Computing Paradigm Comparison: Classical vs Quantum vs ASA
Author: Ian R. C. Buckley Last updated: 2026-06-13 Scope: Three-way comparison of the full computing stack across Classical, Quantum (gate-model), and Adelic Simplicial Architecture (ASA / Resonance / Fano / Origami) paradigms.
The table is not a list of unrelated claims. The Freudenthal-Tits Magic Square provides a single generating structure — the four division algebras $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ tensored with Jordan algebras — that fills in all exceptional Lie groups simultaneously. Classical computing sits at the $\mathbb{R}$-corner; standard quantum computing at the $\mathbb{C}$-corner; ASA at the $\mathbb{O}$-corner. The table is therefore an algebraic ladder read at three different rungs.
Stack Comparison
| Layer | Classical Computing | Quantum (Gate Model) | ASA / Resonance / Fano / Origami |
|---|---|---|---|
| Foundational algebra | Associative Boolean algebra; ring arithmetic over $\mathbb{Z}/\mathbb{R}$ | Associative complex matrix algebra; Hilbert space over $\mathbb{C}$ | Non-associative octonion algebra; $G_2$ exceptional symmetry over $\mathbb{O}$ — Paper 211: Non-Associative Calculus · Paper 207: 731-Calculus |
| Foundational geometry | Euclidean / Cartesian grid; flat metric | Hilbert space projective geometry; Bloch sphere | Fano plane $PG(2,2)$; co-associative $G_2$ structure on $\mathbb{R}^7$ — Paper 200: Fano-Foam Manifold · Paper 263: Magic Square |
| Number system | Real / integer arithmetic | Complex amplitudes | Adelic arithmetic: $\mathbb{R} \times \prod_p \mathbb{Z}_p$ (real + all $p$-adic completions simultaneously) — Paper 000: Adelic Invitation |
| State representation | Bit string ${0,1}^n$ | Qubit density matrix $\rho \in \mathcal{H}^{\otimes n}$ | Rank-1 projector in $\mathfrak{J}_3(\mathbb{O})$; Peirce decomposition $1 + 16 + 10 = 27$ dimensions — Paper 257: NA-QEC · Paper 202: TRS |
| The exceptional core | None | None | The 16-dimensional Peirce-$\tfrac{1}{2}$ subspace $\mathcal{J}_{1/2}(P)$: invisible to all associative devices, spanned by Fano associators, the unforgeable quantum working register — Paper 235: Fano-Token · Paper 257: NA-QEC · Paper 258: Origami ISA |
| Hardware substrate | CMOS transistors; SRAM/DRAM | Superconducting transmons; trapped ions; photonic qubits | Resonance Processing Unit (RPU): PT-symmetric optical cavities, $G_2$ metamaterial lattices, 731-RPU register architecture — Paper 205: RPU |
| Register architecture | 64-bit integer / float registers; cache hierarchy | Qubit state; stabiliser tableau | Peirce registers: $\mathcal{J}_{1/2}$ (16-dim quantum working register) + $\mathcal{J}_1$ (1-dim output) + $\mathcal{J}_0$ (10-dim ancilla) — Paper 258: Origami ISA v2 |
| Gates / elementary operations | NAND, NOR, XOR; Boolean logic | Clifford gates (H, CNOT, S); non-Clifford T gate | Pachner moves (2↔3 and 1↔4 simplicial surgery); octonion left/right multiplication $L_{e_i}$, $R_{e_i}$ — Paper 258: Origami ISA |
| Instruction set architecture | x86 / ARM; RISC-V | OpenQASM; Quil; QASM3 | 731 Origami ISA: ■ Split, ◇ Splat, ▲ Flip, ▷ Flop (Pachner opcodes) + ↻ Twist + SPIN (triality) + BIND (octonion product) — Paper 258: Origami ISA · Paper 411: Pulse Sequences |
| Calculus / differentiation | Real calculus; autodiff (backpropagation); Jacobian | No standard differential calculus; parameter-shift rule | Non-associative octonion calculus: right-division norm $|f’_{(x)}|$, Cauchy-Fueter regularity, chain rule with associator correction — Paper 211: Non-Associative Calculus |
| Diagrammatic calculus | Circuit diagrams; dataflow graphs | ZX-calculus (Coecke & Duncan); ZH-calculus | 731-calculus: magmoidal string diagrams; co-associative 4-form $\psi = \star\varphi$ as rewrite rule — Paper 207: 731-Calculus |
| Routing / interconnect | TCP/IP; PCIe; Ring All-Reduce (GPU clusters) | Quantum teleportation; Bell-pair distribution; quantum repeaters | Secure State Routing (SSR): Fano-plane geometric routing; Excluded Volume Principle rejects non-Fano writes in $O(1)$ — Paper 200: Fano-Foam Manifold · Paper 208: Magmoidal Cipher |
| Memory / storage | RAM; cache hierarchy; virtual memory paging | Quantum RAM (qRAM, proposed); stabiliser tableau | Simplicial Paging: resolved sub-graphs compressed to 0-skeleton pointers; page-in/out via Pachner opcodes — Paper 258: Origami ISA |
