicable","citationCount":66,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Assembly of a π–π stack of ligands in the binding site of an acetylcholine-binding protein"},{"id":"E12","title":"Targeted disruption of pi–pi stacking in Malaysian banana lectin reduces mitogenicity while preserving antiviral activity","author":"Covés-Datson et al.","year":"2021","journal":"Scientific Reports","doi":"10.1038/s41598-020-80577-7","theme":"Epsilon · aromatic stacking","hero":null,"methods":"Engineered plant lectin; structures and cellular assays","finding":"an F84T mutation disrupted aromatic stacking in Malay BanLec, reduced mitogenic activity and retained antiviral activity in the studied assays.","relevance":"a direct structure–function example for Epsilon; also connects molecular interactions with proliferation as a biological outcome.","limit":"structural checks and functional comparisons strengthen the rationale. Mutation can change several properties simultaneously, so this is not a pure isolation of one energetic contribution. Plant-lectin effects do not establish OXTR behaviour or clinical benefit.","access":"primary abstract and main structural/results sections. Published in 2021 despite “2020” appearing within the DOI.","scoreParts":{"design":3,"validation":2,"review":2,"provenance":1},"score":8,"scoreReason":"Targeted mutation and structural / cellular comparisons connect aromatic organisation with biological outcomes.","jadad":"Not applicable","citationCount":24,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Targeted disruption of pi–pi stacking in Malaysian banana lectin reduces mitogenicity while preserving antiviral activity"},{"id":"E13","title":"Reconstruction of molecular orbital densities from photoemission data","author":"Puschnig et al.","year":"2009","journal":"Science","doi":"10.1126/science.1176105","theme":"Valence electron orbitals","hero":null,"methods":"Organic molecular films; photoemission and calculations","finding":"photoemission measurements and Fourier reconstruction yielded orbital-density distributions for pentacene films and p-sexiphenyl monolayers, compared with density-functional calculations.","relevance":"direct physics evidence for studying spatial electronic structure; contextual to the valence-electron theme and more distant from biological significance.","limit":"comparison of measurement and theory is valuable. Reconstruction is model-dependent and the samples are non-living materials. Orbital densities are not classical electron orbits or evidence of unicursal electron motion.","access":"primary abstract and metadata. No biological role, creation-of-life claim or numeric WSS assignment follows from this paper.","scoreParts":{"design":2,"validation":1,"review":1,"provenance":1},"score":5,"scoreReason":"Measurement and reconstruction are compared with calculations; current appraisal is abstract-level.","jadad":"Not applicable","citationCount":334,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"Reconstruction of Molecular Orbital Densities from Photoemission Data"},{"id":"E14","title":"CD38 is critical for social behaviour by regulating oxytocin secretion","author":"Jin et al.","year":"2007","journal":"Nature","doi":"10.1038/nature05526","theme":"Supporting OXT context","hero":null,"methods":"Mice; Cd38 knockout and rescue experiments","finding":"Cd38-knockout mice showed altered oxytocin secretion and behavioural phenotypes. Oxytocin replacement or hypothalamic delivery of human CD38 rescued selected outcomes.","relevance":"direct support for CD38 involvement in the oxytocin system. Retained as supporting hero context without forcing it into a particular sequence family.","limit":"knockout plus rescue strengthens causal interpretation in mice. Global deletion, behavioural measurement and human translation require careful appraisal. This concerns secretion, not