The unified field theorem

Once the realization and symbol are fixed, the two responses belong to one spectral algebra. This chapter also gives the precise conditions under which a family of readouts recovers that algebra.

7. The Unified Field Theorem

Definition 7.1. Canonical Spectral-Curvature Field.

Given an admissible monotone curvature system with canonical generator $K$ and an admissible coherence symbol $h$, the Canonical Spectral-Curvature Field is

$$\mathfrak F_K := \big(\, K,\; h(K),\; e^{-tK},\; e^{-it\,h(K)},\; \Lambda_T(K),\; \Gamma_T(h(K)),\; \mathcal D_T(K),\; \mathfrak B_T(K)\, \big).$$

Theorem 7.2. Unified Spectral-Curvature Field.

Let $K=K_{\mathcal M}\ge0$ be the Friedrichs representative of an admissible curvature system, and fix an admissible symbol $h$. Then:

1. The bounded operators $h(K)$, $e^{-tK}$, $e^{-it h(K)}$, $\Lambda_T(K)$, $\Gamma_T(h(K))$ and $\mathbf C_T$ belong to the abelian algebra $W^*(K)$. The possibly unbounded $K$ is affiliated with it.

2. These bounded operators commute. They act through the same projection-valued measure $E_K$, with scalar mode responses $h(\lambda)$, $e^{-t\lambda}$, $e^{-it h(\lambda)}$, $f_T(\lambda)$, $a_T(h(\lambda))$ and $q_T(h(\lambda))$, where $a_T(u)=(1-e^{-iTu})/(iTu)$ and $a_T(0)=1$.

3. $\mathsf H_0=\ker K$ is fixed pointwise by $S_t$, $U_t$ and $\Lambda_T$.

4. Wherever (D1)–(D3) apply, $\mathcal D_T$ and $\mathfrak B_T$ are unitary-invariant scalar functionals of the same spectral data, with the full resonance convention for $\mathfrak B_T$.

5. A unitary transport $\widetilde K=VKV^*$ carries every bounded readout to $V\phi(K)V^*=\phi(\widetilde K)$ and preserves defined budgets. Thus a specified physical realization of $(K,h)$ realizes both faces on one carrier.

Proof. The bounded Borel functional calculus is a unital $*$-homomorphism. It gives (i), commutation and the spectral integrals in (ii). At $\lambda=0$, $h(0)=0$ and $f_T(0)=1$, proving (iii). The determinant products and trace-log expressions depend only on the spectral multiplicities, proving (iv). The spectral theorem gives $E_{VKV^*}(B)=VE_K(B)V^*$, so all functional calculi and their domains transport as stated. This proves (v). ◻

The theorem identifies one operator architecture with two complementary responses. Physical identification of a measured relaxation generator with $K$ specifies the dissipative realization. Identification of a measured coherent generator with the nominated $h(K)$ specifies a paired physical realization; the mathematical construction already supplies the coherent readout associated to that choice.

Corollary 7.3. Combined attenuating phase evolution.

For $\gamma>0$, the effective operator $H_{\rm eff}=h(K)-i\gamma K$ is normal on $\mathop{\mathrm{dom}}K$, and its forward evolution is

$$E(t)=e^{-\gamma tK}e^{-it h(K)},\qquad t\geq0.$$

It is a contraction semigroup. On a finite retained sector, $[H_{\rm eff},H_{\rm eff}^*]=0$ and $\frac{d}{dt}\left\lVert E(t)u\right\rVert^2=-2\gamma\langle E(t)u,KE(t)u\rangle$.

Proof. The complex spectral multiplier $h(\lambda)-i\gamma\lambda$ defines a normal operator with domain $\mathop{\mathrm{dom}}K$, since $h$ is bounded. Its exponential is $e^{-\gamma t\lambda}e^{-it h(\lambda)}$, proving factorization and contractivity. In finite dimension, commutation and the derivative follow directly from the same spectral multipliers. ◻

Thus the common-carrier non-Hermitian realization is a normal subclass. Reciprocal induction and memory may determine its local carrier through a nonlinear physical model, as in Section 12; arbitrary noncommuting reversible–irreversible dynamics need not have this form.

7.1 Recovery of the spectral algebra from readouts

Theorem 7.4. Spectral algebra recovery from separating readouts.

Let $K = K^*$ be selfadjoint, let $X := \mathop{\mathrm{\sigma}}(K)$, and let $\{m_\alpha:X\to\mathbb C\}_{\alpha\in A}$ be bounded Borel functions. Assume that the sigma-algebra generated by the readouts is the full Borel sigma-algebra of $X$:

$$\sigma(m_\alpha:\alpha\in A) \;=\; \mathfrak B(X).$$

Then

$$W^*(m_\alpha(K):\alpha\in A) \;=\; W^*(K).$$

In particular, if a countable subfamily $\{m_n\}_{n\ge 1}$ separates points of $X$, then the hypothesis holds and

$$W^*(m_n(K):n\ge 1) \;=\; W^*(K).$$

Proof. Let $\mathcal M := W^*(m_\alpha(K):\alpha\in A)$. Since each $m_\alpha(K)$ is a bounded Borel function of $K$, $\mathcal M\subseteq W^*(K)$.