| Compiler / transpiler | LLVM; GCC; MLIR | Qiskit transpiler; tket; circuit optimisation | 731 Compiler: Pachner surgery sequences → iSWAP gate decompositions; MGE tensor contraction compiler; typed DSL with Schur-boundary enforcement — Paper 258: Origami ISA · Paper 412: Typed DSL · Paper 217: LS2.0 |
| Error correction | Hamming codes; RAID; CRC; ECC DRAM | Surface code; Steane $[[7,1,3]]$ code; toric code | Fibrational Tensor Codes (FTC): $G_2$ fibre bundle stabilisers; $\mathcal{J}_{1/2}$ protected by Map Collapse theorem — Paper 206: FTC · Paper 257: NA-QEC · Paper 217: LS2.0 |
| Code switching / runtime | JIT compilation; dynamic dispatch | Magic State Distillation; code switching via lattice surgery | Fibrational Lattice Surgery (LS2.0): 2D→3D Pachner 2-3 move; iSWAP-native, no magic state factory — Paper 217: LS2.0 · Paper 210: Geometric Code Switching |
| Cryptography / security | RSA; AES; SHA-256 (associative group operations) | BB84 / E91 QKD; Kyber/CRYSTALS (lattice-based PQC) | Magmoidal Cipher: non-associative sequential lock — Paper 208: Magmoidal Cipher |
| Quantum money / unforgeability | Classical banknotes (no unforgeability proof) | Wiesner (1970/1983); Aaronson-Christiano hidden subspace money | Fano-Token: NA No-Cloning via Target Obstruction + Map Collapse; $\mathcal{J}_{1/2}$ exceptional core as structurally unforgeable register — Paper 235: Fano-Token |
| Thermodynamics / energy model | von Neumann / Landauer bit erasure; $k_BT\ln 2$ per bit flip | Adiabatic quantum computation; quantum annealing (D-Wave) | Maslov-Gibbs Einsum (MGE): thermodynamic routing via tropical Boltzmann weights; BOIL (explore) → SNAP (crystallise) phase transition — Paper 201: MGE · Paper 202: TRS |
| Phase transitions | None (classical transitions are continuous) | None in gate model; thermal in annealing | Fano-Token frozen/liquid phases: $G_2$-stable (associator $= 0$) vs $SO(7)\setminus G_2$ (associator $= 2$); $\beta$-threshold $\beta^* = \tfrac{3}{8}\ln(1/\varepsilon - 1)$ — Paper 235: Fano-Token · Paper 221: Fano-Fisher |
| Optimisation / learning | SGD; Adam; backpropagation | QAOA; VQE; quantum natural gradient | Tropical SGD: asynchronous gradient routing via MGE; differentiable Fano graph optimisation — Paper 201: MGE |
| AI / fine-tuning | Full-parameter fine-tuning; LoRA; RLHF | (No established quantum equivalent) | URN: Fano-Fisher Topological Immune System; gradient projection onto 10-dim Information Valley; machine-precision zero drift on 4-dim Information Ridge — Paper 203: URN · Paper 221: Fano-Fisher |
| Chaos / scrambling | N/A (classical chaos is deterministic) | SYK model; OTOCs; MSS chaos bound $\lambda_L = 2\pi k_BT/\hbar$ | Fano-SYK model: geometric impedance $R_\infty \approx 2.72$ on pre-thermal scrambling; Fano structure suppresses chaos by factor ~3 — Paper 267: Fano-SYK |
| Holography / bulk-boundary | N/A | AdS/CFT; $p$-adic Bruhat-Tits trees (Gubser 2017) | Bruhat-Tits building as boundary of $G_2$ bulk; non-associative fermionic couplings as holographic impedance — Paper 267: Fano-SYK |
| Foundations of logic | Classical two-valued logic; Boolean satisfiability | Quantum logic (Birkhoff–von Neumann); contextuality | Non-associative logic: Hardy paradox as Fano-line contextuality; Fano Monogamy Paradox; Spacelike Associator Paradox — Paper 268 · Paper 269 · Paper 270 |
| Grand unified structure | No unification framework | No agreed unification | Freudenthal-Tits Magic Square: division algebras $\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ × Jordan algebras → all exceptional Lie groups as one table — Paper 263: Magic Square |
| Number theory connection | Modular arithmetic; discrete logarithm | Quantum phase estimation; hidden subgroup problem | $\mathfrak{J}_3(\mathbb{O})$ linked to Bruhat-Tits cospan; $F_4$–Riemann Hypothesis conjecture; Apéry generalisations — Paper 240 · Paper 265 · Paper 266 |
| Chemistry / materials | Density functional theory; molecular dynamics | Variational quantum eigensolver (VQE) for molecules | Nuclear magic numbers and electron shell structure from exceptional Lie algebras $G_2$, $F_4$, $E_6$ — Paper 209 · Paper 210 |
| Biological molecular machines | Bioinformatics; molecular dynamics; force fields | Quantum biology (photosynthesis coherence debate) | Ribosome, FMO complex, and nitrogenase each run broken-Fano Origami ISA programmes; FMO efficiency $\eta = 0.1825$ is a topological theorem (unique among 7-node graphs); nitrogenase N≡N cleavage requires SPIN opcode ($G_2$ triality) — Paper 413: Molecular Machines · Paper 324: Decoding Engine · Paper 325: Topological Heat Engine · Paper 318: FeMo-Cofactor |
