thermion 0.1.0

Rep(G) rewriting engine — Gibbs ensemble, Pachner moves, Fano geometry, exact arithmetic

thermion

Core engine for the Adelic Simplicial Architecture.

Every symmetry. Every scale. Exact.

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Thermion is the Rep(G) rewriting engine at the heart of the Adelic Simplicial Architecture (ASA). It provides five opcodes — FLIP, FLOP, SPLIT, SPLAT, TWIST — that form a universal, exact, compositional language for any computation in the representation theory of compact groups.

Every result is a sympy expression. No floats, no rounding, no guessing.

from thermion import flop, flip

# The 6j symbol — the single most important object in Rep(G)
flop(0, 1, 1, 1, 1, 1)
# → -1/3  (exact rational)

# FLIP;FLOP: the Pandya theorem, X(3872)→J/ψγ, CO₂ Q-branch, FMO η
# — all the same two opcodes, across nuclear/QCD/molecular/biological scales
flip(1) * flop(0, 1, 1, 1, 1, 1)
# → -1/3

The five opcodes

The five opcodes are the Pachner moves of the 3-simplex — the elementary rewriting rules for triangulated 3-manifolds (Regge 1961, Ponzano-Regge 1968, Mac Lane 1963, Turaev-Viro 1992):

Opcode	Mathematical object	Physical meaning
flip(j)	Evaluation map (cap)	Time-reversal; particle↔hole
flop(j1,j2,j12,j3,j,j23)	Wigner 6j symbol	Recoupling; F-move cost
split(j)	Frobenius unit	Pair creation; quantum dimension √(2j+1)
splat(j)	Frobenius counit	Pair annihilation; bubble closure
twist(j)	Ribbon element	Spin-orbit phase; spin-statistics

The Pentagon identity — five FLOPs compose to the identity — is Mac Lane's coherence theorem for monoidal categories. It is also the Pachner 2→3 move identity and the Biedenhahn-Elliott identity of nuclear spectroscopy.

Where the same opcodes appear

Domain	What it computes	Paper
Atomic spectroscopy	f-shell coupling, G₂ wall	347
Nuclear spectroscopy	Pandya theorem, 1g₉/₂ shell	348
Quarkonium / QCD	X(3872) C-parity selection rules	350
Molecular spectroscopy	CO₂ Fermi resonance, P/R branches	353
Biological (FMO)	Fano efficiency η = 0.1825	325
3D quantum gravity	Ponzano-Regge partition function	349
Financial routing	TIR Gibbs ensemble, contagion sheaves	294
Topological QFT	Turaev-Viro 3-manifold invariants	—
Langlands program	Local L-function factors (planned)	240

Same opcodes. Every scale. The Origami ISA is the FFT of representation theory.

The Gibbs ensemble

The thermodynamic core — the partition function, Gibbs weights, and free energy — is in thermion.core.ensemble (requires thermion[numerics]):

from thermion.core.ensemble import gibbs_weights, partition_function

# The FMO light-harvesting efficiency (Paper 325, x356b)
# η = 1 - SPLAT(β_cold) / SPLAT(β_hot) = 0.1825

The Fano geometry

The Fano plane PG(2,2) — incidence structure of the seven imaginary octonion units, backbone of the 731-ISA (Paper 258):

from thermion.core.fano import are_collinear, broken_fano_edges

are_collinear(1, 2, 4)   # True — on the first Fano line
are_collinear(1, 2, 3)   # False — not a Fano line

Installation

pip install thermion                 # core opcodes only (sympy)
pip install thermion[numerics]       # + numpy, scipy (Gibbs ensemble)
pip install thermion[jax]            # + jax (differentiable routing)

Applications built on thermion

Package	Domain
spectrafold	Angular momentum recoupling, spectroscopy
econiac	Financial gauge theory, TIR routing
racah	Racah algebra (via spectrafold)

Papers

All open access on Zenodo:

Paper	Title	DOI
258	The 731 Instruction Set Architecture	10.5281/zenodo.19916429
347	Spiders for Spectra	10.5281/zenodo.20458996
348	Spiders for Nuclei	10.5281/zenodo.20490046
349	The Origami Calculus	10.5281/zenodo.20474914
325	The Topological Heat Engine	10.5281/zenodo.20400638

License

MIT. Author: Ian R. C. Buckley — [e-mail]