proof of OXTR-expression regulation or geometric correspondence.","access":"primary abstract and figure descriptions. Full methods are needed before formal risk-of-bias grading.","scoreParts":{"design":3,"validation":1,"review":1,"provenance":1},"score":6,"scoreReason":"Knockout and rescue experiments test the secretion mechanism; full methods remain open for appraisal.","jadad":"Not applicable","citationCount":585,"citationSource":"Crossref cited-by count","citationChecked":"17 October 2026","crossrefTitle":"CD38 is critical for social behaviour by regulating oxytocin secretion"}]},"views":[{"id":"alpha","name":"Alpha","svg":"pilot-v1/views/m675/alpha.svg","png":"pilot-v1/views/m675/alpha.png"},{"id":"beta","name":"Beta","svg":"pilot-v1/views/m675/beta.svg","png":"pilot-v1/views/m675/beta.png"},{"id":"gamma","name":"Gamma","svg":"pilot-v1/views/m675/gamma.svg","png":"pilot-v1/views/m675/gamma.png"},{"id":"fibonacci-m675p1800","name":"Fibonacci 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pair of alternating equilateral triangles per distinct regular six-node set on the full Alpha circle. Each set is counted once, with its orientation retained.","period_rule":"The full Alpha count is P / 6 when P is divisible by 6, otherwise zero.","gamma_rule":"Primary Gamma counts only formations whose six vertices belong to the selected Gamma node set.","source":"formation_catalogue; each referenced Hexagon supplies the same six vertices for the paired triangles in Other Geometry.","research_status":"Descriptive geometry. Native doubling is the current starting hypothesis for mitosis; other Beta sequences remain available for future investigation."},"beta_m9_view":{"version":"beta-m9-view-v1","native_modulus":675,"target_modulus":9,"native_values":[4,26,178,545,262,614,661,638,430,347,649,146,373,440,7,284,631,83,625,242,394,491,343,560,202,179,376,428,595,362,589,386,88,230,172,299,571,323,340,32,559,506,283,125,592,644,541,443,535,602,304,176,253,245,112,539,286,113,505,47,499,71,673,590,82,659,481,8,250,392,469,191,193,485,502,329,451,128,445,287,214,536,163,605,22,224,196,473,415,407,409,431,583,275,667,344,391,368,160,77,379,551,103,170,412,14,361,488,355,647,124,221,73,290,607,584,106,158,325,92,319,116,493,635,577,29,301,53,70,437,289,236,13,530,322,374,271,173,265,332,34,581,658,650,517,269,16,518,235,452,229,476,403,320,487,389,211,413,655,122,199,596,598,215,232,59,181,533,175,17,619,266,568,335,427,629,601,203,145,137,139,161,313,5,397,74,121,98,565,482,109,281,508,575,142,419,91,218,85,377,529,626,478,20,337,314,511,563,55,497,49,521,223,365,307,434,31,458,475,167,19,641,418,260,52,104,1,578,670,62,439,311,388,380,247,674,421,248,640,182,634,206,133,50,217,119,616,143,385,527,604,326,328,620,637,464,586,263,580,422,349,671,298,65,157,359,331,608,550,542,544,566,43,410,127,479,526,503,295,212,514,11,238,305,547,149,496,623,490,107,259,356,208,425,67,44,241,293,460,227,454,251,628,95,37,164,436,188,205,572,424,371,148,665,457,509,406,308,400,467,169,41,118,110,652,404,151,653,370,587,364,611,538,455,622,524,346,548,115,257,334,56,58,350,367,194,316,668,310,152,79,401,28,470,562,89,61,338,280,272,274,296,448,140,532,209,256,233,25,617,244,416,643,35,277,554,226,353,220,512,664,86,613,155,472,449,646,23,190,632,184,656,358,500,442,569,166,593,610,302,154,101,553,395,187,239,136,38,130,197,574,446,523,515,382,134,556,383,100,317,94,341,268,185,352,254,76,278,520,662,64,461,463,80,97,599,46,398,40,557,484,131,433,200,292,494,466,68,10,2],"reduced_values":[4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2],"display_cycle":[4,8,7,5,1,2],"repeat_count":75,"native_circle_positions":675,"reduced_circle_positions":9,"original_alignment_preserved":true,"start_interval_clock_positions":[1800,4],"reference_cycle_match":true,"family_audit":{"audit_id":"beta-m9-family-audit-v1","modulus_range":[1,1001],"total_moduli_in_range":1001,"family_members_checked":111,"matching_family_members":111,"coverage_fraction":"111/1001","coverage_percent":11.08891108891109,"coverage_display":"11.1%","classification":"Exact