The chord data is a function of the pairing

Systems with the same boundary pairing have the same chord-crossing count, hence the same path sign: pathSign telescopes freely along pairing-preserving blocks, and around any pairing-returning loop of repairs the crossing count returns — the parity backbone of the holonomy bookkeeping.

theorem RS.EdgeSubset.chordCross_of_samePairing {α : Type} [LinearOrder α] {W : Fragment α} {F : EdgeSubset W} {κ κ' : F.RelTransitionSystem} (h : SamePairing κ κ') (b b' : ↥F.boundaryFlags) : ChordCross κ b b' ↔ ChordCross κ' b b'

The crossing relation only sees the pairing.

theorem RS.EdgeSubset.chordCrossingCount_of_samePairing {α : Type} [LinearOrder α] {W : Fragment α} {F : EdgeSubset W} {κ κ' : F.RelTransitionSystem} (h : SamePairing κ κ') : chordCrossingCount κ = chordCrossingCount κ'

The crossing count is a pairing invariant.

theorem RS.EdgeSubset.pathSign_of_samePairing {α : Type} [LinearOrder α] {W : Fragment α} {F : EdgeSubset W} {κ κ' : F.RelTransitionSystem} (h : SamePairing κ κ') : pathSign κ = pathSign κ'

The path sign is a pairing invariant: around any pairing-returning block the chord signs cancel.