Theorem. Link reflection as a trace square [fdip-0086]
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  • November 1, 2026

Use the periodic gauge law with \(N=2m\), \(m\ge 1\). Reflect time by \(\theta (j)=1-j\) modulo \(2m\); the positive half is \(\{1,\ldots ,m\}\). As in the scalar definition, this is a link reflection: its cuts lie between times \(0,1\) and \(m,m+1\).

Let \(\mathcal A_+\) consist of finite linear combinations of functions \[F(U)=\prod _{j=1}^m f_j(U_j),\] where each \(f_j:X\to \mathbb C\) is bounded, measurable and gauge invariant. These form an algebra of spatial observables. For such a product, on \(\mathcal H_{\mathrm {phys}}\) define \[A_F=T_{\mathrm {phys}}^{1/2}M_{f_1}T_{\mathrm {phys}} M_{f_2}\cdots T_{\mathrm {phys}}M_{f_m} T_{\mathrm {phys}}^{1/2},\] where for \(m=1\) this means \(T_{\mathrm {phys}}^{1/2}M_{f_1}T_{\mathrm {phys}}^{1/2}\). Extend this assignment linearly to finite sums. Then \(A_F\) is Hilbert–Schmidt and \[\mathbb E_{2m}\left [\overline {F(\theta U)}F(U)\right ] =\frac {\operatorname {Tr}(A_F^*A_F)}{Z_{2m}}\ge 0, \qquad (\theta U)_j=U_{1-j}.\] The temporal links have already been integrated in the periodic trace identity. No assertion about arbitrary observables involving individual temporal links is included.

Proof. 

  • November 1, 2026

The positive trace-class operator \(T_{\mathrm {phys}}\) has a Hilbert–Schmidt square root. Every multiplication operator here is bounded, as are all intervening transfer factors, so \(A_F\) is Hilbert–Schmidt; the same holds for a finite sum.

For products \(F=\prod _j f_j(U_j)\) and \(G=\prod _j g_j(U_j)\), the adjoint reverses the order and conjugates each multiplier. The bounded trace formula and cyclicity of a trace-class product give \[\operatorname {Tr}(A_F^*A_G) =Z_{2m}\,\mathbb E_{2m} \left [\overline {F(\theta U)}G(U)\right ].\] The cyclic order of the multipliers is \[\overline f_m,\ldots ,\overline f_1,g_1,\ldots ,g_m,\] with one full transfer between adjacent entries. They occur at times \(m+1,\ldots ,2m,1,\ldots ,m\), where \(2m=0\); hence \(f_j\) is conjugated at time \(1-j\). There are exactly \(2m\) transfer factors after the adjacent half-powers combine.

Expanding finite sums proves the same sesquilinear identity for all \(F,G\in \mathcal A_+\). If two product-sum presentations define the same function, their difference has zero expectation against its reflected conjugate, and the identity makes the corresponding difference of operators have Hilbert–Schmidt norm zero. Thus \(A_F\) is independent of that presentation. Taking \(G=F\) proves the trace-square identity and positivity.

This proof uses the explicitly positive operator of Theorem [fdip-0080] and the exact finite-time law of Theorem [fdip-0083]. It does not reverse the argument to reconstruct a transfer operator from an arbitrary reflection-positive law; the scalar quotient obstruction shows why that distinction matters.