modular reduction; descriptive coverage","summary":"All 111 multiples of nine in M1–M1001 (11.1% of moduli) share this reduced Beta cycle through exact modular reduction.","audit_file":"pilot-v1/beta-m9-family-audit.json"},"operation":"Reduce each native Beta residue modulo 9, preserving order and the original start.","drawing_rule":"Plot the six-value cycle once; the repeat count records overlapping traversals. The complete reduced sequence remains available alongside the diagram and in the data.","native_svg":"pilot-v1/views/m675/beta.svg","native_png":"pilot-v1/views/m675/beta.png","reduced_svg":"pilot-v1/views/m675/beta-m9.svg","reduced_png":"pilot-v1/views/m675/beta-m9.png"},"m9_sequence_views":{"beta":{"version":"beta-m9-view-v1","native_modulus":675,"target_modulus":9,"native_values":[4,26,178,545,262,614,661,638,430,347,649,146,373,440,7,284,631,83,625,242,394,491,343,560,202,179,376,428,595,362,589,386,88,230,172,299,571,323,340,32,559,506,283,125,592,644,541,443,535,602,304,176,253,245,112,539,286,113,505,47,499,71,673,590,82,659,481,8,250,392,469,191,193,485,502,329,451,128,445,287,214,536,163,605,22,224,196,473,415,407,409,431,583,275,667,344,391,368,160,77,379,551,103,170,412,14,361,488,355,647,124,221,73,290,607,584,106,158,325,92,319,116,493,635,577,29,301,53,70,437,289,236,13,530,322,374,271,173,265,332,34,581,658,650,517,269,16,518,235,452,229,476,403,320,487,389,211,413,655,122,199,596,598,215,232,59,181,533,175,17,619,266,568,335,427,629,601,203,145,137,139,161,313,5,397,74,121,98,565,482,109,281,508,575,142,419,91,218,85,377,529,626,478,20,337,314,511,563,55,497,49,521,223,365,307,434,31,458,475,167,19,641,418,260,52,104,1,578,670,62,439,311,388,380,247,674,421,248,640,182,634,206,133,50,217,119,616,143,385,527,604,326,328,620,637,464,586,263,580,422,349,671,298,65,157,359,331,608,550,542,544,566,43,410,127,479,526,503,295,212,514,11,238,305,547,149,496,623,490,107,259,356,208,425,67,44,241,293,460,227,454,251,628,95,37,164,436,188,205,572,424,371,148,665,457,509,406,308,400,467,169,41,118,110,652,404,151,653,370,587,364,611,538,455,622,524,346,548,115,257,334,56,58,350,367,194,316,668,310,152,79,401,28,470,562,89,61,338,280,272,274,296,448,140,532,209,256,233,25,617,244,416,643,35,277,554,226,353,220,512,664,86,613,155,472,449,646,23,190,632,184,656,358,500,442,569,166,593,610,302,154,101,553,395,187,239,136,38,130,197,574,446,523,515,382,134,556,383,100,317,94,341,268,185,352,254,76,278,520,662,64,461,463,80,97,599,46,398,40,557,484,131,433,200,292,494,466,68,10,2],"reduced_values":[4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2,4,8,7,5,1,2],"display_cycle":[4,8,7,5,1,2],"repeat_count":75,"native_circle_positions":675,"reduced_circle_positions":9,"original_alignment_preserved":true,"start_interval_clock_positions":[1800,4],"reference_cycle_match":true,"family_audit":{"audit_id":"beta-m9-family-audit-v1","modulus_range":[1,1001],"total_moduli_in_range":1001,"family_members_checked":111,"matching_family_members":111,"coverage_fraction":"111/1001","coverage_percent":11.08891108891109,"coverage_display":"11.1%","classification":"Exact modular reduction; descriptive coverage","summary":"All 111 multiples of nine in M1–M1001 (11.1% of moduli) share this reduced Beta cycle through exact modular reduction.","audit_file":"pilot-v1/beta-m9-family-audit.json"},"operation":"Reduce each native Beta residue modulo 9, preserving order and the original start.","drawing_rule":"Plot the six-value cycle once; the repeat count records overlapping traversals. The complete reduced sequence remains available alongside the diagram and in the 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