Finite-Window Phase-Commit Dynamics in the CHC Framework

The preceding CHC layers separate several roles that are often compressed in informal language. Propagation-side statements describe phase-linked propagation and accessible wave semantics. Detector-side statements describe opening, threshold behavior, local commit, amplification, and durable readout. Sector-exchange statements describe the covariant response of a declared action to the global phase coordinate, whose cosmological reading is the expansion-response branch where applicable. These roles are adjacent, but they are not the same object.

The present paper isolates the finite reduced dynamics that sits between them. The object is a finite state window equipped with phase-sensitive coherent transport and dissipative channels. It is not a new sector action. It is not a detector microdynamics. It is a reduced open-system class in which phase-link persistence, coherence loss, local commit, and standard recovery are written as one finite-dimensional dynamical system.

The construction uses three standard mathematical facts. First, open quantum states on a finite system are density matrices. Second, Markovian completely positive trace-preserving reduced evolution is represented by generators of Gorini-Kossakowski-Sudarshan-Lindblad type [citation]. Third, phase-link data can be represented by a connection and its holonomy, in the same geometric sense in which adiabatic quantum phase is represented as holonomy of a Hermitian line bundle [citation]. PCD combines these ingredients with the CHC distinction between phase-linked persistence and local commit.

The paper is organized as follows. reference defines finite phase windows and phase connection data. reference defines phase-indexed reduced generators and proves standard recovery and state admissibility. reference gives phase-transport covariance and the finite curvature response. reference defines coherence load and commit forms. reference derives the phase-response formula and its chart covariance. reference states the relation between reduced response and PLE loading. reference gives composition and marginal consistency. reference gives representative finite systems. reference defines observable projection maps.

Let $\Hcal_N$TeX\Hcal_N be a complex Hilbert space with $\dim\Hcal_N=N<\infty$TeX\dim\Hcal_N=N<\infty. The density-state space is

\Scal_N=\{\rho\in\End(\Hcal_N): \rho=\rho^\dagger,\ \rho\ge0,\ \Tr\rho=1\}.

A finite phase window consists of an open phase patch $P\subset\Rbb^d$TeXP\subset\Rbb^d, a time interval $I=[0,T]$TeXI=[0,T], and the state space $\Scal_N$TeX\Scal_N.

definition: Phase connection. A phase connection on $P$TeXP is a smooth matrix-valued one-form

A=\sum_{i=1}^{d}A_i(\vartheta)\,\dd\vartheta^i,
\qquad
A_i(\vartheta)^\dagger=-A_i(\vartheta),

with values in the anti-Hermitian operators on $\Hcal_N$TeX\Hcal_N. For a piecewise smooth path $\Gamma:[0,1]\to P$TeX\Gamma:[0,1]\to P, its phase-link transport is

U_\Gamma=\mathcal P\exp\!\left(-\int_\Gamma A\right),

where $\mathcal P$TeX\mathcal P denotes path ordering.

Since $A_i$TeXA_i is anti-Hermitian, $U_\Gamma$TeXU_\Gamma is unitary. The associated action on states is

\Ad_{U_\Gamma}(\rho)=U_\Gamma\rho U_\Gamma^\dagger.

The curvature of $A$TeXA is

F_A=\dd A+A\wedge A,
\qquad
(F_A)_{ij}=\partial_iA_j-\partial_jA_i+[A_i,A_j].

Flatness of $A$TeXA is not assumed. Nonzero curvature records path dependence of phase-link transport on the declared window.

definition: Finite PCD system. A finite-window PCD system is a tuple

\Xcal=(\Hcal_N,P,I,A,\Ecal,H,M,\rho_{\rm in}),

where $\Ecal\subset\Rbb$TeX\Ecal\subset\Rbb is an interval containing $0$TeX0, $H:P\times\Ecal\to\End(\Hcal_N)$TeXH:P\times\Ecal\to\End(\Hcal_N) is smooth with $H(\vartheta,\varepsilon)=H(\vartheta,\varepsilon)^\dagger$TeXH(\vartheta,\varepsilon)=H(\vartheta,\varepsilon)^\dagger, $M_1,\ldots,M_r:P\times\Ecal\to\End(\Hcal_N)$TeXM_1,\ldots,M_r:P\times\Ecal\to\End(\Hcal_N) are smooth channel maps, and $\rho_{\rm in}\in\Scal_N$TeX\rho_{\rm in}\in\Scal_N.

