The 731 Instruction Set Architecture (Origami ISA): Machine Code, Pachner Opcodes, Geometric Constraint Satisfaction, and Simplicial Paging

Paper: 258 Portfolio C — Quantum Hardware

DOI: 10.5281/zenodo.19916429

Abstract

Specifies the Origami ISA: a machine-code level instruction set for the 731-RPU (Resonance Processing Unit) using Pachner moves as opcodes and simplicial paging for memory management.

v2.0 adds the Peirce Register Architecture: the full 27-dimensional state space $\mathfrak{J}3(\mathbb{O}) = \mathcal{J}_1(P) \oplus \mathcal{J}{1/2}(P) \oplus \mathcal{J}_0(P)$ is identified as three distinct registers — the 16-dimensional Peirce-½ exceptional core as the quantum working register, the 1-dimensional $\mathcal{J}_1$ as the output register, and the 10-dimensional $\mathcal{J}_0$ as the classical ancilla. Algebraic noise protection (from Paper 235 Theorem 3.2) is distinguished from thermodynamic error suppression.

Key Results

  • 4 base opcodes: ■ Split ($1\to 4$ Pachner), ◇ Splat ($4\to 1$), ▲ Flip ($2\to 3$), ▷ Flop ($3\to 2$)
  • Peirce register architecture (v2.0): $\mathcal{J}_{1/2}(P)$ as quantum working register; Fano-Slots ($e_1\ldots e_7$) are its generators
  • Simplicial Paging: saturated Fano-crystals compressed to 0-skeleton pointers; constant VRAM overhead
  • Error suppression: Associator Penalty thermodynamically disfavours non-$PSL(2,7)$ states; reinforced by algebraic protection of $\mathcal{J}_{1/2}(P)$

Opcode Symbol Table

The ISA uses a Unicode visual alphabet. Each symbol encodes its semantics: filled shapes (■ ▲) are creation operators ($\alpha^\dagger$, add geometric mass); hollow shapes (◇ ▷) are annihilation operators ($\alpha$, erase geometric mass). The outer shape encodes the Pachner type: 4-sided (diamond/square) = 1↔4 stellar move, 3-sided (triangle) = 2↔3 bistellar flip.

Symbol Unicode Verb Move Academic home
■ U+25A0 Split $1 \to 4$ mesh refinement, subdivision surfaces
◇ U+25C7 Splat $4 \to 1$ 3D Gaussian splatting, volume rendering
▲ U+25B2 Flip $2 \to 3$ PL topology, Delaunay, Mori MMP
▷ U+25B7 Flop $3 \to 2$ Mori minimal model programme
↻ U+21BB Twist BOIL/SNAP thermodynamic scheduling

Mnemonic: filled shapes (■ ▲) have a flat horizontal base — they sit and build. Hollow shapes (◇ ▷) have a rightward point — they lean and release. Split/Splat rhyme (stellar pair); Flip/Flop rhyme (bistellar pair).

The self-duality of $G_2$ is encoded directly in the symbols: the coroot isomorphism $\sigma$ acts as “hollow out and tip rightward”, sending ■ $\mapsto$ ◇ and ▲ $\mapsto$ ▷. The Pachner unitarity identities are:

\[◇ \circ ■ = \mathrm{id}, \qquad ▷ \circ ▲ = \mathrm{id}\]

See Paper 271 for the full algebraic development.

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