| Quantum gravity | General relativity; lattice QCD (background) | Loop quantum gravity (spin foams); string theory | Every LQG spin foam model is an Origami ISA circuit: SPLAT = 6j vertex amplitude, Pentagon identity = Biedenhahn-Elliott identity; $G_2$ extension may fix Barbero-Immirzi parameter $\gamma_\mathrm{BI}^{G_2} = \sqrt{8/3}/(2\pi\ln 14) \approx 0.09848$ — Paper 410: Spin Foams as Origami |
| Resource hierarchy (quantum) | N/A | Stabiliser states / magic states (T-gate resource) | Four-level hierarchy: stabiliser / dark magic (TV=1) / genuine magic (TV<1) / associamancy ($\nu_2 = 0$, SPIN opcode); Schur boundary separates Levels 1a/1b from Level 2; G₂ holonomic gates topologically protected (Weyl group $D_6$, order 12) — Paper 469: ISA Completeness · Paper 470: Hot Logic · Paper 407: Associamancy · Paper 408: Fano Primer |
| Network / consensus protocol | Paxos; Raft; Spanner (leader-based total order) | Quantum Byzantine agreement (proposed) | Simplicial consensus: Excluded Volume Principle as $O(1)$ geometric conflict rejection; no leader required — Paper 200: Fano-Foam Manifold |
| Introductory / survey | Knuth, TAOCP; Sipser, Theory of Computation | Nielsen & Chuang, QC & QI | Paper 000: An Adelic Invitation |
Rows Not Yet Covered by Published Papers
| Layer | Classical | Quantum | ASA (gap) |
|---|---|---|---|
| Operating system | Linux / Windows kernel; process scheduling | No agreed quantum OS | TRS acts as scheduler (BOIL/SNAP cycle), but no dedicated OS-layer paper |
| Networking stack (layers 1–7) | OSI model; Ethernet; HTTP | BB84 at layer 1; no full quantum OSI | SSR covers layers 3–4; layer 1–2 for RPU hardware not yet formalised |
| Formal verification | Coq; Lean; TLA⁺ | Quantum Hoare logic; QPMC | Partial: typed ISA DSL (Paper 412) gives runtime type checking; Pentagon identity is a typing rule; Schur boundary is a compile-time TypeError; full static checker (Agda/Idris formalisation) is future work |
| Biological computing | DNA computing; neuromorphic (Intel Loihi) | Quantum biology (photosynthesis, avian compass) |
Notes
The ZX-calculus comparison. ZX-calculus (quantum row, diagrammatic calculus) is associative: spider fusion and the bialgebra law are equations between associative string diagrams. The 731-calculus is genuinely non-associative: Pachner move rewrite rules do not commute, and the magmoidal string diagram formalism captures this. The two calculi are not competitors — ZX lives at the $\mathbb{C}$ rung, 731 at $\mathbb{O}$.
The Peirce-½ subspace as the common thread. The row labelled “The exceptional core” is the single concept that unifies the security (Paper 235), error correction (Paper 257), register architecture (Paper 258), and information geometry (Paper 221) columns. In every case the result is the same: the 16-dimensional subspace $\mathcal{J}_{1/2}(P)$ is the part of $\mathfrak{J}_3(\mathbb{O})$ that no associative device can reach. Classical noise, classical measurement, and classical cloning attempts all fail at the same algebraic boundary.
The symmetry chain. The security, thermodynamic, and quantum gravity rows share a common geometric picture: \(PSL(2,7) \;\hookrightarrow\; G_2 \;\hookrightarrow\; \mathrm{Spin}(8) \;\hookrightarrow\; SO(7) \;\hookrightarrow\; F_4 = \mathrm{Aut}(\mathfrak{J}_3(\mathbb{O}))\) The key structural fact: $G_2 = \mathrm{Fix}(\text{triality in }\mathrm{Spin}(8))$ — $G_2$ is precisely the subgroup of $\mathrm{Spin}(8)$ that fixes a chosen triality decomposition. Triality is the order-3 outer automorphism of $\mathrm{Spin}(8)$ that cyclically permutes its three 8-dimensional representations ($8_v$, $8_s$, $8_c$); it is unique to $\mathrm{Spin}(8)$ among all Spin groups. The SPIN opcode implements triality restricted to $G_2$. The 731 ISA is triality-complete; the Origami ISA (regime 2) is not.
$G_2$-stable transformations preserve all 7 Fano lines (frozen phase, associator $= 0$); transformations in $SO(7)\setminus G_2$ break at least one line (liquid phase, associator $= 2$). The $PSL(2,7)$ discrete skeleton (168 elements, 56 order-3 triality generators) is the fingerprint of any genuine octonionic state. No associative algebra has a symmetry group with this discrete-inside-continuous structure.