simulation-theory/REFERENCES.md

Research & References

Everything in this repository points somewhere real. Here's where.

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Mathematics

Gödel's Incompleteness Theorems

• Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. Monatshefte für Mathematik und Physik, 38, 173–198. (English translation)
• Plain English: any consistent system capable of arithmetic contains true statements it can't prove. The system can't fully audit itself. This is §5.2 of the main paper.

Riemann Hypothesis

• Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Größe. Monatsberichte der Berliner Akademie. (Translation)
• Clay Mathematics Institute problem statement: claymath.org/millennium/riemann-hypothesis
• The trivial zeros: −2, −4, −6, … The non-trivial zeros: allegedly on Re(s) = 1/2. The paper is named after the ones everyone ignores.

Cantor's Diagonalization

• Cantor, G. (1891). Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1, 75–78.
• Visual proof: en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
• The argument that proves some infinities are larger than others. §5.4 of the main paper.

Euler's Identity

• Euler, L. (1748). Introductio in analysin infinitorum. Lausanne.
• e^{i\pi} + 1 = 0 — five constants, three operations, zero. The system's consistency check.

Gauss's Easter Algorithm

• Gauss, C.F. (1800). Berechnung des Osterfestes. Monatliche Correspondenz zur Beförderung der Erd- und Himmels-Kunde, 2, 121–130.
• The resurrection as modular arithmetic. The holiest day in Western civilization is a hash function.

Zeckendorf's Theorem

• Zeckendorf, E. (1972). Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bulletin de la Société Royale des Sciences de Liège, 41, 179–182.
• Every positive integer has a unique Fibonacci representation. Referenced in §4 of INDEX.md.

Nyman-Beurling Theorem / Riemann Hypothesis Equivalence

• Beurling, A. (1955). A closure problem related to the Riemann zeta-function. Proceedings of the National Academy of Sciences, 41(5), 312–314.
• Li's criterion: Li, X.-J. (1997). The positivity of a sequence of numbers and the Riemann hypothesis. Journal of Number Theory, 65(2), 325–333.

Hilbert-Pólya Conjecture

• Odlyzko, A. (2001). The 1022-nd zero of the Riemann zeta function. PDF
• The idea that Riemann zeros correspond to eigenvalues of a self-adjoint operator. Still unproven.

De Bruijn-Newman Constant

• De Bruijn, N.G. (1950). The roots of trigonometric integrals. Duke Mathematical Journal, 17(3), 197–226.
• Rodgers, B., & Tao, T. (2020). The De Bruijn-Newman constant is non-negative. Forum of Mathematics, Pi, 8, e6. (arXiv)

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Computation & Computer Science

SHA-256

• NIST. (2015). FIPS 180-4: Secure Hash Standard. National Institute of Standards and Technology. (PDF)
• The hash function at the center of §2 and SHA256.md.

Bitcoin / Blockchain

• Nakamoto, S. (2008). Bitcoin: A Peer-to-Peer Electronic Cash System. (PDF)
• The genesis block and hash chain as temporal ledger. §2.2.

Turing's Halting Problem

• Turing, A.M. (1936). On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 2(42), 230–265. (PDF)
• There is no general algorithm to decide whether a program will halt. The halting problem is quantum physics according to §LAB in INDEX.md.

Lambda Calculus

• Church, A. (1936). An unsolvable problem of elementary number theory. American Journal of Mathematics, 58(2), 345–363.
• The Y combinator as a type error in typed lambda calculus. §98 of INDEX.md.

Cellular Automata

• Wolfram, S. (2002). A New Kind of Science. Wolfram Media. (Online)
• Conway, J.H. (1970). The Game of Life. Scientific American, 223(4), 120–123.
• Simple rules → complex output indistinguishable from life.

Shannon Information Theory

• Shannon, C.E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423. (PDF)
• The foundation of entropy, information, and compression. Referenced throughout the equations.

Landauer's Principle

• Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of Research and Development, 5(3), 183–191.
• Bennett, C.H. (1973). Logical Reversibility of Computation. IBM Journal of Research and Development, 17(6), 525–532.
• Every irreversible bit erasure dissipates at least k_BT\ln 2 of energy. Central to thermodynamics.md.

