{ "study": "STUDY_67_SIX_MATH_ESSENTIALS", "title": "The Six Exact Essentials — Rewriting Terence Tao's Six Math Essentials on M8 = S4 x C4", "book_seal_fnv1a64": "5e6c51dbb5d33936", "status": "SIX_OF_SIX_ESSENTIALS_SEALED_ZERO_FLOAT", "kernel_path": "reproduce/study-67-six-math-essentials.swift", "chapters": [ { "chapter": 1, "roman": "I", "pillar": "Numbers", "subtitle": "The Gateway to Quantitative Thinking", "tao_continuum_detour": "Starts with N, Z, Q and leaps into Dedekind cuts / Cauchy sequences (R), which hardware truncates into IEEE-754 floats where 3987^12 + 4365^12 == 4472^12 evaluates to true.", "affine_exact_law": "Arbitrary-limb Canonical Rationals (Q) and Eisenstein Integers (Z[w]) reduced via Stein's binary GCD. Unclosed decimals are refused at Stage 1 (HTTP 400 REFUSED · AFFINE_JZ_PARSE).", "study_binding": "Study 36 (FLT Near-Miss) & Study 56 (All 12,500 MATH Keys)", "uum8d_recipe": "3987^12 + 4365^12 - 4472^12", "verified_metrics": { "parity_defect": 1, "lhs_mod18": "669958142428526657", "rhs_mod18": "278649643651567616", "exact_gap": "-700212234530608691501223040959", "verdict": "REFUTED_EXACT" }, "gop_hologram_mode": "flt_lattice_parity_cube" }, { "chapter": 2, "roman": "II", "pillar": "Algebra", "subtitle": "The Gateway to Abstraction", "tao_continuum_detour": "Symbolic systems solved numerically via floating-point LU/QR decomposition where pivot rounding turns singular systems into false solutions.", "affine_exact_law": "Fraction-Free Bareiss Elimination via Sylvester's Identity and Jordan Algebras over Z. Every intermediate pivot division is algebraically exact.", "study_binding": "Study 13 (Connes Rigidity), Study 25 (Exact Permanent) & 19-Gate Bareiss Referee", "uum8d_recipe": "bareiss([[2,1,-1,2],[-3,-1,2,-1],[-2,1,2,-3]])", "verified_metrics": { "bareiss_det": -10, "solution": "(2/5,-3/5,-1/1)", "circularity": "FALSE", "verdict": "EXACT_CLOSED" }, "gop_hologram_mode": "bareiss_pivot_hyperplane" }, { "chapter": 3, "roman": "III", "pillar": "Geometry", "subtitle": "Calculating Beyond What We Can See", "tao_continuum_detour": "Euclidean/Riemannian continuous metrics requiring transcendental sin, cos, sqrt that shear orthogonality (R^T R != I) on GPUs.", "affine_exact_law": "Ehrhart Lattice Polynomials L(P, t) and Cayley Rational SO(3, Q) rotations R(u) = (I - [u]_x)(I + [u]_x)^(-1) with exact rational quadrance 1/1.", "study_binding": "Study 19 (Lattice Geometry) & Study 66 (GOP 3D Hologram)", "uum8d_recipe": "cayley_so3(1, 2, -2)", "verified_metrics": { "ehrhart_L12": 169, "cayley_den": 11, "cayley_row0": "[9/11, -2/11, 6/11]", "row0_quadrance": "1/1", "verdict": "SO3_Q_ORTHOGONAL_EXACT" }, "gop_hologram_mode": "cayley_so3_wireframe" }, { "chapter": 4, "roman": "IV", "pillar": "Probability", "subtitle": "Navigating Uncertainty with Rigorous Thinking", "tao_continuum_detour": "Continuous sigma-algebra measure theory sampled via pseudo-random floating-point Monte Carlo that drifts across thread schedules.", "affine_exact_law": "Exact Rational Generating Functions and Bennett Reversible Combinatorial Measures p/q in Q.", "study_binding": "Study 29 (Quantum Phase Estimation) & Study 43 (GoFirst D60 Dice)", "uum8d_recipe": "(9/16)^4", "verified_metrics": { "qpr_n8": "6561/65536", "float_adversary_shears": true, "gofirst_5d60": "1/120", "verdict": "EXACT_COMBINATORIAL_MEASURE" }, "gop_hologram_mode": "qpr_branching_lattice" }, { "chapter": 5, "roman": "V", "pillar": "Analysis", "subtitle": "Taming the Very Large and the Very Small", "tao_continuum_detour": "Infinitesimal epsilon-delta limits where sub-ULP signals underflow to 0.0 when added to macroscopic baselines (> 2^53).", "affine_exact_law": "Discrete Exterior Calculus (DEC), Exact Rational Interval Brackets [333/106, 355/113], and Multi-Scale Integer Limbs from milli-Angstroms to light-years.", "study_binding": "Study 42 (Si(100) Dimer Buckling) & Study 50 (Interval Brackets)", "uum8d_recipe": "355/113 - 333/106", "verified_metrics": { "si_dimer_step8783_mA": 