Conversely, define

$$\mathfrak S \;:=\; \{B\in\mathfrak B(X) : E_K(B)\in\mathcal M\}.$$

This is a sigma-algebra. For each $\alpha$ and each Borel set $B\subseteq\mathbb C$,

$$E_K(m_\alpha^{-1}(B)) \;=\; E_{m_\alpha(K)}(B) \;\in\; \mathcal M.$$

Hence $\mathfrak S$ contains the sigma-algebra generated by all $m_\alpha$. By hypothesis, $\mathfrak S = \mathfrak B(X)$. Therefore every spectral projection $E_K(B)$ belongs to $\mathcal M$. Since

$$W^*(K) \;=\; W^*(E_K(B):B\in\mathfrak B(X)),$$

$W^*(K)\subseteq\mathcal M$. Thus equality holds.

It remains only to justify the countable point-separating clause. Suppose $\{m_n\}_{n\ge 1}$ separates points of $X$. The map

$$m:X\to\mathbb C^{\mathbb N},\qquad m(\lambda) = (m_1(\lambda), m_2(\lambda),\dots),$$

is Borel and injective. Since $X\subseteq\mathbb R$ is a standard Borel space, the Lusin–Souslin theorem implies that $m(X)$ is Borel in $\mathbb C^{\mathbb N}$ and that $m^{-1}:m(X)\to X$ is Borel. Hence every Borel subset of $X$ is the inverse image under $m$ of a Borel subset of $m(X)$, and therefore belongs to $\sigma(m_n:n\ge 1)$. Thus $\sigma(m_n:n\ge 1) = \mathfrak B(X)$, and the first part applies. ◻

Corollary 7.5. CSCF readout recovery.

Let $K\ge 0$ be selfadjoint.

1. For every fixed $T_0 > 0$,

   $$W^*(\Lambda_{T_0}(K)) \;=\; W^*(K).$$

   Indeed, $f_{T_0}(\lambda) = T_0^{-1}\int_0^{T_0}e^{-t\lambda}\,dt$ is bounded Borel and strictly decreasing on $[0,\infty)$, hence separates points of $\mathop{\mathrm{\sigma}}(K)$. This is an exact operator-algebraic statement; it does not imply that finitely sampled or noisy empirical estimates of $\Lambda_{T_0}(K)$ recover $K$ without a separate inverse/stability analysis. Theorem 5.4 is in fact stronger than this fixed-scale algebra-recovery statement: it gives the explicit operator-level inversion $K = \psi_{T_0}(\Lambda_{T_0}(K))$ rather than mere equality of generated von Neumann algebras.

2. If the coherence symbol $h$ is injective on $\mathop{\mathrm{\sigma}}(K)$, then

   $$W^*(h(K)) \;=\; W^*(K).$$

   This injectivity condition is sufficient. Algebra recovery is insensitive to changes on $E_K$-null sets, so pointwise injectivity on the entire topological spectrum is not a necessary condition.

3. Let $q_T(u) := \mathop{\mathrm{sinc}}^2(Tu/2)$. If a countable family of coherent compression readouts $\{q_{T_n}\circ h\}_{n\ge 1}$ separates points of $\mathop{\mathrm{\sigma}}(K)$, then

   $$W^*(\Gamma_{T_n}(h(K))^*\Gamma_{T_n}(h(K)):n\ge 1) \;=\; W^*(K).$$

4. More generally, any declared bounded Borel CSCF readout family among $f_T$, $h$, $q_T\circ h$, or their countable combinations, recovers $W^*(K)$ whenever it generates $\mathfrak B(\mathop{\mathrm{\sigma}}(K))$. More generally, recovery is governed by the generated measurable algebra modulo $E_K$-null sets. Readouts that identify spectral distinctions of positive spectral measure recover a corresponding proper observable subalgebra.

7.2 The common spectral calculus

The dissipative face reads the positive-real axis of a single spectral calculus. The coherent face reads the phase / imaginary axis of the same calculus. There is one carrier $K$, two readouts.

For the unregularized generator, the semigroup $e^{-zK}$ is holomorphic for $\operatorname{Re}z>0$ and has the unitary boundary group $e^{-itK}$ as a strong limit. Thus a literal restriction to real and imaginary time gives

$$\begin{aligned} z=t\in\mathbb R_+ &: \quad e^{-zK}=e^{-tK} &&\text{(dissipative)},\\ z=it &: \quad e^{-itK} &&\text{(unregularized coherent)}. \end{aligned}$$

The strong boundary limit follows by dominated convergence in each vector’s spectral measure. A general symbol $h$ instead selects the spectral reparameterization $C=h(K)$ within the same Borel calculus. Its unitary group is bounded-generator and norm-continuous. When $h(\lambda)=\lambda$ on a retained spectral window, it agrees there with the imaginary-time boundary group; otherwise the scalar phase frequencies are reparameterized, while the common spectral projections are preserved.

The physical realization and source-selection questions are treated separately from the common-carrier theorem.