The parameter $\varepsilon$TeX\varepsilon labels the phase-response displacement from the selected standard member. The point $\varepsilon=0$TeX\varepsilon=0 is not a limiting approximation to be inferred; it is part of the definition of the finite system.

For $X\in\End(\Hcal_N)$TeXX\in\End(\Hcal_N) define

\Dcal[X](\rho)=X\rho X^\dagger-\frac12\{X^\dagger X,\rho\}.

The PCD generator associated with reference is

\Lcal_{\vartheta,\varepsilon}(\rho)
=-\frac{\ii}{\hbar}[H(\vartheta,\varepsilon),\rho]
+\sum_{\alpha=1}^{r}\Dcal[M_\alpha(\vartheta,\varepsilon)](\rho).

The standard member is

\Lcal_{\vartheta,0}(\rho)
=-\frac{\ii}{\hbar}[H(\vartheta,0),\rho]
+\sum_{\alpha=1}^{r}\Dcal[M_\alpha(\vartheta,0)](\rho).

The form reference includes a closed system as the case $M_\alpha=0$TeXM_\alpha=0 for all $\alpha$TeX\alpha. It also includes the common separated representation

H(\vartheta,\varepsilon)=H_0(\vartheta)+\varepsilon H_1(\vartheta),

M_\alpha(\vartheta,\varepsilon)=L_\alpha(\vartheta)+\varepsilon K_\alpha(\vartheta),

whenever the displaced channel itself remains the declared channel. Alternatively one may write an explicitly nonnegative rate form

\Lcal_{\vartheta,\varepsilon}
=\Lcal_{\vartheta,0}
-\frac{\ii\varepsilon}{\hbar}[H_1(\vartheta),\cdot]
+\sum_{a=1}^{s}\kappa_a(\vartheta,\varepsilon)\Dcal[K_a(\vartheta)],
\qquad \kappa_a\ge0,

with $\kappa_a(\vartheta,0)=0$TeX\kappa_a(\vartheta,0)=0. The two forms are distinct parametrizations of finite reduced dynamics, not distinct principles.

lemma: Dissipator identities. For every $X\in\End(\Hcal_N)$TeXX\in\End(\Hcal_N) and every Hermitian $\rho$TeX\rho,

\Tr\Dcal[X](\rho)=0,
\qquad
\Dcal[X](\rho)^\dagger=\Dcal[X](\rho).

proof. Cyclicity gives

\Tr(X\rho X^\dagger)=\Tr(X^\dagger X\rho)
=\frac12\Tr(X^\dagger X\rho)+\frac12\Tr(\rho X^\dagger X).

The adjoint identity follows by taking the adjoint of reference.

theorem: State admissibility. For every $(\vartheta,\varepsilon)\in P\times\Ecal$TeX(\vartheta,\varepsilon)\in P\times\Ecal, the semigroup $\ee^{t\Lcal_{\vartheta,\varepsilon}}$TeX\ee^{t\Lcal_{\vartheta,\varepsilon}} maps $\Scal_N$TeX\Scal_N into $\Scal_N$TeX\Scal_N for all $t\ge0$TeXt\ge0. It is trace preserving, Hermiticity preserving, positivity preserving, and completely positive.

proof. The Hamiltonian part is a commutator with a Hermitian operator. Each dissipative term is of GKSL form. In finite dimension, the GKSL representation generates a completely positive trace-preserving semigroup. The preceding lemma gives trace and Hermiticity preservation at the generator level, and the GKSL theorem gives complete positivity of the semigroup.

corollary: Standard recovery. At $\varepsilon=0$TeX\varepsilon=0, PCD evolution is exactly the selected standard reduced evolution:

\rho(t;\vartheta,0)=\ee^{t\Lcal_{\vartheta,0}}\rho_{\rm in}.

