The Origami ISA: EconIAC's Computational Engine
EconIAC's cohomological risk computations — bilateral, triangular, systemic — are not implemented as ad-hoc algorithms. They are instances of a single five-opcode instruction set called the Origami ISA, whose opcodes correspond exactly to the operations of Čech cohomology on a sheaf.
This page explains the connection. No prior knowledge of the ISA or of cohomology is required; the bilateral · triangular · systemic page covers the financial meaning, and the primer develops it from scratch.
The five opcodes and their cohomological meaning
The Origami ISA has five primitive operations. Each has a precise mathematical identity and a concrete financial implementation in EconIAC.
| Opcode | Move | Cohomology operation | Financial meaning | EconIAC |
|---|---|---|---|---|
SPLIT |
\(1 \to 4\) Pachner | \(\delta^0: H^0 \to C^1\) — coboundary | Bilateral price → triangular obstruction (the \(H^1\) class) | finance.cohomology.split() |
SPLAT |
\(4 \to 1\) Pachner | \(\int_\text{fibre}: C^1 \to H^0\) — integration over fibre | Triangular risk → price (conditional expectation, SPLAT = pricing map) | finance.cohomology.splat() |
TWIST |
— | Gauge transformation on \(H^1\) | Numeraire change; measure change; change of collateral posting convention | finance.cohomology.twist() |
FLIP |
\(1 \to 3\) | Sheaf dualisation \(\mathcal{F} \to \mathcal{F}^\vee\) | Time reversal; ket → bra; asset → liability in double-entry | finance.cohomology.flip() |
FLOP |
\(3 \to 1\) | Trace \(\mathcal{F}^\vee \otimes \mathcal{F} \to \mathbf{1}\) | Discounting; probability rule; taking expectation | finance.cohomology.flop() |
The Pentagon identity — the consistency condition that governs the ISA — is the statement \(d^2 = 0\) for the Čech complex: the coboundary of a coboundary is zero. In finance this is:
- The HJM no-arbitrage condition (discount factors compose consistently)
- The no-static-arbitrage condition on the volatility surface
- The tower property of conditional expectations (martingale pricing)
- The \(H^2 = 0\) stability condition (no systemic cascade)
All four are the same equation. The ISA enforces it by construction.
Why this matters for EconIAC
Model-free by construction
Because the ISA opcodes are basis-independent Čech operations, EconIAC's cohomological risk computations are model-free: they derive from the topology of the financial interaction diagram and the prices of observable instruments, not from parametric assumptions about distributions or dynamics.
Standard XVA engines fit a Gaussian copula and compute CVA from it.
EconIAC's SPLIT applies the coboundary map to the bilateral credit spread
matrix and reads off the \(H^1\) class directly from market prices. No copula.
No calibration. The \(H^1\) class is the correlation structure.
Compositionality
The opcodes compose. A sequence of ISA operations is a circuit on the financial interaction diagram. Econiac's Pacioli Combinator Library (PCL) provides the typed DSL for writing such circuits:
from econiac.pcl import split, splat, twist, flip, flop, sequence
# Price a triangular risk: split bilateral prices → H¹ class → price
xva_circuit = sequence(
split(bilateral_exposures), # H⁰ → H¹: find the triangular obstruction
twist(numeraire="risk_neutral"), # gauge transform to pricing measure
splat(conditional_expectation) # H¹ → H⁰: collapse to price
)
Every circuit preserves the Pacioli identity (conservation of value) by construction — the types enforce it.
The Pentagon as a runtime check
The Pentagon identity is checkable at runtime. If a sequence of ISA operations
violates \(d^2 = 0\), the EconIAC runtime raises a PentagonViolation — the
financial equivalent of a type error. This catches:
- Calendar spread arbitrage in a volatility surface
- HJM drift misspecification in an interest rate model
- No-arbitrage violation in a credit spread matrix
- \(H^2 \neq 0\) system fragility in a stress scenario
The ISA across scales
The Origami ISA is not specific to finance. Paper 370 (The Origami ISA as Nature's Universal Computer) shows the same five opcodes govern physical processes across 20 orders of magnitude — from nuclear spectroscopy to quantum computing to molecular energy transfer.
The reason: all of these systems share the same mathematical structure (a representation sheaf on an interaction diagram with a Pentagon topology). The ISA is the universal computational language for this structure.
EconIAC is the financial instantiation. The five opcodes, the Pentagon identity, and the cohomological hierarchy (\(H^0/H^1/H^2\)) are the same objects in finance as in physics — just with different sheaves:
| System | Sheaf | \(H^0\) | \(H^1\) | \(H^2 = 0\) condition |
|---|---|---|---|---|
| Nuclear spectroscopy | \(SU(2)\) representation sheaf | Selection rules | Racah 6j symbol | Biedenharn–Elliott |
| Quantum computing | Stabiliser sheaf on \(W(5,2)\) | Pauli syndromes | Magic valence | Pentagon identity |
| Interest rates | Discount factor sheaf | Bilateral prices | Convexity (HJM) | HJM no-arbitrage |
| Systemic risk | Pricing sheaf on interaction diagram | Bilateral stress | Triangular risk | \(H^2\) stability |
| Supply chains | Input-output sheaf | Sector balances | Cross-sector coupling | Stock-flow consistency |
The universality is not an analogy. It is the same theorem — the 6j symbol is \(H^1\) of the relevant sheaf — instantiated for different sheaves.
The connection to Paper 395 (IBAP as Origami ISA)
The Brody–Hughston–Macrina information-based asset pricing (IBAP) framework is an Origami ISA programme:
- SPLIT: hidden market factor \(X\) → noisy signal \(\sigma t X\) + Brownian bridge \(\beta_{tT}\) (the information process)
- SPLAT: information field \(\xi_t\) → asset price \(H_t\) (conditional expectation)
- TWIST: numeraire change (risk-neutral → terminal measure)
- FLIP/FLOP: time reversal and Born rule
The Brownian bridge \(\beta_{tT}\) is parallel transport on the prequantum bundle; its holonomy is the convexity adjustment — the same \(H^1\) class that appears in HJM. Paper 395 (in preparation) develops this in full.
Further reading
| Paper | Content |
|---|---|
| 370 — The Origami ISA as Nature's Universal Computer | Five opcodes across 20 orders of magnitude; universality proof |
| 396 — The Unhedgeability Theorem | ISA opcodes as Čech cohomology operations (§6); five-instance table |
| 397 — Systemic Risk as \(H^2\) | ISA applied to systemic risk; Pentagon = HJM = stability |
| 303 — Pacioli Combinator Library | The typed DSL for ISA circuits in EconIAC |
| 393 — Projective Geometry as the Mother Tongue of QM | ISA as the finite-field limit of the Penrose transform |
| Paper 395 — IBAP as Origami ISA (in preparation) | Brownian bridge = parallel transport; SPLIT/SPLAT = BHM information process |