Appendix: Recomputable Tables and Cross-Civilization Derivations
The mathematical appendix compiles planetary period alignments, Fuxi binary trigram encodings, decan-tarot mappings, and 78-card combinatorial state spaces for public verification.
Introduction to Recomputable Tables and Mathematical Auditability
Scientific research demands transparency, auditability, and independent recomputability. This appendix brings together the foundational constants, algebraic permutation matrices, combinatorial probability figures, and algorithmic lookup tables referenced throughout the monograph. Every table includes clear derivation methods and citations, ensuring third-party researchers can replicate all calculations.
Table A-1: Five Planets: Periods and Classical Chinese Correspondences
Constants determined from NASA Planetary Fact Sheets: Mercury (Chenxing / Water) sidereal period 88 days, synodic period 116 days; Venus (Taibai / Metal) sidereal 225 days, synodic 584 days; Mars (Yinghuo / Fire) sidereal 687 days, synodic 780 days; Jupiter (Suixing / Wood) sidereal 11.86 years, synodic 399 days; Saturn (Zhenxing / Earth) sidereal 29.46 years, synodic 378 days. Sidereal periods lock chronological cycles, while synodic periods pace observational omen windows.
Table A-2: Bagua Three-Bit Binary Values (Fuxi Order)
Following the Bouvet-Leibniz binary interpretation (yang = 1, yin = 0, read top-down): Qian 111 corresponds to decimal 7, Dui 110 to 6, Li 101 to 5, Zhen 100 to 4, Xun 011 to 3, Kan 010 to 2, Gen 001 to 1, and Kun 000 to 0. Shao Yong's circular and square arrangements illustrate ascending and descending binary progressions, proving 64 hexagrams exhaust the 6-bit state space.
Table A-3 and A-3b/c: Book T Tarot Correspondences and Thirty-Six Decans
Sourced from the 1888 Hermetic Order of the Golden Dawn Book T manuscripts: 0 Fool corresponds to Aether/Air, 1 Magician to Mercury, 2 High Priestess to Moon, 3 Empress to Venus, 4 Emperor to Aries, through 21 World to Saturn. Egyptian thirty-six decans divide the zodiac into 10-degree intervals, providing a recursive 3x12 subdivision identical to astronomical timekeeping practices across Persia and Hellenistic Greece.
Table A-4a/b/c: Tarot 78-Card Combinatorial State Spaces and Permutations
Across 78 cards with independent upright/reversed orientations: 1-card draw yields 156 states; 3-card spread yields 456,456 permutations (3,651,648 with orientations); 5-card spread yields 2,533,330,800 permutations (8.10x10¹⁰ with orientations); 10-card Celtic Cross spans 4.67x10²¹ states. Popular literature erroneously applying combinations underestimates spread complexity by 3,628,800-fold (10!). Exact probabilities: 3-card contains major arcana: 63.56%; 5-card: 81.91%; 10-card: 97.17%.
Figure A-1: Liuren Cosmic Board Information Flow and Modular Lookup
The ancient Chinese Liuren cosmic board maps celestial mechanics through rotating heaven and earth plates. The derivation executes modular arithmetic: solar month plus hour establishes the heaven plate; Four Lessons identify elemental relationships; Three Transmissions extract initial, middle, and final pivots. The algorithm functions as a deterministic finite state machine with zero random sources.
Figure A-2: Sexagenary Sixty-Cycle Congruence Generation Rule
Ten heavenly stems (mod 10) and twelve earthly branches (mod 12) pair via i ↦ (i mod 10, i mod 12). Because gcd(10, 12) = 2 and lcm(10, 12) = 60, the pairs remain distinct across steps 0 through 59, closing precisely at step 60. This simple algebraic rule demonstrates the mathematical inevitability of the sixty-year calendrical cycle without arbitrary empirical intervention.
Appendix Bibliography I: Classical Astronomy and Archaeological Sources
Foundational astronomical calculations and archaeological source editions: Bretagnon & Francou (1988) VSOP87 planetary theories; Espenak & Meeus (2006) NASA Five Millennium Canon of Solar Eclipses (-1999 to +3000); Hunger & Steele (2019) MUL.APIN: An Astronomical Treatise and Its Contexts (c. 1000 BCE); Justeson & Lowry (2025) Maya Dresden Codex Eclipse Table (Science Advances 11(43): eadt9039, 69 stations across 1,655 lunar cycle); Li et al. (2026) OBIMD multimodal dataset of 10,077 oracle bone rubbings (Scientific Data); Neugebauer (1955, 1975) Astronomical Cuneiform Texts and A History of Ancient Mathematical Astronomy; Sivin (1969, 2009) Cosmos and Computation in Early Chinese Astronomy & Shoushi Calendar; Steele (2000) Observations and Predictions of Eclipse Times; Ptolemy (1940) Tetrabiblos (Loeb Classical Library 435).