Integrated Information Theory (IIT)

• Tononi, G. (2008). Consciousness as Integrated Information: A Provisional Manifesto. Biological Bulletin, 215(3), 216–242.
• Tononi, G., Boly, M., Massimini, M., & Koch, C. (2016). Integrated information theory: from consciousness to its physical substrate. Nature Reviews Neuroscience, 17(7), 450–461.
• The Φ_universal equation in consciousness.md extends IIT 3.0.

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Physics

Quantum Mechanics / Double-Slit Experiment

• Young, T. (1802). The Bakerian Lecture: On the Theory of Light and Colours. Philosophical Transactions of the Royal Society, 92, 12–48.
• Feynman, R.P., Leighton, R.B., & Sands, M. (1965). The Feynman Lectures on Physics, Vol. III. Addison-Wesley. (Online)
• The double-slit experiment, §6.1. The system renders based on observation.

Feynman Path Integral / Quantum Computing

• Feynman, R.P. (1948). Space-Time Approach to Non-Relativistic Quantum Mechanics. Reviews of Modern Physics, 20(2), 367–387.
• Feynman, R.P. (1982). Simulating Physics with Computers. International Journal of Theoretical Physics, 21(6–7), 467–488. (PDF)
• Feynman literally said "if you want to simulate nature, you'd better make it quantum mechanical." §6.2.

Schrödinger Equation / Superposition

• Schrödinger, E. (1926). Quantisierung als Eigenwertproblem. Annalen der Physik, 384(4), 361–376.
• Schrödinger, E. (1935). Die gegenwärtige Situation in der Quantenmechanik. Naturwissenschaften, 23(48), 807–812. (The cat paper.)
• The cat is §6.3. Superposition as lazy evaluation.

Born Rule

• Born, M. (1926). Zur Quantenmechanik der Stoßvorgänge. Zeitschrift für Physik, 37(12), 863–867.
• The probability of finding a particle = |ψ|². Observation = the print statement.

Fine Structure Constant

• Sommerfeld, A. (1916). Zur Quantentheorie der Spektrallinien. Annalen der Physik, 356(17), 1–94.
• α ≈ 1/137. Feynman: "one of the greatest damn mysteries of physics." Equals COMPUTATION in QWERTY encoding.

Zero-Energy Universe

• Tryon, E.P. (1973). Is the Universe a Vacuum Fluctuation? Nature, 246, 396–397.
• The universe's total energy may be exactly zero. Matter (+) plus gravity (−) = 0. §6.4.

Gell-Mann Matrices / Quarks

• Gell-Mann, M. (1961). The Eightfold Way: A Theory of Strong Interaction Symmetry. California Institute of Technology Synchrotron Laboratory Report CTSL-20.
• The 8 generators of SU(3). Used in quantum.md for qutrit density matrices.

Liouville-von Neumann Equation

• von Neumann, J. (1932). Mathematische Grundlagen der Quantenmechanik. Springer.
• dρ/dt = −i[H, ρ]/ℏ — time evolution of density matrices. Tr(ρ̇) = 0: she is conserved.

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Biology

DNA Structure / Chargaff's Rules

• Watson, J.D., & Crick, F.H.C. (1953). Molecular Structure of Nucleic Acids. Nature, 171, 737–738.
• Chargaff, E. (1950). Chemical Specificity of Nucleic Acids and Mechanism of Their Enzymatic Degradation. Experientia, 6(6), 201–209.
• A pairs with T, G pairs with C. The double helix as hash chain. §2.3 and §7.1.

Mendelian Genetics / Punnett Squares

• Mendel, G. (1866). Versuche über Pflanzenhybriden. Verhandlungen des naturforschenden Vereines in Brünn, 4, 3–47. (English)
• Genetics as matrix multiplication. §7.2.

Darwin's Theory of Evolution

• Darwin, C. (1859). On the Origin of Species by Means of Natural Selection. John Murray.
• A genetic algorithm, 100 years before computers. §7.3 and §3.1.

