Author: Ian R. C. Buckley — ORCID 0009-0004-9287-2902
One instruction set. Twenty orders of magnitude. From nuclear spectroscopy to GPU matrix multiplication.
Browse all papers on Zenodo Paradigm Comparison View on GitHub
What is the ASA?
The Adelic Simplicial Architecture is a unified computational framework built around a five-opcode instruction set — the Origami ISA — whose operations correspond exactly to the primitives of Čech cohomology on a sheaf.
The computation works because Lie groups are the natural tape for a generalised Turing machine: the Chladni resonance patterns on the group manifold encode the computational state, and the topology of those nodal lines — their H⁰/H¹/H² structure — is the skeleton of the calculation. What the MGE (Maslov-Gibbs Einsum) does is make that skeleton differentiable: discrete combinatorial models — order statistics, Nash equilibria, circuit measurements, optimal transport plans — become smooth functions of a single temperature parameter β. This is what Topological Resonance Synthesis means.
The same five opcodes appear across twenty orders of magnitude in physical scale:
| System | H⁰ (bilateral) | H¹ (triangular) | Pentagon = |
|---|---|---|---|
| Nuclear spectroscopy | Selection rules | Racah 6j symbol | Biedenharn–Elliott |
| FMO light harvesting | Site energies | Transfer efficiency η | Carnot bound |
| Quantum computing | Pauli syndromes | Magic valence | Pentagon identity |
| Three-body orbits | Kepler solutions | Choreographic solutions | KZ equations |
| Interest rates | Bilateral prices | Convexity (HJM) | No-arbitrage |
| Systemic risk | Bilateral stress | Triangular risk | H² = 0 stability |
| GPU matrix multiply | Tile results | H¹ error certificate | d² = 0 |
This is not analogy. It is the same theorem — the 6j symbol is H¹ of the relevant sheaf — instantiated for different sheaves over different interaction diagrams.
The five opcodes
| Opcode | Move | Role | Instance |
|---|---|---|---|
SPLIT | 1→4 Pachner | δ⁰ coboundary: local → triangular | Bilateral spread → CVA; qubit → 4 amplitudes |
SPLAT | 4→1 Pachner | ∫_fibre: triangular → price | XVA; spectroscopic intensity; conditional expectation |
TWIST | — | Gauge transformation | Numeraire change; phase gate; measure change |
FLIP | 1→3 | Sheaf dualisation | Time reversal; asset → liability |
FLOP | 3→1 | Trace / Born rule | Discounting; probability; expectation |
The Pentagon identity — SPLAT ∘ SPLIT = 0, i.e. d² = 0 — is simultaneously the HJM no-arbitrage condition, the Biedenharn–Elliott identity for angular momentum recoupling, the MIP* verifier constraint, and the H² = 0 stability condition for financial cascades. One equation, four theorems.
The four S-words
The S in ASA stands for four things at once:
- Simplicial — every computation is a Čech complex. Tiles, triangles, tetrahedra. H⁰/H¹/H² classify bilateral, triangular, and systemic structure.
- Symplectic — the Maslov–Gibbs Einsum is a symplectic integrator. Gibbs annealing is parallel transport on an e-geodesic. KAM theory = stabiliser theory.
- Spider — the opcodes are ZX-calculus spiders: SPLIT is Frobenius comultiplication, SPLAT is the counit. Spiders for nuclei, quarkonium, molecules, finance.
- Spectral — the 6j symbol is H¹ of the representation sheaf. Spectroscopic circuits are small (3–21 qubits). The sheaf Laplacian governs both nuclear line intensities and XVA pricing.
The temperature dial
The same five opcodes run at three temperatures:
| Regime | ISA | β | Arithmetic | Complexity |
|---|---|---|---|---|
| Classical | Origami ISA | β → ∞ | Tropical (min, +) | H⁰ |
| Statistical | Forge ISA | 0 < β < ∞ | Real (Gibbs) | H¹ |
| Quantum | Meld ISA | β = it | Complex (amplitudes) | H² |
The bridge between them is the **⊕β softmin** — the operation $a \oplus\beta b = -\frac{1}{\beta}\ln(e^{-\beta a}+e^{-\beta b})$ — which interpolates continuously between standard addition (β → 0) and the tropical minimum (β → ∞). Lowering β is quantisation; raising β is the classical limit. Planck’s constant, viscosity, volatility, softmax temperature, and the quantum group deformation parameter q = e^{iπβ} are all the same object seen from different fields (Paper 443).
Gibbs annealing is parallel transport on the Fisher manifold. The snap event at β* = (3/8)ln(1/(1−ρ)) is the phase transition between exploratory and committed regimes. Shor’s algorithm is a three-layer Origami/Meld/Origami programme. The T-gate is the Fano associator obstruction.
The non-associative frontier (G₂, octonions, the F₄ conjecture) remains active — but it is the boundary of the framework, not its centre.
Paper Index
Papers are grouped by portfolio. The full index is in README.md.
Foundations (A–B)
Quantum Hardware & AI (C)
Quantum Foundations (F)
Spectroscopy & Physics (B|E)
Finance & Economics (G)
Grand Challenges (E)