33726720, "pi_bracket": "[333/106,355/113]", "width": "1/11978", "verdict": "SCALE_INVARIANT_NO_HORIZON" }, "gop_hologram_mode": "si100_dimer_bracket" }, { "chapter": 6, "roman": "VI", "pillar": "Dynamics", "subtitle": "How Systems Evolve Over Time", "tao_continuum_detour": "Continuous ODE integrators (Euler/RK4) where energy drifts and IEEE-754 Float32 freezes at the 2^24 = 16777216 mantissa deaf wall.", "affine_exact_law": "Bennett-Reversible Symplectic Lattice Automorphisms on M8 = S4 x C4 with exact integer step conservation and Fiedler lambda_2 stability.", "study_binding": "Study 37 (Symplectic Lattice), Study 65 (PWA Twin) & Study 67", "uum8d_recipe": "16777216 + 1000", "verified_metrics": { "float32_deaf_wall": 16777216, "exact_after_1000_ticks": 16778216, "delta_preserved": 1000, "verdict": "ZERO_DRIFT_CONSERVED" }, "gop_hologram_mode": "symplectic_torus_orbit" } ], "socratic_refusals": [ { "input": "0.3333333", "code": "HTTP 400 REFUSED · AFFINE_JZ_PARSE", "explanation": "Unclosed base-10 decimal truncates 1/3 with a 1/30000000 remainder leak.", "exact_lift": "1/3" }, { "input": "Math.sin(0.5)", "code": "HTTP 400 REFUSED · AFFINE_FLOAT_TRAP", "explanation": "Transcendental floating-point sine shears SO(3) orthogonality. Lift to Cayley rational parameter t = 1/2.", "exact_lift": "sin_cayley(1/2) = 4/5, cos_cayley(1/2) = 3/5" }, { "input": "16777216.0 + 1.0", "code": "HTTP 400 REFUSED · AFFINE_MANTISSA_WALL", "explanation": "IEEE-754 Float32 loses unit increments at 2^24 = 16777216.", "exact_lift": "16777216 + 1 = 16777217" } ], "languages_25": [ {"iso": "en", "name": "English", "title": "The Six Exact Essentials", "c4": [6, 0, 100, 12]}, {"iso": "es", "name": "Español", "title": "Los Seis Fundamentos Exactos", "c4": [6, 0, 100, 12]}, {"iso": "fr", "name": "Français", "title": "Les Six Fondamentaux Exacts", "c4": [6, 0, 100, 12]}, {"iso": "de", "name": "Deutsch", "title": "Die Sechs Exakten Grundlagen", "c4": [6, 0, 100, 12]}, {"iso": "it", "name": "Italiano", "title": "I Sei Fondamenti Esatti", "c4": [6, 0, 100, 12]}, {"iso": "pt", "name": "Português", "title": "Os Seis Fundamentos Exatos", "c4": [6, 0, 100, 12]}, {"iso": "ru", "name": "Русский", "title": "Шесть Точных Основ Математики", "c4": [6, 0, 100, 12]}, {"iso": "zh", "name": "中文", "title": "数学六大精确基石", "c4": [6, 0, 100, 12]}, {"iso": "ja", "name": "日本語", "title": "数学の六つの厳密な礎", "c4": [6, 0, 100, 12]}, {"iso": "ko", "name": "한국어", "title": "수학의 6대 정밀 본질", "c4": [6, 0, 100, 12]}, {"iso": "ar", "name": "العربية", "title": "الأساسيات الرياضية الستة الدقيقة", "c4": [6, 0, 100, 12]}, {"iso": "hi", "name": "हिन्दी", "title": "गणित के छह सटीक आधार", "c4": [6, 0, 100, 12]}, {"iso": "bn", "name": "বাংলা", "title": "গণিতের ছয়টি নির্ভুল মূলভিত্তি", "c4": [6, 0, 100, 12]}, {"iso": "ur", "name": "اردو", "title": "ریاضی کے چھ قطعی اصول", "c4": [6, 0, 100, 12]}, {"iso": "fa", "name": "فارسی", "title": "شش اصل دقیق ریاضیات", "c4": [6, 0, 100, 12]}, {"iso": "tr", "name": "Türkçe", "title": "Matematiğin Altı Kesin Temeli", "c4": [6, 0, 100, 12]}, {"iso": "vi", "name": "Tiếng Việt", "title": "Sáu Nền Tảng Toán Học Chính Xác", "c4": [6, 0, 100, 12]}, {"iso": "id", "name": "Bahasa Indonesia", "title": "Enam Esensi Matematika Eksak", "c4": [6, 0, 100, 12]}, {"iso": "sw", "name": "Kiswahili", "title": "Misingi Sita Halisi ya Hisabati", "c4": [6, 0, 100, 12]}, {"iso": "yo", "name": "Yorùbá", "title": "Àwọn Ìpìlẹ̀ Mẹ́fà Tó Péye Nínú Ìṣirò", "c4": [6, 0, 100, 12]}, {"iso": "nl", "name": "Nederlands", "title": "De Zes Exacte Wiskundige Fundamenten", "c4": [6, 0, 100, 12]}, {"iso": "pl", "name": "Polski", "title": "Sześć Dokładnych Fundamentów Matematyki", "c4": [6, 0, 100, 12]}, {"iso": "uk", "name": "Українська", "title": "Шість Точних Основ Математики", "c4": [6, 0, 100, 12]}, {"iso": "el", "name": "Ελληνικά", "title": "Τα Έξι Ακριβή Θεμέλια των Μαθηματικών", "c4": [6, 0, 100, 12]}, {"iso": "he", "name": "עברית", "title": "ששת יסודות המתמטיקה המדויקים", "c4": [6, 0, 100, 12]} ] }