The phase connection and the reduced generator refer to the same finite state space. Their compatibility is expressed by unitary change of frame on the phase patch. Let

W:P\to U(\Hcal_N)

be smooth. The transformed connection is

A^W=WAW^\dagger-(\dd W)W^\dagger,

and the transformed generator coefficients are

H^W=W H W^\dagger,
\qquad
M_\alpha^W=W M_\alpha W^\dagger.

The state and observable representatives transform by

\rho^W=W\rho W^\dagger,
\qquad
B^W=W B W^\dagger.

proposition: Generator covariance. For each fixed $(\vartheta,\varepsilon)$TeX(\vartheta,\varepsilon),

\Lcal^W_{\vartheta,\varepsilon}(W\rho W^\dagger)
=W\Lcal_{\vartheta,\varepsilon}(\rho)W^\dagger.

Consequently,

\ee^{t\Lcal^W_{\vartheta,\varepsilon}}(W\rho W^\dagger)
=W\ee^{t\Lcal_{\vartheta,\varepsilon}}(\rho)W^\dagger.

proof. The commutator term transforms by conjugation because $H^W=W H W^\dagger$TeXH^W=W H W^\dagger. For the dissipator,

\Dcal[WMW^\dagger](W\rho W^\dagger)=W\Dcal[M](\rho)W^\dagger.

Summing the terms gives reference. Exponentiation gives the semigroup statement.

corollary: Observable invariance. With $B^W=WBW^\dagger$TeXB^W=WBW^\dagger and $\rho_{\rm in}^W=W\rho_{\rm in}W^\dagger$TeX\rho_{\rm in}^W=W\rho_{\rm in}W^\dagger,

\Tr\!\left(B^W\ee^{t\Lcal^W_{\vartheta,\varepsilon}}\rho_{\rm in}^W\right)
=\Tr\!\left(B\ee^{t\Lcal_{\vartheta,\varepsilon}}\rho_{\rm in}\right).

proof. Use the semigroup covariance and cyclicity of trace.

The curvature of $A$TeXA records finite-window path dependence. Let $R_{ij}(\delta_i,\delta_j)$TeXR_{ij}(\delta_i,\delta_j) be a small coordinate rectangle based at $\vartheta$TeX\vartheta in the $i,j$TeXi,j directions. Its holonomy has the expansion

U_{R_{ij}}=\one-F_{ij}(\vartheta)\delta_i\delta_j+O(|\delta|^3).

For a state $\rho$TeX\rho this gives

\Ad_{U_{R_{ij}}}(\rho)-\rho
=-[F_{ij}(\vartheta),\rho] \delta_i\delta_j+O(|\delta|^3).

Thus curvature is a finite-window obstruction to path-independent phase-link transport.

A reduced phase statement requires a declared resolution. Let

\Pi=\{P_1,\ldots,P_p\}

be a family of orthogonal projectors with $\sum_{a=1}^{p}P_a=\one$TeX\sum_{a=1}^{p}P_a=\one. Define the block-diagonal and block-off-diagonal projections

\Delta_\Pi(\rho)=\sum_{a=1}^{p}P_a\rho P_a,
\qquad
\Off_\Pi(\rho)=\rho-\Delta_\Pi(\rho).

definition: Coherence load. The coherence load of $\rho$TeX\rho relative to $\Pi$TeX\Pi is

C_\Pi(\rho)=\normHS{\Off_\Pi(\rho)},
\qquad
\normHS{X}=(\Tr X^\dagger X)^{1/2}.

proposition: Block-unitary invariance. If $U$TeXU is unitary and $UP_a=P_aU$TeXUP_a=P_aU for all $a$TeXa, then

C_\Pi(U\rho U^\dagger)=C_\Pi(\rho).

