thermion

Core engine for the Adelic Simplicial Architecture.

Every symmetry. Every scale. Exact.

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Thermion is the computational engine for the Adelic Simplicial Architecture (ASA) — a unified framework in which representation theory, thermodynamics, information geometry, and exceptional Lie algebra all reduce to the same five rewriting operations on typed wires.

A thermion is a charged particle emitted by thermal energy. The name is apt: thermion computes by emitting exact amplitudes from thermal distributions, wherever symmetry acts.

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The five Origami ISA opcodes

The core of thermion is five operations — the Pachner moves of the 3-simplex — that generate all of representation theory:

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

Every result is an exact sympy expression — never a float.

from thermion import flop, flip

flop(0, 1, 1, 1, 1, 1)          # → -1/3  (exact rational)
flip(1) * flop(0, 1, 1, 1, 1, 1)  # → -1/3  (FLIP;FLOP chain)

The Pentagon identity — five FLOPs compose to the identity — is simultaneously Mac Lane's coherence theorem (1963), the Biedenhahn-Elliott identity of nuclear spectroscopy (Racah 1942), and the Pachner 2→3 move of triangulated topology (Pachner 1991). Three communities. One equation.

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The three computational frameworks

MGE — Maslov-Gibbs Einsum

The Maslov-Gibbs Einsum is the thermodynamic bridge between continuous optimisation and discrete logic. At inverse temperature β → ∞, it recovers the tropical (max-plus) semiring — exact discrete logic. At β = 0, it gives the uniform Gibbs distribution — maximum entropy. At finite β, it interpolates: a differentiable, thermally-annealed tensor contraction.

from thermion.core.ensemble import gibbs_weights, partition_function, free_energy

# Route probability mass across 7 channels at inverse temperature β
weights = gibbs_weights(utilities, beta=2.0)

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

MGE unifies eight independent rediscoveries of the same theorem: McFadden (discrete choice), Jaynes (maximum entropy), Gibbs (statistical mechanics), Maslov (tropical geometry), Sims (rational inattention), McKelvey-Palfrey (quantal response), Friston (free energy principle), and Goel (information equilibrium). They all derived the same routing primitive.

Key paper: The Maslov-Gibbs Einsum (doi:10.5281/zenodo.17981393)

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TRS — Topological Resonance Synthesis

Topological Resonance Synthesis is the computational mode of the ASA in which information geometry, holomorphic relaxation, and thermodynamic flow are unified into a single engine. The TRS processor operates on the statistical manifold (𝒫₊, g_Fisher) via Gibbs annealing — parallel transport along the e-geodesic toward the Gibbs fixed point.

The TRS framework identifies three regimes of computation:

• Regime 1 (associative): standard quantum mechanics, Pentagon trivial
• Regime 2 (Origami): Rep(G) rewriting, Pentagon holds, 6j symbols exact
• Regime 3 (Frog): octonion associator, Pentagon fails, PSL(2,7) symmetry

from thermion.core.geometry import FanoGeometry, G2Geometry, AbelianGeometry

# Admissibility filter for Fano-structured routing
geom = FanoGeometry()
masked_utilities = geom.mask(utilities, source=0)

Key paper: Topological Resonance Synthesis (doi:10.5281/zenodo.19858021)

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URN — Unitary Resonance Network

The Unitary Resonance Network is the hardware architecture that implements the Origami ISA at the physical level. The 731-ISA (Paper 258) specifies the machine code; the URN is the resonant substrate that executes it — a network of coupled oscillators whose natural resonances implement the five opcodes.

The Fano plane PG(2,2) is the switching fabric of the 731-register:

from thermion.core.fano import are_collinear, broken_fano_edges, FANO_LINES

# The 7 Fano lines — the connectivity of the 731-ISA register
FANO_LINES  # [(1,2,4), (2,3,5), (3,4,6), (4,5,7), (5,6,1), (6,7,2), (7,1,3)]

# Broken-line 6-731 topology (Paper 325, topological heat engine)
H = broken_fano_edges(source=0, r=0.18)  # J_weak/J_strong = 0.18

Key paper: The 731 ISA (doi:10.5281/zenodo.19916429)

