Appendix D: Fejér uniqueness

D. Fejér Uniqueness: Proof of Theorem 8.2

Proof of Theorem 8.2. Fix a determinant-admissible spectral class and a positive selfadjoint $K$ in that class. Put

$$\mathsf H_{\mathrm{diss}}= (\ker K)^\perp,\qquad \Lambda_{T,d} := \Lambda_T(K)\big|_{\mathsf H_{\mathrm{diss}}}.$$

By (A4), an admissible scalar has determinant/log-volume form

$$F(K,T) \;=\; -\log\det\nolimits_*\!\big(\Phi_T(K)\big|_{\mathsf H_{\mathrm{diss}}}\big).$$

By the Fejér calibration axiom (A5),

$$\Phi_T(K)\big|_{\mathsf H_{\mathrm{diss}}} \;=\; c(T)\Lambda_T(K)\big|_{\mathsf H_{\mathrm{diss}}} \;=\; c(T)\Lambda_{T,d},$$

where $c(T)>0$ and $\log c$ is bounded on the admissible scale range.

The determinant convention is part of the declared admissible class. Thus, in finite rank, in the ordinary Fredholm class when the scalar factor is admissible, the determinant factorizes as

$$\det\nolimits_*\!\big(c(T)\Lambda_{T,d}\big) \;=\; \det\nolimits_*\!\big(c(T)I_{\mathsf H_{\mathrm{diss}}}\big)\,\det\nolimits_*\!\big(\Lambda_{T,d}\big).$$

Using the definition of $\mathcal D_T(K)$,

$$\begin{aligned} F(K,T) &= -\log\det\nolimits_*\!\big(\Lambda_{T,d}\big) - \log\det\nolimits_*\!\big(c(T)I_{\mathsf H_{\mathrm{diss}}}\big) \\ &= \mathcal D_T(K) + B_c(K,T), \end{aligned}$$

where

$$B_c(K,T) \;:=\; -\log\det\nolimits_*\!\big(c(T)I_{\mathsf H_{\mathrm{diss}}}\big).$$

By (A5), this modulation counterterm is bounded in $T$ on the fixed spectral class.

In the finite-rank case, if $r = \dim\mathsf H_{\mathrm{diss}}< \infty$, the formula is explicit:

$$\det\nolimits_*\!\big(c(T)\Lambda_{T,d}\big) \;=\; c(T)^r\det\nolimits_*\!\big(\Lambda_{T,d}\big),$$

and therefore

$$B_c(K,T) \;=\; -r\log c(T).$$

Since $\log c \in L^\infty$, $B_c(K,T)$ is bounded in $T$ for fixed $r$.

In the ordinary infinite-dimensional Fredholm determinant class, a nontrivial scalar multiple $c(T)I_{\mathsf H_{\mathrm{diss}}}$ is not of the form $I + \mathcal L^1$ unless $c(T) = 1$. Hence the Fredholm-admissible infinite-dimensional case has $B_c(K,T) = 0$. Other determinant regularizations are outside this theorem.

Thus every admissible scalar in the declared Fejér-calibrated determinant class has the form

$$F(K,T) \;=\; \mathcal D_T(K) + B(K,T),\qquad B(K,T) := B_c(K,T),$$

with $B$ bounded in $T$ on each fixed spectral class. Axiom (A6) then says that the admissible scalar has genuine unbounded scale dependence as $T \to \infty$ on the fixed nonzero dissipative sector. Since $B$ is bounded, the unbounded part is precisely the Fejér determinant budget $\mathcal D_T(K)$, up to bounded modulation.

No statement is made about dissipative spectral scalars outside (A1)–(A6) or outside the determinant conventions (D1)–(D2). ◻