The exact ordered trace bridge
Abstract
The ordered integer matrix trace is exactly the Boolean phase sum for every positive length.
Theorem 1.1 (The ordered trace is the exact Boolean phase sum).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/CyclicBoolean/CyclicSignTrace/Trace.sourceTrace_eq_phase_sum (✓ std3). ∎
Source. Repository-derived.
Acknowledgement. ricardorr (MathOverflow user 49534; real-name attribution unverified) (2026). Does the support of a cyclic trace of 2x2 sign matrices always span a hyperplane in Z_2^n?. URL: https://mathoverflow.net/questions/512655.
Commentary.
At every positive length n and for all words b,a, the literal ordered matrix trace is the sum over every x of sign(dot(b,x)+dot(a,chi(x))). Length one and all coincident cyclic indices are included. This identity also applies to even lengths; oddness is needed only for the inverse construction.
For a list of r+1 matrices, expand its (i,j) entry over all r intermediate states. The row tuple is i followed by the intermediate states; the column tuple is those states followed by j. Induction uses the first matrix factor and the ordinary Matrix.mul entry formula, preserving the complete list order. Taking the trace identifies j with i. Joining the initial state to the intermediate tuple gives every cyclic path exactly once; the final column is the initial state, including for a one-factor product.
The path product is the sign of the sum of its entry exponents. Reindex the a_(k-2)x_k term by k=i+2 and the a_(k-1)x_k x_(k+1) term by k=i+1. The coefficient of a_i becomes x_i+x_(i+2)+x_(i+1)x_(i+2), exactly chi(x)_i. These are bijections of the actual cyclic index set, so repeated indices need no separate independence assumption.