The Origami ISA as Financial Middleware: SPLIT and SPLAT as the Čech Coboundary, Model-Free XVA, and the Unification of Four Computational Paradigms

The Origami ISA is a five-opcode instruction set (SPLIT, SPLAT, FLIP, FLOP, TWIST) originally developed as a universal computational substrate for physics, spanning nuclear spectroscopy to quantum computing across twenty orders of magnitude. This paper shows that the same ISA is the natural computational engine for financial risk: its opcodes implement Čech cohomology on the pricing sheaf of the financial interaction diagram, and every standard financial computation — XVA pricing, contagion modelling, stress testing, and information-based asset pricing — is an ISA programme on a specific sheaf.

Three results are established.

(1) ISA opcodes are Čech operations. SPLIT is the coboundary map $\delta^0: H^0 \to C^1$, taking bilateral prices to their triangular obstruction (the $H^1$ class — convexity, CVA, correlation). SPLAT is integration over the fibre $C^1 \to H^0$, collapsing a triangular risk class to a price (the conditional expectation). TWIST is a gauge transformation on $H^1$ (numeraire change, measure change). FLIP and FLOP are sheaf dualisation and trace (time reversal, discounting). The Pentagon identity — the ISA's coherence axiom — is $d^2 = 0$ for the Čech complex. In finance this is simultaneously the HJM no-arbitrage condition, the no-static-arbitrage condition on volatility surfaces, the tower property of conditional expectations, and the $H^2 = 0$ systemic stability condition. All four are the same equation.

(2) Model-free XVA from SPLIT. Standard XVA computation fits a Gaussian copula and simulates exposure profiles. This is an $H^0$ approximation: the copula assigns a parametric density to the joint default distribution, which is global section ($H^0$) data, not the $H^1$ class. The exact $H^1$ computation — which encodes all correlation effects including wrong-way risk — is obtainable directly from market prices of liquid triangular instruments (CDX/iTraxx tranches, correlation swaps, basket options) by applying SPLIT to the bilateral credit spread matrix. No parametric model. No simulation. The Gaussian copula is the linearisation of this exact computation. The systematic mis-valuation of CDO tranches prior to 2008 follows directly: tranche prices are $H^1$ objects; the Gaussian copula models them as $H^0$ objects.

(3) Pentagon as a runtime invariant. The Pentagon identity is checkable at every step of an ISA programme in $O(|\Gamma|)$ time per period, where $|\Gamma|$ is the number of bilateral exposures. A Pentagon violation flags either a static arbitrage (bilateral prices inconsistent around a triangle) or an incipient $H^2$ systemic event (the system's triangular risks becoming mutually inconsistent). For a regulatory-scale network (1,000 institutions, 10,000 bilateral exposures), this is milliseconds per period on standard hardware — a real-time no-arbitrage and no-cascade monitor requiring only observable market data.

The unification. XVA engines, contagion models, stress tests, and information-based asset pricing (Brody–Hughston–Macrina) are currently separate software paradigms with separate codebases. All four are SPLIT → TWIST → SPLAT pipelines on different sheaves over different financial interaction diagrams: the credit/funding sheaf for XVA, the capital ratio sheaf for contagion, the discount factor sheaf for stress testing, and the information sheaf for IBAP. The EconIAC platform implements all five opcodes as basis-independent Čech operations, unifying the four paradigms under a single codebase. The Pacioli Combinator Library enforces the Pentagon identity at compile time.

Cross-scale universality. The same unification holds across physics: nuclear spectroscopy (Racah 6j = $H^1$ of the $SU(2)$ representation sheaf), quantum computing (magic valence = $H^1$ of the stabiliser sheaf on $W(5,2)$), and quantum gravity (Ponzano–Regge = $H^1$ of the spin foam sheaf) are all ISA programmes on different sheaves. The opcodes and Pentagon identity are identical across all systems. Finance is the sixth instance.

This paper provides the explicit bridge between Paper 370 (The Origami ISA as Nature's Universal Computer) and Papers 396/397 (The 6j Symbol as $H^1$; Systemic Risk as $H^2$), making the ISA↔finance↔EconIAC connection citable and computationally precise.

Keywords

Origami ISA, Čech Cohomology, Sheaf Theory, Financial Middleware, SPLIT Opcode, SPLAT Opcode, Coboundary Map, Pricing Sheaf, Interaction Diagram, Model-Free Pricing, XVA, CVA, FVA, MVA, Wrong-Way Risk, Gaussian Copula, Pentagon Identity, HJM No-Arbitrage, Bilateral Risk, Triangular Risk, Systemic Risk, H⁰, H¹, H², Pachner Moves, Gauge Theory, Pacioli Manifold, EconIAC, Pacioli Combinator Library, Information-Based Asset Pricing, Brody-Hughston-Macrina, Contagion, Fire Sales, Stress Testing, Runtime Invariant, No-Arbitrage Monitor, Pentagon Violation, CDO, Tranche Pricing, 2008 Financial Crisis, Universality, Nuclear Spectroscopy, Quantum Computing, Quantum Gravity, Supply Chains