Proposition. The corner branches of the centered Dirac symbol [fdip-009R]
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  • November 1, 2026

Extend the free Fourier symbol to the continuous momentum torus \((\mathbb R/2\pi \mathbb Z)^d\): \[\mathcal D(p)=i\sum _j\Gamma _j\sin p_j.\] Its zero matrices occur exactly at the \(2^d\) corners \(p^\epsilon =\pi \epsilon \), with \(\epsilon \in \{0,1\}^d\). Near each corner, \[\begin {aligned} \mathcal D(p^\epsilon +k) &=i\sum _j\widetilde \Gamma _j k_j+O(\|k\|^3),\\ \widetilde \Gamma _j&=(-1)^{\epsilon _j}\Gamma _j, \end {aligned}\] where the remainder is measured in the operator norm on \(S\otimes F\). The matrices \(\widetilde \Gamma _j\) obey the same Clifford anticommutation relations as \(\Gamma _j\).

Proof. 

  • November 1, 2026

The identity \(\mathcal D(p)^2=-(\sum _j\sin ^2p_j)I\) holds for every real momentum. If \(\mathcal D(p)=0\), its square forces each sine to vanish; conversely those vanishing sines make the matrix zero. Modulo \(2\pi \), each coordinate is \(0\) or \(\pi \). The expansion follows from \(\sin (\pi \epsilon _j+k_j)=(-1)^{\epsilon _j}\sin k_j\) and \(\sin k_j=k_j+O(|k_j|^3)\). The gamma matrices have norm one, and there are finitely many coordinates, so summing the remainders gives the stated operator-norm bound. Finally \[\widetilde \Gamma _i\widetilde \Gamma _j +\widetilde \Gamma _j\widetilde \Gamma _i =2\delta _{ij}I,\] since the two signs cancel when \(i=j\), while the off-diagonal anticommutator was already zero.

Thus the centered symbol has \(2^d\) local first-order branches on the same specified spinor/internal carrier. This is the algebraic doubling mechanism behind the sine propagator and opposite-slope discussion in [wingate2025lattice, Sections 9.1–9.2, Equations (9.6)–(9.7), (9.13)–(9.14), pp. 20–21]. The finite periodic grid samples only \(2^e\) of these corners exactly; an odd period need not sample \(\pi \). Fixed boundary twists alter the allowed momenta, as in Example [fdip-006J], without removing the corners of this symbol.

The chosen \(S\) is reducible, so the number of corners is not a count relative to a minimal physical spinor representation. Nor does the local Taylor expansion construct a quantum continuum limit. The precisely scoped Nielsen–Ninomiya comparison concerns a different theorem: free local translation-invariant Hamiltonians on an infinite odd-dimensional lattice, with isolated relativistic zeros and compatible smooth charge projections. Under its hypotheses the total chiral index vanishes separately for each charge [friedan1982nielsen, Sections 1–2, pp. 481–484; Equations (6), (10)–(11)]. The finite count and the Taylor calculation here do not remove any of those hypotheses or supply that theorem's conclusion.