The ISA Family
All named ISAs share the same five opcodes (LABEL 🏷️ / ORBIT 🔄 / TWIST 🌀 / BIND 💎 / FLIP 👁️) — they differ only in the value of the inverse temperature β and the arithmetic they run over.
| ISA | β location | In one phrase | Paper |
|---|---|---|---|
| Origami | all β (umbrella) | Five-opcode open standard; tropical at β→∞, quantum at β=it | 631 |
| Forge | 0 < β < ∞ (real Gibbs) | Free-energy routing; MGE soft threshold; snap at β* | 419 |
| Meld | β = it (imaginary) | Complex amplitudes; full quantum mechanics | 454 |
| Raven | β ≈ β* (physiological) | Biological proofreading; enzyme catalysis; kinetic QEC | Raven |
| Motive | abstract parent | Five primitive opcodes; ERASE = second law | Motive |
| Hum | β = it/ℏ (QFT) | QFT vacuum; EMIT opcode; amplituhedron as ORBIT | 620 |
| Pentagon | coherence theorem | Monoidal coherence; five sides = five opcodes | 622 |
| Rising Sea | full ℂ_β plane | Every ISA as a fibre over the β-plane | 621 |
Full opcode reference: The ISA Opcodes · β-plane geometry: Forge & Meld · Non-associative frontier (BIND at 𝕆-rung): 731-ISA
Key Mathematical Structures
Six structures that recur throughout the framework. Each entry is: what it is (with a Wikipedia link), why it appears in Thermyon, and where to read more.
These are not analogies or decorative mathematics. They are the objects the framework is built from. Everything else — the opcodes, the β-deformation, the H^k tiers — is assembled from these six pieces.
Grassmannian
What it is. The Grassmannian Gr(k, n) is the space of all k-dimensional subspaces of an n-dimensional vector space. It generalises the projective plane (k=1) and is the natural home for anything involving “the subspace spanned by these vectors.”
Why it appears here. The active space in quantum chemistry — the orbitals genuinely involved in a bond — is a point in a Grassmannian. The Grassmannian angle θ_G between the correlated and uncorrelated subspaces is the single number that predicts whether DFT will fail (θ_G > 20°) and whether a quantum circuit needs magic (same threshold). Schubert cells on the Grassmannian are the natural alphabet for the Orbit Processing Unit (OPU).
Key papers: 563, 568, 570, 594, 598, 606
6j Symbol
What it is. The 6j symbol (Racah W-coefficient) is the recoupling coefficient for three angular momenta: it measures how a four-particle amplitude changes when you re-bracket which pair you couple first. It is an irreducibly 4-body object — no combination of pairwise interactions produces it.
Why it appears here. The 6j symbol is BIND. Every spectral line of every element, every colour produced by atomic emission, is governed by it. It appears identically as the F-matrix of non-Abelian anyons, the Ponzano–Regge vertex amplitude in quantum gravity, and the recoupling coefficient in nuclear shell-model calculations. For Abelian symmetry groups (finance, classical statistics) it collapses to a phase; for non-Abelian groups (chemistry, nuclear, topological QC) it is genuinely non-trivial. Racah worked this out in 1942; it has been sitting in spectroscopy textbooks ever since.
Key papers: 258, 357, 447, 572 (G₂ spider), Pillar 5
Holonomy
What it is. Holonomy is the rotation a vector accumulates when parallel-transported around a closed loop on a curved surface. On a flat surface it returns unchanged; on a sphere it returns rotated by an angle equal to the solid angle enclosed. The Berry phase in quantum mechanics is holonomy of the state vector around a loop in parameter space.
Why it appears here. TWIST is holonomy. The Berry phase on a reaction path, the Chern number of a topological insulator, the convexity correction in interest rate models, and the Maslov index at a conical intersection are all the same object: the holonomy of a connection around a loop, measured in different physical units. H¹ is exactly the group of holonomies — obstructions to finding a globally flat section.