proof. The commutation assumption gives

\Delta_\Pi(U\rho U^\dagger)=U\Delta_\Pi(\rho)U^\dagger,

hence $\Off_\Pi(U\rho U^\dagger)=U\Off_\Pi(\rho)U^\dagger$TeX\Off_\Pi(U\rho U^\dagger)=U\Off_\Pi(\rho)U^\dagger. The Hilbert-Schmidt norm is unitarily invariant.

proposition: Resolution coarsening. Let $\Pi'$TeX\Pi' refine $\Pi$TeX\Pi. Then

C_\Pi(\rho)\le C_{\Pi'}(\rho)
\qquad
\text{for all }\rho\in\Scal_N.

proof. The $\Pi$TeX\Pi-off-diagonal subspace is an orthogonal subspace of the $\Pi'$TeX\Pi'-off-diagonal subspace. Orthogonal projection onto a smaller subspace cannot increase the Hilbert-Schmidt norm.

definition: Commit form. For a channel operator $X$TeXX define the $\Pi$TeX\Pi-commit form

Q_{\Pi,X}(\rho)=-\operatorname{Re}\inner{\Off_\Pi(\rho)}{\Off_\Pi(\Dcal[X](\rho))}.

The form $Q_{\Pi,X}$TeXQ_{\Pi,X} measures the instantaneous dissipative contribution to decay of squared coherence load:

\left.\frac{\dd}{\dd t}\frac12 C_\Pi(\rho+t\Dcal[X](\rho))^2\right|_{t=0}
=-Q_{\Pi,X}(\rho).

It need not be nonnegative for a general channel. It is nonnegative for channels aligned with the declared resolution.

theorem: Aligned dephasing. Let

X=\sum_{a=1}^{p}x_aP_a,
\qquad x_a\in\Cbb.

Then, for $a\ne b$TeXa\ne b,

P_a\Dcal[X](\rho)P_b=
\left(
-\frac12|x_a-x_b|^2
+\ii\,\operatorname{Im}(x_a\overline{x_b})
\right)P_a\rho P_b.

Consequently, the real dissipative contribution to the commit form is

Q_{\Pi,X}(\rho)=\frac12\sum_{a\ne b}|x_a-x_b|^2\normHS{P_a\rho P_b}^2\ge0.

proof. Since $XP_a=x_aP_a$TeXXP_a=x_aP_a and $X^\dagger XP_a=|x_a|^2P_a$TeXX^\dagger XP_a=|x_a|^2P_a,

P_a\Dcal[X](\rho)P_b
=x_a\overline{x_b}P_a\rho P_b
-\frac12(|x_a|^2+|x_b|^2)P_a\rho P_b

=\left(
-\frac12|x_a-x_b|^2
+\ii\,\operatorname{Im}(x_a\overline{x_b})
\right)P_a\rho P_b.

The imaginary coefficient does not contribute to the real part in reference. Substitution into reference therefore gives reference.

Let $B=B^\dagger$TeXB=B^\dagger be an observable. The finite-window observable projection is

\Ocal_B(t;\vartheta,\varepsilon)=\Tr\left(B\,\ee^{t\Lcal_{\vartheta,\varepsilon}}\rho_{\rm in}\right).

Assume $\Lcal_{\vartheta,\varepsilon}$TeX\Lcal_{\vartheta,\varepsilon} is differentiable in a parameter $u$TeXu. The phase-response derivative is the one-form component

\partial_u\Ocal_B(t;\vartheta,\varepsilon).

theorem: Duhamel response formula. For any differentiable parameter $u$TeXu in $\vartheta$TeX\vartheta or $\varepsilon$TeX\varepsilon,

\partial_u\Ocal_B(t;\vartheta,\varepsilon)
=\int_{0}^{t}\Tr\!\left(
B\,\ee^{(t-s)\Lcal_{\vartheta,\varepsilon}}
(\partial_u\Lcal_{\vartheta,\varepsilon})
\ee^{s\Lcal_{\vartheta,\varepsilon}}\rho_{\rm in}
\right)\dd s.

proof. For finite-dimensional linear operators,

\partial_u\ee^{t\Lcal_u}=\int_0^t\ee^{(t-s)\Lcal_u}(\partial_u\Lcal_u)\ee^{s\Lcal_u}\dd s.

