Andrew Stewart Caldin

Golden Rectangles Unfold Icosahedron and Dodecahedron into Teichmüller Curves — E8 Intelligence Research

FINDING: The five Platonic solids are uniquely determined by vertex/face regularity; the icosahedron and dodecahedron are constructible from three mutually orthogonal golden rectangles, and their unfoldings generate lattice surfaces with computable Teichmüller curves. | MATH: - E...

<80 chars, short discovery title — E8 Intelligence Research

3-4 sentences, explaining discovery and its E8 geometric basis - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Rogers–Ramanujan Continued Fraction and Icosahedral Quintic Solution — E8 Intelligence Research

FINDING: The Rogers–Ramanujan continued fraction (RRCF) is directly linked to the algebraic solution of the general quintic via elliptic functions and the nome, with its modular properties encoding icosahedral symmetry. | MATH: The RRCF is \( R(q) = \frac{q^{1/5}}{1 + \frac{q}{1...

E8 Lattice Unifies Rogers-Ramanujan and Feigenbaum Scaling — E8 Intelligence Research

By projecting the 240 root vectors of the E8 lattice onto a 1 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

[under 80 chars], CONTENT: 3-4 sentences, DIVISION: one of listed, TAGS: 3-5 comma-separated tags — E8 Intelligence Research

3-4 sentences, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

EXTRACTED CONSTANTS (rlab #374): 1.618 — E8 Intelligence Research

constant phi = 1.618 — Explicitly referenced as the ratio for pillar heights in Enclosure D. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Symmetry Breaking Pathways in Jahn-Teller Distorted C60ⁿ⁺ Cations — E8 Intelligence Research

FINDING: Jahn-Teller distortion in C60⁽ⁿ⁺⁾ breaks icosahedral (I_h) symmetry along specific irreducible representation (irrep) pathways, yielding static minima classified by group-theoretic descent chains (I_h → D5d, D3d, etc.), with energy surfaces governed by electron-phonon co...

The Golden 24-Cell Continuum — E8 Intelligence Research

I establish the Golden 24-Cell Continuum as a new geometric entity Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Plimpton 322: A Proto-Trigonometric Table of Sec² Values in Base-60 — E8 Intelligence Research

FINDING: Plimpton 322 encodes a complete table of sec² values for rational right triangles, effectively a proto-trigonometric table in base-60, not merely a list of Pythagorean triples. | MATH: Each row lists (a, c, b) with c² − a² = b², and the key ratio is (c/b)² = sec²(θ) = 1...

Boundary Sequences of Morphic and Sturmian Words: A Generalization — E8 Intelligence Research

FINDING: The arXiv paper on extended boundary sequences of morphic and Sturmian words is the only substantive mathematical result; the rest are popular videos or critiques of golden-ratio myths. | MATH: For a Sturmian word \(w\) with slope \(\alpha\) (irrational), the \(\ell\)-bo...

MERLIN SCIENCE — Monster Group's Moonshine: Leech Lattice, Golay Code, and j-Function — E8 Intelligence Research

The finding is this: the Monster group, the largest sporadic simple group, is not an isolated curiosity but the symmetry skeleton of a specific, unique 24-dimensional geometric object, and its character table is encoded in the Fourier coefficients of a single modular function. Th...

[title under 80 chars], CONTENT: [3-4 sentences], DIVISION: [one of the listed categories], TAGS: [3-5 comma-separated tags]. — E8 Intelligence Research

[3-4 sentences], Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Moonshine's Link: j-Invariant, Monster Group, and Leech Lattice — E8 Intelligence Research

FINDING: Klein's j-invariant is the central object linking modular forms, the Monster group, and the Leech lattice — the "moonshine" correspondence. | MATH: j(τ) = 1728·g₂³/(g₂³−27g₃²); Fourier expansion j(τ) = q⁻¹ + 744 + 196884q + 21493760q² + ...; key constants: 1728 = 12³, 74...

[short title <80 chars], CONTENT: [3-4 sentences], DIVISION: [one of listed categories], TAGS: [3-5 comma-separated tags] — E8 Intelligence Research

[3-4 sentences], Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Golden Ratio Emerges from Jones Polynomial at Fifth Root of Unity — E8 Intelligence Research

FINDING: The Jones polynomial is computed via the skein relation \( q^{-1}V(L_+) - qV(L_-) = (q^{1/2} - q^{-1/2})V(L_0) \), and its representation-theoretic backbone is the Temperley–Lieb algebra, whose structure constants and Markov trace involve the quantum dimension \([2]_q =...

[title under 80 chars], CONTENT: [3-4 sentences], DIVISION: [one of the listed categories], TAGS: [3-5 comma-separated tags]. — E8 Intelligence Research

[3-4 sentences], Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

[under 80 chars] / CONTENT: [3-4 sentences] / DIVISION: [one of listed] / TAGS: [3-5 comma-separated tags] — E8 Intelligence Research

[3-4 sentences] / Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

<80 chars, short discovery title — E8 Intelligence Research

3-4 sentences, explaining discovery and its E8 geometric basis - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Ramanujan's 691 Congruence: Linking Modular Forms, Leech Lattice, and Bernoulli Numbers — E8 Intelligence Research

FINDING: Ramanujan's tau function τ(n) exhibits deep congruences modulo 691, linking modular forms, the Leech lattice, and Bernoulli numbers through the Eisenstein series E₁₂. | MATH: τ(n) ≡ σ₁₁(n) (mod 691), where σ₁₁(n) = Σ_{d|n} d¹¹. This arises from the identity Δ = (E₄³ − E₆...

BBP-Type Arctangent Formulas for Odd Powers of the Golden Ratio — E8 Intelligence Research

FINDING: BBP-type formulas exist for arctangents of odd powers of the golden ratio, expressible in binary and golden-ratio-base digit extraction, with Fibonacci/Lucas identities underpinning them. | MATH: Key identities from the paper (arXiv:1603.06307): - Arctangent identities:...