Exact Trace of Signed Two-Strand Braids
Abstract
Every signed power of the two-strand Artin generator has an exact weighted path-trace value.
Theorem 1.1 (All integer crossing exponents).
Proof. Machine-checked in Lean as D5/S3/Quantum/Algebra/Knot/TwoStrandTrace.two_strand_trace (✓ std3). ∎
Source. Repository-derived.
Commentary.
Let L be any finite decidable vertex type, R a decidable symmetric relation, d strictly positive real weights, and delta strictly positive with the exact Perron equality at every vertex. Fix any start s and nonzero complex A satisfying delta = -A squared minus A inverse squared. twoStrand(m) is the m-th integer power of the sole generator in the repository’s actual two-strand Artin presentation. Empty endpoint fibers are retained. The theorem applies to every integer m, including zero and negative values.
Write P for the existing lastProjection on the two-edge path space. The actual represented generator is A inverse times (1-P) plus -A cubed times P. Its inverse has the reciprocal two coefficients. Idempotence gives two orthogonal algebraic sectors P and 1-P, so their coefficients multiply independently. Integer induction through the actual unit and its inverse yields the signed-power formula. No diagonalization, nonempty matrix block or extra matrix-inverse assumption is used.
The existing Perron Markov law gives tau(P) = delta inverse squared and tau(1) = 1. Combining these with the two-sector formula and the exact integer writhe m proves the displayed value. This supplies a concrete evaluation interface downstream of the existing representation and Markov descent. It computes every two-strand torus word and its mirror directly from scalar powers.
The mathematics is classical Temperley-Lieb and Jones theory. The crossing and writhe convention is the one already fixed by WeightedPathArtinRepresentation and its Aharonov-Jones-Landau source record, Library/Topology/aharonovjoneslandau2005jones.md. The result evaluates the formal algebraic Markov invariant. A geometric closure map and convention-preserving Jones identification remain separate obligations. This theorem does not construct the Aiello-Baader-Ferretti modulo-three witness or solve their Question 1.
The four-slot audit is unfinished. Its Reg mirror carries compiled actual-law, variation, sensitivity and dependence proofs, but the current source binding reports IE-C050 source.dictionary_or_proof_coordinate. The exact original theorem telescope and failed binding evidence are tracked in https://github.com/the-omega-institute/trureturing/issues/16285. This is an unresolved registration, not a validated source correspondence.