Multiplying by $B$TeXB, applying to $\rho_{\rm in}$TeX\rho_{\rm in}, and taking the trace gives the result.

definition: Response one-form. For an observable family $\Bcal=\{B_1,\ldots,B_q\}$TeX\Bcal=\{B_1,\ldots,B_q\}, the response one-form on $P\times\Ecal$TeXP\times\Ecal is

\Omega_{\Bcal,t}
=\sum_{j=1}^{q}\sum_{i=1}^{d}
\partial_{\vartheta^i}\Ocal_{B_j}(t;\vartheta,\varepsilon)\,\beta^j\otimes\dd\vartheta^i
+\sum_{j=1}^{q}
\partial_\varepsilon\Ocal_{B_j}(t;\vartheta,\varepsilon)\,\beta^j\otimes\dd\varepsilon,

where $\{\beta^j\}$TeX\{\beta^j\} is the coordinate basis of $\Rbb^q$TeX\Rbb^q.

The one-form formulation is important: scalar components depend on the phase chart, whereas the covector transforms canonically.

proposition: Phase-chart covariance. Let $\widetilde\vartheta=f(\vartheta)$TeX\widetilde\vartheta=f(\vartheta) be a smooth regular change of phase coordinates. Then the response components obey

\partial_{\widetilde\vartheta^a}\Ocal_B
=\sum_{i=1}^{d}\frac{\partial\vartheta^i}{\partial\widetilde\vartheta^a}\,
\partial_{\vartheta^i}\Ocal_B.

Hence $\sum_i\partial_{\vartheta^i}\Ocal_B\,\dd\vartheta^i$TeX\sum_i\partial_{\vartheta^i}\Ocal_B\,\dd\vartheta^i is chart independent as a one-form.

proof. This is the chain rule for a scalar function on the phase patch.

PLE assigns a phase-loading representative only after a sector action, stress-tensor convention, phase chart, and variational domain have been specified. PCD begins at a different level: a finite reduced generator is already given. A bridge between the two levels is possible only when the finite generator is obtained as a reduction of a variational sector.

definition: Variational lift. A PCD system admits a variational lift on a phase patch $P$TeXP if there exist a sector action $S[g,\Psi,\mathcal H]$TeXS[g,\Psi,\mathcal H], a finite reduction map $R$TeXR, and a family of state-observable pairs such that the reduced Euler response of $S$TeXS to $\mathcal H$TeX\mathcal H induces the response one-form reference for the PCD generator.

This definition is intentionally asymmetric. A variational sector may reduce to a PCD system, but a finite PCD system does not by itself determine a unique sector action.

proposition: Reduced loading criterion. Let $\Xcal$TeX\Xcal be a finite PCD system. Its phase-response one-form represents a PLE loading object on $P$TeXP only if $\Xcal$TeX\Xcal admits a variational lift whose phase Euler response and normalization agree with the PLE conventions on the same patch. Without such a lift, reference is a reduced response one-form and not a Noether loading representative.

proof. PLE loading is defined by variational response of a declared sector action to the phase coordinate. A PCD response one-form is defined by differentiating a finite reduced generator and its observable projections. Equality of the two objects requires a map identifying the finite generator derivative with the action-level Euler response under the same phase convention and normalization. That is precisely the variational lift. In its absence, the two objects have different domains of definition.

Finite windows should behave coherently under independent product composition and under reductions that discard an uncoupled factor.

