Classification under explicit axioms

The two faces have different uniqueness results. The dissipative axioms select a transmission law; the reversible axioms still leave a symbol to declare.

8. Classification and Uniqueness

8.1 Dissipative classification

Axioms 8.1. Admissible dissipative scalar.

A functional $F:(K,T)\mapsto F(K,T)\in\mathbb R$ is admissible dissipative if:

• (A1) $F$ is determined by $\Lambda_T(K)$ on $\mathsf H_{\mathrm{diss}}$.

• (A2) Spectral invariance: $F(UKU^*,T)=F(K,T)$.

• (A3) Direct-sum additivity.

• (A4) Determinant/log-volume form: $F = -\log\det_*\Phi_T(K)$.

• (A5) Fejér calibration with bounded nondegenerate modulation: on each declared determinant-admissible spectral class there is a scalar modulation

  $$c:(0,\infty)\to(0,\infty)$$

  such that, on $\mathsf H_{\mathrm{diss}}$,

  $$\Phi_T(K) \;=\; c(T)f_T(K) \;=\; c(T)\Lambda_T(K).$$

  The modulation is bounded in logarithmic scale:

  $$\log c \in L^\infty(0,\infty)$$

  on the scale range under consideration. In the infinite-dimensional Fredholm convention, the scalar factor $c(T)I_{\mathsf H_{\mathrm{diss}}}$ is admitted only when its determinant counterterm

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

  is defined and bounded in $T$ on the fixed spectral class. This forces $c(T)=1$ whenever $\dim\mathsf H_{\mathrm{diss}}=\infty$. Other regularizations are outside the present classification.

• (A6) Scale dependence: unbounded as $T\to\infty$ on each fixed nonzero dissipative sector.

Theorem 8.2. Dissipative classification.

Under (A1)–(A6) and (D1)–(D2), every admissible dissipative scalar takes the form

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

where $B(K,T)$ is bounded in $T$ on each fixed spectral class. In particular, $\mathcal D_T(K)$ is unique up to the stated bounded modulation within this calibrated determinant class.

Proof. See Appendix D. The proof is internal to the declared Fejér-calibrated determinant class and does not classify arbitrary dissipative spectral scalars. ◻

8.2 Coherent classification

Axioms 8.3. Admissible reversible response.

A one-parameter family $(V_t)$ is admissible if:

• (R1) Each $V_t$ unitary, $(V_t)$ a strongly continuous one-parameter group.

• (R2) $V_t$ commutes with every spectral projection of $K$.

• (R3) Every spectral subspace of $K$ is $V_t$-invariant.

• (R4) Generator $\widetilde C$ bounded selfadjoint, $\widetilde C|_{\ker K} = 0$.

• (R5) $\widetilde C \in W^*(K)$.

Theorem 8.4. Coherent classification.

Under (R1)–(R5), $V_t = e^{-it\psi(K)}$ for some real bounded Borel $\psi$ with $\psi(0)=0$.

Proof. By (R1) and Stone’s theorem, the strongly continuous one-parameter unitary group $(V_t)_{t\in\mathbb R}$ has a selfadjoint generator $A$ such that $V_t = e^{itA}$. With the sign convention of this paper, write $\widetilde C := -A$, so

$$V_t \;=\; e^{-it\widetilde C}.$$

Axiom (R4) assumes that this generator is bounded selfadjoint and that

$$\widetilde C\big|_{\ker K} \;=\; 0.$$

By (R5), $\widetilde C \in W^*(K)$. The spectral theorem identifies the abelian von Neumann algebra generated by $K$ with the bounded Borel functional calculus of $K$:

$$W^*(K) \;=\; \{\varphi(K):\varphi \in L^\infty(\mathop{\mathrm{\sigma}}(K),E_K)\}.$$

Hence there exists a bounded Borel function $\psi:\mathop{\mathrm{\sigma}}(K)\to\mathbb R$ such that

$$\widetilde C \;=\; \psi(K).$$

The function may be chosen real-valued because $\widetilde C$ is selfadjoint.

Let $P_0 = E_K(\{0\})$ be the projection onto $\ker K$. From (R4),

$$0 \;=\; \widetilde C P_0 \;=\; \psi(K)P_0.$$

Thus the value of $\psi$ on the $0$-eigenspace is zero. If $0$ is not an eigenvalue, the value of a Borel representative at the single point $0$ is immaterial to the functional calculus; we choose the representative with $\psi(0) = 0$. Therefore

$$V_t \;=\; e^{-it\psi(K)}$$

for a real bounded Borel $\psi$ with $\psi(0) = 0$.

Finally, once $\widetilde C = \psi(K)$, every spectral projection $E_K(B)$ commutes with $\widetilde C$ and with $V_t$, so (R3) is automatic. This proves the classification under (R1)–(R5). ◻

Corollary 8.5. Selection of canonical $C$.

Once a specific admissible coherence symbol $h$ is selected, $C=h(K)$ and $U_t=e^{-ith(K)}$ are uniquely determined. Membership in a symbol class and normalization alone need not select one function.

8.3 Unified classification

Proposition 8.6. Scope of the response classification.

The bounded operator responses selected by the stated dissipative and reversible axioms belong to $W^*(K)$. Their scalar budgets are spectral invariants. The reversible classification uses the substantive hypothesis $\widetilde C\in W^*(K)$.

Proof. The dissipative responses are specified scalar functions of $K$. The reversible statement is Theorem 8.4. The budget statement follows from the spectral determinant formula. ◻

For example, $K=I$ on $\mathbb C^2$ commutes with $\widetilde C=\operatorname{diag}(1,2)$, and every spectral subspace of $K$ is preserved, but $\widetilde C$ is not a scalar function of $K$. Spectral-algebra membership excludes unresolved dynamics inside degenerate carrier eigenspaces. The classification determines the form of admitted responses; it does not select the physical symbol from commutation alone.

Theorem 8.6 classifies under the stated axioms. Responses violating any of (A1)–(A6) or (R1)–(R5) are excluded.