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The regime ladder

Thermion's regime taxonomy is parametrised by the normed division algebra filling the tetrahedron interior (Hurwitz 1898 — the ladder terminates at rung 3):

Rung	Interior	Symmetry	Pentagon	Calculus	Physical instances
0	±1 scalar	ℤ/2ℤ	Trivial	ZX	Qubit circuits, Clifford gates
1	ℝ (6j)	S₄ (order 24)	Holds	Origami	All spectroscopy; spin networks
1+	ℂ (q-6j)	S₄ over ℂ	Holds	q-Origami	Turaev-Viro TQFT; topological QC
3	𝕆 (associator)	PSL(2,7) (order 168)	Fails	Frog/731	FeMo-cofactor; non-assoc. QEC

ZX ⊂ Origami ⊂ Frog (strict inclusions).

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Where the same opcodes appear

The five opcodes compute the same objects across every physical scale:

Domain	Computation	Scale	Paper
Atomic spectroscopy	f-shell coupling, G₂ wall	eV	347
Nuclear spectroscopy	Pandya theorem, ⁹²Mo	MeV	348
Quarkonium / QCD	X(3872) C-parity, FLIP;FLOP	GeV	350
Molecular spectroscopy	CO₂ Fermi resonance	10⁻³ eV	353
Biological (FMO)	Fano efficiency η = 0.1825	10⁻⁶ eV	325
Topological heat engine	Carnot cycle, broken Fano line	—	325
3D quantum gravity	Ponzano-Regge partition function	Planck	349
Financial routing	TIR Gibbs ensemble	—	294
Topological QFT	Turaev-Viro 3-manifold invariants	—	—
Langlands program	Local L-function factors (planned)	—	240

The flop(0,1,1,1,1,1) = -1/3 identity appears in nuclear spectroscopy and QCD charmonium decays at 3 GeV. Same exact rational. The universality is only visible because the result is exact — floats return two slightly different approximations and the identity is obscured.

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Installation

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

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Applications built on thermion

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

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Selected papers

The ASA framework spans 50+ papers. Key foundational works:

The engine:

Paper	DOI
The Maslov-Gibbs Einsum (MGE)	10.5281/zenodo.17981393
Topological Resonance Synthesis (TRS)	10.5281/zenodo.19858021
The 731 ISA (Origami ISA)	10.5281/zenodo.19916429
The Origami Calculus (foundations)	10.5281/zenodo.20474914
Thermodynamic Information Routing (TIR)	10.5281/zenodo.20237288

Spectroscopy (spectrafold):

Paper	DOI
Spiders for Spectra (atomic)	10.5281/zenodo.20458996
Spiders for Nuclei (nuclear)	10.5281/zenodo.20490046
Spiders for Quarkonium (QCD)	10.5281/zenodo.20490294
The Topological Heat Engine (FMO/ribosome)	10.5281/zenodo.20400638

Thermodynamics & economics (econiac):

Paper	DOI
Thermal Economics	10.5281/zenodo.20318505
Economic Gauge Theory	10.5281/zenodo.20259495
Temperature of Rationality	10.5281/zenodo.20234841
EconIAC: MONIAC for the 21st Century	10.5281/zenodo.20315689

Non-associative geometry (rung 3):

Paper	DOI
The Frog Calculus, Part 1	10.5281/zenodo.19713350
The Frog Calculus, Part 2	10.5281/zenodo.20139448
Non-Associative Calculus	10.5281/zenodo.20025384
Virtual Monopoles in FeMo-Cofactor	10.5281/zenodo.20346650

Quantum computing (hardware):

Paper	DOI
The Resonance Processing Unit (RPU)	10.5281/zenodo.19743800
Fibrational Tensor Codes (FTC)	10.5281/zenodo.19821692
In Praise of Qudits	10.5281/zenodo.20269991

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License

MIT. Author: Ian R. C. Buckley — ian.r.c.buckley@gmail.com

"The same equation — the Pentagon identity — was independently discovered and named three times: the Biedenhahn-Elliott identity (spectroscopy, 1952), Mac Lane's coherence condition (category theory, 1963), and the Pachner 2→3 move (topology, 1991). Thermion names the calculus they all describe."