Appendix Bibliography II: Chinese Divination, Near Eastern Astrology, and Binary History
Comparative mathematical models and primary textual sources: Cullen (2011) Wu xing zhan: Prognostics of the Five Planets from Mawangdui (SCIAMVS 12: 193-249); Ho Peng-Yoke (2003) Chinese Mathematical Astrology: Reaching Numbers to the Heavens (RoutledgeCurzon); Needham (1959) Science and Civilisation in China, Vol. 3; Pingree (1973) The Mesopotamian Origin of Early Indian Mathematical Astronomy; Sun & Kistemaker (1997) The Chinese Sky during the Han; Standard histories (Shiji, Hanshu, Yuanshi, Zhonghua Book Company); Bouché-Leclercq (1899) L'astrologie grecque; Cauville (1997) Le zodiaque d'Osiris; Catalogus Codicum Astrologorum Graecorum (CCAG 1898-1953); Rochberg (2004, 2010) The Heavenly Writing & In the Path of the Moon; West (1880) Pahlavi Texts: Bundahishn; Leibniz (1703) Explication de l'arithmétique binaire; Ryan (1996) Leibniz' Binary System and Shao Yong's Yijing (Philosophy East & West); Schumann (2021) Logical Reasoning for Forecasting in Mesopotamia.
Appendix Bibliography III: Tarot History, Statistical Verification, and Preregistration Protocol
Sampling mechanism history, statistical methodologies, and audit protocols: Dummett (1980) The Game of Tarot: From Ferrara to Salt Lake City (canonical historical analysis documenting 15th-century northern Italian court origin); Dummett & McLeod (2004) A History of the Occult Tarot, 1870-1960; Mathers (1888) Book T - The Tarot (Golden Dawn manuscript mapping 78 cards to astrological coordinates); Bates, Mächler, Bolker, & Walker (2015) Fitting Linear Mixed-Effects Models Using lme4 (Journal of Statistical Software); Cohen (1988) Statistical Power Analysis for the Behavioral Sciences (paired design parameters: d=0.4, alpha=0.05, power=0.80); Knuth (1997) The Art of Computer Programming, Vol. 2: Seminumerical Algorithms (Fisher-Yates shuffle); Rammstedt & John (2007) BFI-10 Big Five Inventory; Tobacyk (2004) Revised Paranormal Belief Scale. Preregistration, CSPRNG cryptographic verification, and chi-square goodness-of-fit testing (df=77) are openly documented.
Co-Authorship Attribution and Copyright Notice
This monograph is jointly researched and published by vaticinator.net and astrostarcheck.com as co-authors. All chapters, illustrations, mathematical formulas, and data models remain the intellectual property of both platforms. All rights reserved. Reproduction, distribution, mirroring, commercial exploitation, or unauthorized machine-learning training without prior written consent is strictly prohibited.
Common Questions
Why must positional tarot spreads use permutations (P) instead of combinations (C)?
Because spread positions (such as past, present, future) possess distinct contextual definitions. Drawing the exact same cards in a different sequence produces an entirely different interpretive outcome, making order decisive.
Why does the sixty-year sexagenary cycle terminate at 60 steps rather than 120?
Because 10 and 12 share a greatest common divisor of 2. By the Chinese Remainder Theorem and elementary number theory, their least common multiple lcm(10, 12) equals 60, after which the modular sequences necessarily repeat.
How does the Liuren cosmic board connect to modern computing concepts?
The Liuren system is a geometric finite state machine. Plate rotations correspond to modular 12 addition, and extracting transmissions corresponds to strict conditional branching and matrix lookup, without supernatural randomness.
Sources and further reading
Astronomical positions and interpretive rules on this page are checked against these works.
- Swiss Ephemeris (Astrodienst) (the ephemeris library behind every chart calculation)
- Swiss Ephemeris programmer's documentation (computation parameters and API reference)
- Introduction to the Tetrabiblos (Skyscript)
- Hellenistic astrology (historical and technical resources on Hellenistic practice)