The Anyon Amplitude Is an Algebraic Integer: Exact, Cross-Validated, Kernel-Checked Topological Quantum Computation

Abstract

A Fibonacci-anyon quantum computer evaluates the Jones polynomial of a knot at a fifth root of unity; the value is the amplitude of a braiding process. We show that this value is an algebraic integer—it lives in the cyclotomic ring Z[ζ5 ]—and that a classical machine can therefore compute it exactly, with no floating point and no Hilbert space, and a proof kernel can certify that it did. The braiding gates become 2 × 2 matrices over Z[ζ5 ] in which the golden ratio satisfies φ2 = φ + 1 as an arithmetic identity; the Kauffman loop value is φ on the nose. We build the amplitude two independent ways—an algebraic Temperley–Lieb trace and a topological state sum that counts loops in a closed braid diagram—and the two agree, bit for bit, on the unknot, the Hopf link, the trefoil, and the figure-eight (the first knot that is not a torus link). Writhe normalization turns the framed value into a genuine link invariant, and we prove its invariance: under Reidemeister I and Markov conjugation—in general, for every braid, by showing the matrix trace over Z[ζ5 ] is cyclic—and under Markov stabilization of either sign for every braid, since the union–find loop count is sign-independent and the spectator surgery closes σ and σ −1 alike (framing factors −A3 and −A−3 ). A diagrammatic runtime recovers the Temperley–Lieb relations not as axioms but as theorems that fall out of contracting wires and collapsing loops. We prove, in the kernel, that the loop count the topological engine returns is the number of connected components of the closed braid diagram—a verified union–find theory whose load-bearing fact is that one merge step drops the component count by exactly one—which reduces the last step of the general invariance theorem to a single component-count comparison. Against an independent floating-point recomputation the exact values agree to 1.3 × 10−15 , the residual being the float error, not ours. The naive state sum is exponential, but it need not be: a confluent reduction that carries the whole Temperley–Lieb element and merges equal diagrams computes the same amplitude in time polynomial in the crossings for braids of bounded strand width, and the kernel certifies it equals the exponential sum—for a torus link σ1k , a collapse from 2k to O(k). Every result is machine-checked in Cubical Agda. We do not claim a quantum speedup, and we do not overturn the #P-hardness of the general problem; what we show is that the classical amplitude is an exact object, that exactness can be certified, and that the certified computation is—in the bounded-width regime—exponentially faster than the simulation it replaces. The same construction carries the Ising anyon over the ring Z[ζ16 ]: the two theories are two evaluation points of one bracket polynomial, and the gap between the universal model and √ the classically-simulable one is, at the level of the ring, the gap between φ2 = φ + 1 and ( 2)2 = 2. Because equality in Z[ζ5 ] is decidable, the exact invariant decides topology a float cannot: we prove the trefoil is chiral—its invariant differs from its mirror's, exactly—and the figure-eight amphichiral. Published by The Institute for Applied Ontological Mathematics.