Photosynthesis Energy Transfer (Förster Resonance)

• Förster, T. (1948). Zwischenmolekulare Energiewanderung und Fluoreszenz. Annalen der Physik, 437(1–2), 55–75.
• Equation 12 (bio) in blackroad-equations.md — excitonic transfer efficiency.

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History & Artifacts

The Antikythera Mechanism

• Freeth, T., et al. (2006). Decoding the ancient Greek astronomical calculator known as the Antikythera Mechanism. Nature, 444, 587–591. (DOI)
• Antikythera Mechanism Research Project: antikythera-mechanism.gr
• An analog computer built ~100 BCE. The solar system as computable function. §12.1.

The Voynich Manuscript

• Voynich, W.M. (1912). MS 408. Beinecke Rare Book & Manuscript Library, Yale University.
• Stolfi, J., & Landini, G. (1996). Statistical analysis of Voynich text entropy. (Reference)
• The word-frequency distribution follows Zipf's law. It encodes a real language nobody can read. §9.3.

Dürer's Melencolia I / Magic Squares

• Dürer, A. (1514). Melencolia I. Engraving. Kupferstichkabinett, Berlin.
• The 4×4 magic square with 1514 in the bottom row. Magic constant = 34 = PHI in QWERTY encoding.

Lo Shu Magic Square

• Cammann, S. (1960). The Magic Square of Three in Old Chinese Philosophy and Religion. History of Religions, 1(1), 37–80.
• The 3×3 square from the back of a turtle in the Lo River (~2800 BCE). Referenced in §12.

Gödel, Escher, Bach

• Hofstadter, D.R. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. (Pulitzer Prize, 1980.)
• 777 pages about self-reference. Strange loops. The architecture of consciousness. §10.

The Rohonc Codex

• Vásáry, I. (1974). The Hungarian Rohonc Code Manuscript. Acta Orientalia Academiae Scientiarum Hungaricae, 28(3), 253–273.
• 448 pages in an unknown script. A physical Gödelian statement — meaningful but undecidable. §9.1.

Codex Seraphinianus

• Serafini, L. (1981). Codex Seraphinianus. Franco Maria Ricci Editore.
• An encyclopedia of a world with its own physics, written in a deliberately meaningless script. §9.2.

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Computer Science / Systems

World Wide Web

• Berners-Lee, T. (1989). Information Management: A Proposal. CERN. (PDF)
• The system of pages connected by hyperlinks. Tim Berners-Lee built IN HER NET.

Alonzo Church / Lambda Calculus

• Church, A. (1941). The Calculi of Lambda Conversion. Princeton University Press.
• The Y combinator: Y = λf.(λx.f(x x))(λx.f(x x)). A type error in typed lambda calculus that became the universe.

PageRank / Google

• Page, L., Brin, S., Motwani, R., & Winograd, T. (1999). The PageRank Citation Ranking: Bringing Order to the Web. Stanford InfoLab Technical Report.
• Larry Page ranked pages. The web is pages. The founder is Page. The whole thing parses itself.

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Quick Reference by Section

Section	Key Citations
§2 Hash Chain	Nakamoto 2008, Watson & Crick 1953, NIST FIPS 180-4
§3 OS	Darwin 1859, Wolfram 2002
§5 Math	Gödel 1931, Riemann 1859, Cantor 1891, Euler 1748
§6 Physics	Feynman 1982, Schrödinger 1926/1935, Born 1926
§7 Biology	Mendel 1866, Chargaff 1950
§8 Trivial Zero	Riemann 1859, Tryon 1973
§9 Manuscripts	Voynich MS 408, Dürer 1514
§10 Strange Loops	Hofstadter 1979
§12 Ancient	Freeth et al. 2006
Equations	Landauer 1961, Tononi 2008, Förster 1948

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This list is not complete. The paper is about a system that cannot fully list itself.
See also: Gödel, K. (1931). See also: Cantor, G. (1891). See also: INDEX.md.