W3 / kappa_u Sign Correlation

Purpose

This note closes the finite calculation that was left open in the topological kernel:

  • how the sign of the DIII winding number W3 tracks the sign of the transport coupling kappa_u.

The result is narrower than “topology derives kappa_u > 0.” What it establishes is:

W3 and kappa_u flip under the same global orientation reversal, so their relative sign convention is fixed once one physical orientation is chosen.

That is enough to remove the sign-correlation item as a free ambiguity. It is not enough to derive which orientation is physically selected.

Input 1: the DIII winding sign flips under q -> q^\dagger

In the DIII paper, the reduced chiral Hamiltonian is

\[Q(X)= \begin{pmatrix} 0 & q(X) \\ q(X)^\dagger & 0 \end{pmatrix},\]

with

\[q(X)=X_0\mathbf{1}_2-i(X_1\sigma^1+X_2\sigma^2+X_3\sigma^3).\]

The paper already proves that, in the current winding convention,

\[W_3[q] = -1,\]

and also records that:

  • replacing q by q^\dagger flips the sign;
  • reversing the orientation on S^3 flips the sign.

So

\[W_3[q^\dagger] = +1\]

in the same convention.

This is the topological Z2 we need.

Input 2: the transport sign flips under u -> -u

The discrete-symmetry ledger already records:

\[u \mapsto -u \qquad \Longrightarrow \qquad \kappa_u \mapsto -\kappa_u.\]

This is the transport-side Z2.

So the remaining issue is whether the topological sign flip and the transport sign flip are the same global orientation reversal in the corpus sign dictionary.

Reduced orientation flip sends q to q^\dagger

The reduced sector split is the J^{01} decomposition

\[\mathbf 4 = T1 \oplus T2.\]

In the chiral basis used in the DIII paper, the Hamiltonian is off-diagonal with upper-right block q and lower-left block q^\dagger.

Now swap the two chiral sectors by the permutation matrix

\[P = \begin{pmatrix} 0 & \mathbf 1_2 \\ \mathbf 1_2 & 0 \end{pmatrix}.\]

Then

\[P^{-1}Q(X)P = \begin{pmatrix} 0 & q(X)^\dagger \\ q(X) & 0 \end{pmatrix}.\]

So the reduced orientation flip that swaps the two J^{01} sectors sends the topological block

\[q \longleftrightarrow q^\dagger.\]

Because W_3[q^\dagger] = -W_3[q], the reduced orientation flip reverses the winding sign.

Global orientation dictionary

The corpus already treats the remaining sign ambiguity as a common global orientation issue involving:

  1. the upstream orientation of u,
  2. the reduced sector naming / orientation,
  3. the readout arrow.

Within that dictionary:

  • u -> -u flips kappa_u;
  • the induced reduced orientation flip swaps T1 <-> T2;
  • the reduced topological block changes q <-> q^\dagger;
  • therefore the winding sign flips W3 -> -W3.

So under the common global orientation reversal,

\[(W_3,\kappa_u) \mapsto (-W_3,-\kappa_u).\]

This is the precise sign-correlation statement.

What is now established

The following statement is now closed:

once the corpus fixes a physical orientation convention, the sign of W3 and the sign of kappa_u are no longer independent; they track the same global Z2.

Equivalently:

  • constructive and inverted transport are exchanged by the same global reversal that exchanges W3 = +1 and W3 = -1;
  • the relative sign convention between topology and transport is fixed.

In particular, if one adopts the operational orientation rule

\[\kappa_u > 0\]

for the physical readout branch, then one should normalize the topological sign so that the same physical branch is assigned

\[W_3 = +1\]

even though the current paper’s raw convention gives W_3[q] = -1 for the un-reoriented block.

This is not a contradiction. It is just the statement that the paper’s raw winding convention and the physical readout orientation need one common sign choice.

What is not established

This note does not prove:

  • that topology alone derives kappa_u > 0;
  • that the constructive branch is physically selected without extra readout / ambient / stability input;
  • that the operational orientation rule has become a theorem.

Those stronger statements still belong to the orientation-upgrade files.

So the clean division is:

  • closed here: relative sign tracking W3 <-> kappa_u;
  • still open elsewhere: why the physical branch must be the constructive one.

Bottom line

The sign-correlation problem is now reduced to a convention-matching statement rather than a missing calculation.

  • W3[q] = -1 and W3[q^\dagger] = +1 in the current winding convention.
  • u -> -u sends kappa_u -> -kappa_u.
  • the reduced orientation flip swaps q <-> q^\dagger.
  • therefore the common global orientation reversal sends \((W_3,\kappa_u) \mapsto (-W_3,-\kappa_u).\)

So the repo can now safely say: the sign of W3 tracks the sign of kappa_u once the physical orientation is fixed, but topology does not yet determine which orientation is physical.