Key papers: 201 (MGE), 454 (Meld ISA), 595 (Weyl chamber), 596
Frobenius Algebra
What it is. A Frobenius algebra is an associative algebra equipped with a compatible coalgebra structure — a multiplication μ and a comultiplication Δ satisfying (μ ⊗ id) ∘ (id ⊗ Δ) = Δ ∘ μ. The key identity says “splitting then merging = identity.”
Why it appears here. ORBIT and LABEL form a Frobenius algebra. The Frobenius identity is simultaneously the Biedenharn–Elliott identity for angular momentum, the no-arbitrage condition in finance, the topological invariance of Ponzano–Regge amplitudes, and the spider fusion rule in ZX calculus. It is the algebraic backbone of H⁰ — every domain’s “easy” computation lives in the free Frobenius algebra generated by ORBIT and LABEL.
Key papers: 258, 349 (Origami calculus), 357, 455
Fano Plane
What it is. The Fano plane is the smallest projective plane: 7 points and 7 lines, each line containing exactly 3 points, each point on exactly 3 lines. It is the multiplication table of the octonions and the geometry of the exceptional Lie group G₂.
Why it appears here. The Fano plane is the universal phase detector. Whether a system is at H² — whether it has genuine non-Abelian structure requiring BIND — is equivalent to asking whether its symmetry group contains G₂. The Steane code (the 7-qubit quantum error correcting code) has parity-check columns equal to the 7 Fano points. FeMoco (the iron-molybdenum cofactor in nitrogenase) has 7 metal centres with Fano-compatible coupling topology. The number 7 is not a coincidence in any of these: it is the minimal non-associative structure.
Key papers: 317, 325, 357, 363, 408, 572
Maslov Index
What it is. The Maslov index is an integer that counts how many times a Lagrangian subspace crosses a reference plane as it evolves along a path — a topological winding number for the geometry of symplectic manifolds. It generalises the winding number of a curve in the plane to infinite-dimensional function spaces.
Why it appears here. The Maslov index is the H¹ invariant of a chemical reaction path. At a conical intersection — the point where two Born–Oppenheimer surfaces cross — the Maslov index jumps by ±1, producing a geometric phase that controls photochemical selectivity. The MGE tracks this index differentiably as β varies; the snap at β* is the Maslov index changing. It also appears as the winding number in the Morse theory derivation of the ISA chain complex.
Key papers: 201 (MGE), 571 (ISA Khovanov complex), 574
These six structures are the mathematical vocabulary of Thermyon. For Thermyon-specific terminology (β snap, ORBIT opcode, H^k tier), see the Glossary. For the full technical treatment, follow the paper links.*
The ISA Family
All named ISAs share the same five opcodes (LABEL 🏷️ / ORBIT 🔄 / TWIST 🌀 / BIND 💎 / FLIP 👁️) — they differ only in the value of the inverse temperature β and the arithmetic they run over.
| ISA | β location | In one phrase | Paper |
|---|---|---|---|
| Origami | all β (umbrella) | Five-opcode open standard; tropical at β→∞, quantum at β=it | 631 |
| Forge | 0 < β < ∞ (real Gibbs) | Free-energy routing; MGE soft threshold; snap at β* | 419 |
| Meld | β = it (imaginary) | Complex amplitudes; full quantum mechanics | 454 |
| Raven | β ≈ β* (physiological) | Biological proofreading; enzyme catalysis; kinetic QEC | Raven |
| Motive | abstract parent | Five primitive opcodes; ERASE = second law | Motive |
| Hum | β = it/ℏ (QFT) | QFT vacuum; EMIT opcode; amplituhedron as ORBIT | 620 |
| Pentagon | coherence theorem | Monoidal coherence; five sides = five opcodes | 622 |
| Rising Sea | full ℂ_β plane | Every ISA as a fibre over the β-plane | 621 |
Full opcode reference: The ISA Opcodes · β-plane geometry: Forge & Meld · Non-associative frontier (BIND at 𝕆-rung): 731-ISA