Let $\Xcal_A$TeX\Xcal_A and $\Xcal_B$TeX\Xcal_B be PCD systems on $\Hcal_A$TeX\Hcal_A and $\Hcal_B$TeX\Hcal_B with generators $\Lcal_A$TeX\Lcal_A and $\Lcal_B$TeX\Lcal_B. The independent product generator on $\Hcal_A\otimes\Hcal_B$TeX\Hcal_A\otimes\Hcal_B is

\Lcal_{A\otimes B}=\Lcal_A\otimes\id_B+\id_A\otimes\Lcal_B,

where the notation denotes the induced action on operators.

proposition: Product composition. For product initial states,

\rho_{AB}(0)=\rho_A(0)\otimes\rho_B(0),

the solution of reference is

\rho_{AB}(t)=\rho_A(t)\otimes\rho_B(t).

proof. The right-hand side satisfies the product equation and the same initial condition. Uniqueness for finite-dimensional linear ordinary differential equations gives the result.

proposition: Marginal consistency. Let

\Lcal_{AB}=\Lcal_A\otimes\id_B+\id_A\otimes\Lcal_B

and let $\rho_{AB}(t)$TeX\rho_{AB}(t) solve $\dot\rho_{AB}=\Lcal_{AB}\rho_{AB}$TeX\dot\rho_{AB}=\Lcal_{AB}\rho_{AB}. Then

\frac{\dd}{\dd t}\Tr_B\rho_{AB}(t)=\Lcal_A(\Tr_B\rho_{AB}(t)).

proof. The term $\Lcal_A\otimes\id_B$TeX\Lcal_A\otimes\id_B commutes with $\Tr_B$TeX\Tr_B in the stated way. The partial trace of $\id_A\otimes\Lcal_B$TeX\id_A\otimes\Lcal_B vanishes because $\Lcal_B$TeX\Lcal_B is trace preserving on the $B$TeXB factor.

If an interaction term is added to reference, marginal consistency requires that the interaction be retained in the reduced declaration or absorbed into a new effective generator. A finite PCD window is therefore stable under reduction only after the reduced generator has been specified.

The following examples are representative members of the class. They are not special axioms.

Two-level phase-commit system

Let $\Hcal_2=\Cbb^2$TeX\Hcal_2=\Cbb^2 with Pauli matrices $\sigma_x,\sigma_y,\sigma_z$TeX\sigma_x,\sigma_y,\sigma_z and lowering operator $\sigma_-$TeX\sigma_-. Consider

H(\vartheta,\varepsilon)=\frac{\hbar\omega(\vartheta)}{2}\sigma_z
+\frac{\hbar\varepsilon\alpha(\vartheta)}{2}\sigma_x,

and channels

M_1=\sqrt{\gamma(\vartheta)}\,\sigma_- ,
\qquad
M_2=\sqrt{\kappa(\vartheta,\varepsilon)}\,\sigma_z,
\qquad
\gamma\ge0,
\quad \kappa\ge0,
\quad \kappa(\vartheta,0)=\kappa_0(\vartheta).

Write

\rho=\frac12(\one+x\sigma_x+y\sigma_y+z\sigma_z).

Then the coherent part rotates the Bloch vector around

(\varepsilon\alpha,0,\omega),

while $M_1$TeXM_1 relaxes the excited population and $M_2$TeXM_2 damps transverse coherence. Relative to the energy resolution $\Pi=\{ |0\rangle\langle0|,|1\rangle\langle1|\}$TeX\Pi=\{ |0\rangle\langle0|,|1\rangle\langle1|\},

C_\Pi(\rho)^2=\frac12(x^2+y^2).

For the aligned dephasing channel $\sigma_z$TeX\sigma_z, reference gives

\Dcal[\sigma_z](\rho)=-x\sigma_x-y\sigma_y,
\qquad
\frac{\dd}{\dd t}C_\Pi(\rho(t))^2=-4\kappa C_\Pi(\rho(t))^2

when the other terms are suppressed. The finite system therefore separates coherent phase relocation from aligned commit damping.

Truncated oscillator with phase-indexed damping

Let $\Hcal_N$TeX\Hcal_N be the span of number states $|0\rangle,\ldots,|N-1\rangle$TeX|0\rangle,\ldots,|N-1\rangle. Let $a_N$TeXa_N be the truncated lowering operator and $n_N=a_N^\dagger a_N$TeXn_N=a_N^\dagger a_N. A phase-indexed oscillator member is

H(\vartheta,\varepsilon)=\hbar\omega(\vartheta)n_N
+\varepsilon q(\vartheta)(a_N+a_N^\dagger),

with channels

M_1=\sqrt{\gamma(\vartheta)}\,a_N,
\qquad
M_2=\sqrt{\kappa(\vartheta,\varepsilon)}\,n_N.

