Adelic Simplicial Architecture (ASA)
Author: Ian R. C. Buckley
ORCID: 0009-0004-9287-2902
Zenodo community: asa-research
What ASA is
The A is Adelic: the framework operates simultaneously at every prime and at the archimedean place, treating the division algebra ladder ℝ → ℂ → ℍ → 𝕆 as a single graduated object rather than four separate theories.
The S is four things at once:
| S-word | What it means in ASA |
|---|---|
| Simplicial | Every computation is a Čech complex on a nerve. Tiles, triangles, tetrahedra. H⁰ = bilateral, H¹ = triangular, H² = systemic. The Pentagon identity is d² = 0. |
| Symplectic | The Maslov–Gibbs Einsum is a symplectic integrator. Gibbs annealing is parallel transport on an e-geodesic. The ASA phase space is symplectic; KAM theory = stabiliser theory. |
| Spider | The Origami ISA opcodes are spiders in the ZX calculus sense: SPLIT is the Frobenius comultiplication, SPLAT is the counit, TWIST is the phase. 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 A is Architecture: a unified instruction set (the Origami ISA) that compiles all four S-structures into the same five opcodes — SPLIT, SPLAT, TWIST, FLIP, FLOP — executable on quantum hardware, classical GPUs, or financial risk engines.
The framework grew from non-associative computing across the ℝ→ℂ→ℍ→𝕆 division algebra ladder. The associative sector (regime 2 of the ISA) now dominates the application papers, but the non-associative residual — the 0.32% of 3NF that lives in the G₂ sector, the F₄ conjecture on J³(𝕆), the octonion calculus — remains an active frontier.
The five opcodes
| Opcode | Pachner move | Mathematical role | Instances |
|---|---|---|---|
SPLIT | 1→4 | δ⁰ coboundary | Bilateral → triangular risk; qubit → 4 amplitudes; nucleus → Racah 6j |
SPLAT | 4→1 | ∫_fibre integration | XVA price; conditional expectation; spectroscopic intensity |
TWIST | — | Gauge transformation | Numeraire change; phase gate; measure change |
FLIP | 1→3 | Sheaf dualisation | Time reversal; ket → bra; asset → liability |
FLOP | 3→1 | Trace / Born rule | Discounting; probability; expectation |
Pentagon identity (SPLAT ∘ SPLIT = 0, i.e. d² = 0) is simultaneously: the HJM no-arbitrage condition · the Biedenharn–Elliott identity · the MIP* verifier constraint · the H² = 0 stability condition for financial cascades.
Paper Index
Foundations (Portfolio A–B)
Quantum Hardware & AI (Portfolio C)
Quantum Foundations (Portfolio F)
Spectroscopy & Physics (Portfolio B|E)
Finance & Economics (Portfolio G)
Grand Challenges (Portfolio E)
| # | Title | DOI |
|---|---|---|
| 240 | Structural Observations on J³(O) | 19824028 |
| 252 | Quantum Fano-Biology (draft) | — |
| 265 | The ζ(21) Apéry Generalisation | 20029647 |
| 266 | Geometric Shadows in Apéry’s Polynomial | 20031913 |
| 297 | Euler-Mascheroni Geometry | 20139443 |
Portfolio Map
| Portfolio | Theme | Representative papers |
|---|---|---|
| A — Core Engine | MGE, TRS, non-associative calculus | 201, 202, 211, 267, 370 |
| B — Foundations | Algebra, simplicial topology, category theory | 200, 207, 258, 263, 386, 393 |
| C — Hardware & AI | RPU, FTCs, Q-VM, Origami ISA registers | 199, 205, 206, 217, 303, 385 |
| D — Protocols | Non-associative cryptography | 208 |
| E — Grand Challenges | Riemann hypothesis, number theory, biology | 240, 252, 265, 266, 297 |
| F — Quantum Foundations | Paradoxes, magic, contextuality, self-tests | 268–270, 357–390 |
| G — Finance | Gauge theory of risk, XVA, cohomological stress tests | 289–316, 396–399 |
The universality table
The same five opcodes appear across twenty orders of magnitude in scale:
| System | Sheaf | H⁰ | H¹ | Pentagon = |
|---|---|---|---|---|
| Nuclear spectroscopy | SU(2) repr. sheaf | Selection rules | Racah 6j (line intensities) | Biedenharn–Elliott |
| FMO light harvesting | 7-site Fano sheaf | Site energies | Transfer efficiency η=0.1828 | Carnot bound |
| Quantum computing | Stabiliser sheaf | Pauli syndromes | Magic valence | Pentagon identity |
| Three-body problem | Braid group sheaf | Kepler orbits | Choreographic solutions | KZ equations |
| Interest rates | Discount factor sheaf | Bilateral prices | Convexity (HJM) | HJM no-arbitrage |
| Systemic risk | Pricing sheaf | Bilateral stress | Triangular risk | H² = 0 stability |
| GPU matrix multiply | Tiling sheaf | Tile results | H¹ error certificate | d² = 0 |
The last row — GPU matrix multiplication — is Paper 400 (in preparation): the tiling sheaf of a blocked GEMM has H¹ = 0 for exact arithmetic and H¹ ≠ 0 for approximate arithmetic (FP8, INT8, Gibbs sampling). The H¹ norm is a cheap O(mn/p) runtime error certificate, tight to 1/√3 across all 225 GEMMs in a LLaMA-7B forward pass.
Repository Structure
papers/ ← Markdown explainers (one folder per concept DOI)
code/ ← Experiment code (one subfolder per concept DOI)
_data/ ← papers.csv, papers.json (machine-readable index)
README.md ← This index
Total papers: ~400 (Papers 200–399 in the main series; Papers 000 and 000-series as introductions).
Full machine-readable index: _data/papers.csv · _data/papers.json