The Origami ISA as a Structural Model for MIP* Verification: Pentagon Identity, Frobenius Axiom, and Fano Geometry as Analogues of Entangled-Prover Rigidity Tests

The landmark theorem MIP* = RE establishes that a classical verifier interacting with multiple quantum provers sharing entanglement can verify any recursively enumerable language. The algebraic rigidity underlying this result — exact-or-fail constraint tests with no partial credit — is structurally reminiscent of the coherence conditions governing the Origami Instruction Set Architecture.

This paper articulates three precise structural parallels:

• The Frobenius axiom (SPLIT(j) ∘ SPLAT(j) = 1) shares the all-or-nothing character of the local Pauli basis test in MIP* verification.
• The Pentagon identity shares the global path-independence character of the low-degree test.
• The Fano plane PG(2,2) is the natural geometry of the GHZ stabiliser group, which forms a quantum strategy satisfying all seven Fano constraints simultaneously while no classical strategy can.

These parallels are verified numerically: Pentagon unitarity holds exactly for all j ∈ {1/2,...,15/2}; Frobenius fidelity degrades as (1-p)^{2(2j+1)} under depolarising noise; G₂ non-associativity produces a nonzero excluded-volume fraction above j = 7/2.

The paper does not claim to prove that the Origami ISA is MIP* verification — it conjectures a precise structural correspondence and presents numerical evidence as motivation for a future theorem. The practical implication is a conjecture that fault-tolerant quantum computing overhead is governed by the regime of the code's F-move amplitudes, with MSD-free computation possible at Rung-1 and higher.

Keywords

MIP* Equals RE, Interactive Proof Systems, Quantum Entanglement, Pentagon Identity, Frobenius Axiom, Fano Plane, PG(2,2), GHZ Stabiliser, Origami ISA, Wigner 6j Symbol, Fault-Tolerant Quantum Computing, Magic State Distillation, Regime Ladder, Complexity Theory, Quantum Prover, Local Pauli Test, Low-Degree Test, G₂ Non-Associativity, Excluded Volume, Depolarising Noise, Rung-1, Quantum Verification, Adelic Simplicial Architecture