The number-state resolution gives

C_\Pi(\rho)^2=\sum_{m\ne n}|\rho_{mn}|^2.

The channel $n_N$TeXn_N is aligned with the number resolution, and

P_m\Dcal[n_N](\rho)P_n=-\frac12(m-n)^2P_m\rho P_n.

Thus long-range number coherence is damped more strongly than adjacent-number coherence. The Hamiltonian displacement term changes the phase relation among neighboring number states, while the aligned channel gives an ordered commit form.

Boundary-interface reduction

Let

\Hcal_N=\Hcal_{\rm bulk}\oplus\Hcal_{\Sigma}\oplus\Hcal_{\rm out}

with corresponding projectors $P_{\rm bulk}$TeXP_{\rm bulk}, $P_\Sigma$TeXP_\Sigma, and $P_{\rm out}$TeXP_{\rm out}. A boundary-interface PCD member has Hamiltonian

H=H_{\rm bulk}\oplus H_{\Sigma}\oplus H_{\rm out}
+\varepsilon(V_{\rm bulk,\Sigma}+V_{\rm bulk,\Sigma}^\dagger)

and commit channels

K_a(\vartheta)=P_{\rm out}W_a(\vartheta)P_\Sigma.

The dissipator $\Dcal[K_a]$TeX\Dcal[K_a] transfers boundary-localized amplitude into the outgoing sector while preserving total trace. Relative to the three-block resolution, the channel is not a pure dephasing direction; the corresponding commit form can mix coherence damping with population transfer. This is the finite reduced form of an interface-local commit channel.

Let $\Bcal=\{B_1,\ldots,B_q\}$TeX\Bcal=\{B_1,\ldots,B_q\} be Hermitian operators on $\Hcal_N$TeX\Hcal_N. The observable projection map of a PCD system is

\Phi_{\Bcal}:P\times\Ecal\to C(I,\Rbb^q),
\qquad
\Phi_{\Bcal}(\vartheta,\varepsilon)(t)
=\left(\Ocal_{B_1}(t;\vartheta,\varepsilon),\ldots,
\Ocal_{B_q}(t;\vartheta,\varepsilon)\right).

The standard contrast map is

\Delta\Phi_{\Bcal}(\vartheta,\varepsilon)(t)
=\Phi_{\Bcal}(\vartheta,\varepsilon)(t)-\Phi_{\Bcal}(\vartheta,0)(t).

A weighted response bilinear form on a compact subwindow $J\subset I$TeXJ\subset I is

\mathfrak I_{ab}(\vartheta,\varepsilon)
=\sum_{j=1}^{q}\int_J
w_j(t)\,
\partial_a\Ocal_{B_j}(t;\vartheta,\varepsilon)
\partial_b\Ocal_{B_j}(t;\vartheta,\varepsilon)
\dd t,

where $w_j(t)\ge0$TeXw_j(t)\ge0 and $a,b$TeXa,b range over the chosen coordinates of $P\times\Ecal$TeXP\times\Ecal. Observable channels with zero weight on $J$TeXJ are removed before the response rank is assigned.

proposition: Response span rank. Assume the retained weights define a positive-definite weighted inner product on the retained observable family on $J$TeXJ. The rank represented by $\mathfrak I(\vartheta,\varepsilon)$TeX\mathfrak I(\vartheta,\varepsilon) is invariant under regular coordinate changes on $P\times\Ecal$TeXP\times\Ecal. It is also invariant under nonsingular linear recombination of the retained observable family when the weighted inner product is carried to the recombined basis.

proof. After zero-weight channels are removed, $\mathfrak I$TeX\mathfrak I is the Gram matrix of the retained response covectors with respect to a positive-definite weighted inner product. A regular coordinate change multiplies these covectors by an invertible Jacobian, and a nonsingular recombination of retained observables changes only the chosen basis of the same weighted response span when the induced inner product is carried along. In both cases the dimension of the response span, and hence the rank represented by the Gram matrix, is preserved.

The map reference is the point where a finite PCD system becomes comparable to a chosen readout family. No readout family is canonical. Different choices of $\Bcal$TeX\Bcal, $I$TeXI, and $w_j$TeXw_j define different finite windows of the same generator.

Finite-window phase-commit dynamics gives a reduced open-system object for CHC phase structure. Its state is a density matrix on a finite Hilbert space. Its phase-link data are represented by a connection on a phase patch. Its time evolution is generated by a phase-indexed GKSL operator. Its undeformed member is the selected standard reduced dynamics. Its coherence and commit quantities are finite functionals of the density state and declared resolution. Its phase response is a covariant one-form on the phase window. Its relation to PLE loading requires a variational lift; without such a lift it remains a reduced response object. Product composition, marginal reduction, and observable projection maps are fixed at the finite-generator level.

The construction therefore supplies a mathematical layer between phase-linked propagation, local commit, and action-level phase loading. It keeps standard reduced dynamics as an exact member while allowing phase-sensitive coherent and dissipative structure to be represented inside the same finite open-system class.

Let $\operatorname{vec}$TeX\operatorname{vec} stack columns. For matrices $A,X,B$TeXA,X,B,

\operatorname{vec}(AXB)=(B^T\otimes A)\operatorname{vec}(X).

Thus reference has matrix representative

\mathbf L_{\vartheta,\varepsilon}
=-\frac{\ii}{\hbar}\left(\one\otimes H-H^T\otimes\one\right)

\quad +\sum_{\alpha=1}^{r}\left(
\overline{M_\alpha}\otimes M_\alpha
-\frac12\one\otimes M_\alpha^\dagger M_\alpha
-\frac12(M_\alpha^\dagger M_\alpha)^T\otimes\one
\right),

where all coefficients are evaluated at $(\vartheta,\varepsilon)$TeX(\vartheta,\varepsilon). The finite-dimensional response formula reference follows equivalently from differentiating $\exp(t\mathbf L_{\vartheta,\varepsilon})$TeX\exp(t\mathbf L_{\vartheta,\varepsilon}).

Assume

\Lcal_{\vartheta,\varepsilon}=\Lcal_{\vartheta,0}+\varepsilon\Kcal_\vartheta+O(\varepsilon^2).

Then

\Ocal_B(t;\vartheta,\varepsilon)
=\Ocal_B(t;\vartheta,0)
+\varepsilon\int_0^t\Tr\!\left(B\,\ee^{(t-s)\Lcal_{\vartheta,0}}
\Kcal_\vartheta\ee^{s\Lcal_{\vartheta,0}}\rho_{\rm in}\right)\dd s
+O(\varepsilon^2).

If $\Kcal_\vartheta=-\ii[H_1(\vartheta),\cdot]/\hbar$TeX\Kcal_\vartheta=-\ii[H_1(\vartheta),\cdot]/\hbar, this is the coherent phase-response term. If $\Kcal_\vartheta=\sum_a\mu_a(\vartheta)\Dcal[K_a(\vartheta)]$TeX\Kcal_\vartheta=\sum_a\mu_a(\vartheta)\Dcal[K_a(\vartheta)] with $\mu_a\ge0$TeX\mu_a\ge0, this is the dissipative commit-response term.

No new observational or experimental data are introduced by this paper. The manuscript is a mathematical framework paper; all definitions, assumptions, and representative finite-window constructions used in the argument are contained in the text. Supplementary PCD/WPL companion statements record formal/numeric PCD gates, including finite-state trace, Hermiticity, positivity-proxy, aligned-dephasing, and product/marginal consistency checks. These are theorem-witness and finite-window consistency checks, not observational empirical tests. The local label PCD-WPL-VP1-FORMAL-AND-PUBLIC-CLOCK-METRIC-COMPATIBILITY-SUPPORT denotes this declared support comparison only; it is not a separate public manuscript claim.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.