Common Propagation Classes and Positive-Frequency Wave Mechanics in the CHC Framework

Given a declared coherent propagation sector of CHC, the question is whether the quadratic fluctuation operator of that sector already determines a common propagation class, discrete spectral hierarchies under admitted confinement data, and a positive-frequency propagation-envelope sector. Only the coherent-branch quadratic pullback of the unreduced CHC phase-field action is used [citation]. Accessible-event probabilities are not part of the present propagation result. Born-type event assignment, detector thresholds, readout laws, objectivity criteria, and record semantics are not inferred from the propagation sector.

The discussion is restricted to a stationary coherent window. On that window the reduced Hessian coefficients are time-independent, the spatial operator is elliptic, and the admitted family varies only through domain, self-adjoint boundary realization, and lower-order confinement. This is the regime addressed by current work on elliptic realizations with boundary, stationary or globally hyperbolic Klein--Gordon spatial operators, and operator-theoretic propagators [citation].

One carrier scale $\Ombar>0$TeX\Ombar>0 is fixed on the declared stationary window. It is held fixed across one admissible benchmark family and serves only as the reference shift used to state shifted positivity and the low-spectral expansion. It is not benchmark-wise retuned inside that family.

The reduced single-field quadratic form is the coherent-branch pullback of the unreduced CHC Hessian. After isolation of the mixed time--space term, the same Hessian fixes one drift-corrected elliptic core whose principal symbol is common across the admitted family. The positive-frequency sector is asserted only on realizations for which $\Ombar^2+\Lchc^{(b)}$TeX\Ombar^2+\Lchc^{(b)} is strictly positive, and the Schr\"odinger-type reduction is asserted only on declared shifted-positive low-spectral support. The two-field reduction below is used only to display the descent of the quadratic coefficients from the same branch data.

Throughout, natural units $c=\hbar=1$TeXc=\hbar=1 are used.

Minimal input and stationary coherent windows

Let the unreduced CHC coherent sector be described by a real phase field $\HH$TeX\HH and auxiliary fields $\Phi^A$TeX\Phi^A on a spacetime $(\mathcal M,g_{\mu\nu})$TeX(\mathcal M,g_{\mu\nu}). Assume an unreduced action

S_{\mathrm{full}}[\HH,\Phi]
=
\int d^4x\,\sqrt{-g}\,\mathcal L_{\mathrm{full}}(\HH,\Phi,\nabla\HH,\nabla\Phi;g).

On the declared coherent branch the auxiliary sector is slaved to $\HH$TeX\HH:

\Phi^A=f^A(\HH,\nabla\HH,\ldots)+O(\epsad).

The reduced action is obtained by substitution,

\Sred[\HH]=S_{\mathrm{full}}[\HH,f(\HH,\nabla\HH,\ldots)].

definition: Stationary coherent window. A stationary coherent window consists of a propagation region and a background class satisfying the following conditions.

- the reduced Hessian coefficients obtained from reference are independent of the laboratory time coordinate on the declared window; - gravitational backreaction on the propagation region is negligible; - the spatial region is either compact with declared boundary data or effectively confining through a lower-order profile; - the reduced coefficients are in $W^{1,\infty}$TeXW^{1,\infty} and satisfy the positivity and ellipticity hypotheses stated below.

All downstream coherent-reduction statements are read only on this declared stationary coherent window.

definition: Family-fixed carrier scale. Fix $\Ombar>0$TeX\Ombar>0 on a declared stationary coherent window. For one admissible benchmark family $\Bfam$TeX\Bfam, $\Ombar$TeX\Ombar is held fixed across the family and serves only as the reference shift used in shifted positivity and in the low-spectral expansion. Benchmark-wise retuning of $\Ombar$TeX\Ombar is not part of the admitted family data. All downstream shifted-positivity, low-spectral, and propagation-class statements are read only relative to this fixed carrier scale on the declared stationary coherent window and within the admitted benchmark family.

proposition: Coherent-branch pullback of the quadratic sector. Let $\HH_0$TeX\HH_0 be a background branch inside a stationary coherent window and write $\HH=\HH_0+\eta$TeX\HH=\HH_0+\eta. Assume that the slaving map $f^A$TeXf^A is $C^2$TeXC^2 on a branch neighborhood of the declared coherent window and that the omitted branch-mismatch terms remain uniformly $O(\epsad\|\eta\|_{H^1(\Sig)}^2)$TeXO(\epsad\|\eta\|_{H^1(\Sig)}^2) on that window. Then, up to the same declared remainder order inherited from the slaving relation reference, the quadratic fluctuation action of the reduced theory equals the pullback of the unreduced CHC quadratic action to the same coherent branch:

\delta^2\Sred[\HH_0](\eta,\eta)
=
\delta^2 S_{\mathrm{full}}\!\left[(\HH_0,f(\HH_0))\right]\!\bigl((\eta,\delta f[\eta]),(\eta,\delta f[\eta])\bigr)
+O(\epsad\|\eta\|_{H^1(\Sig)}^2).

proof. Differentiate reference twice and use the chain rule. The $C^2$TeXC^2 regularity of the slaving map justifies the second variation, the first variation contributes linearly through $\delta f[\eta]$TeX\delta f[\eta], and the omitted branch-mismatch terms are of the declared $O(\epsad\|\eta\|_{H^1(\Sig)}^2)$TeXO(\epsad\|\eta\|_{H^1(\Sig)}^2) order on the coherent window.

Representative coherent reduction

The abstract pullback statement becomes concrete in one admissible coherent sector. The two-field reduction below is used only to display the descent of the quadratic coefficients from one coherent branch. We use the metric signature $(-,+,+,+)$TeX(-,+,+,+) in this example and define

X_H:=g^{\mu\nu}\partial_\mu H\,\partial_\nu H,
\qquad
X_\Phi:=g^{\mu\nu}\partial_\mu \Phi\,\partial_\nu \Phi .

proposition: Representative two-field coherent reduction. On the declared stationary coherent window and within the admitted benchmark family carrying the fixed carrier scale $\Ombar$TeX\Ombar, consider the full density

\mathcal L_{\mathrm{full}}
=
-\frac12 M_H^2\,X_H
-\frac12 M_\Phi^2\,X_\Phi
-
U(H,\Phi),

with constant positive $M_H,M_\Phi$TeXM_H,M_\Phi and a coherent branch

\Phi=f(H)+O(\epsad).

Then the reduced density is

\Lred
=
-\frac12 M_{\mathrm{eff}}(H)^2\,X_H
-
U_{\mathrm{red}}(H)
+O(\epsad),

where

M_{\mathrm{eff}}(H)^2=M_H^2+M_\Phi^2\,[f'(H)]^2,
\qquad
U_{\mathrm{red}}(H)=U(H,f(H)).

With the stated signature, one has $X_H=-(\partial_t H)^2+h^{ij}\nabla_i H\,\nabla_j H$TeXX_H=-(\partial_t H)^2+h^{ij}\nabla_i H\,\nabla_j H on a stationary background, so the quadratic density below has the representative positive time-kinetic and negative spatial-gradient reading inherited from the same branch. For a stationary background $H_0(\mathbf x)$TeXH_0(\mathbf x) and $H=H_0+\eta$TeXH=H_0+\eta, the quadratic density takes the form

\Lred^{(2)}
=
\frac12 M_{\mathrm{eff}}(H_0)^2(\partial_t\eta)^2
-
\frac12 M_{\mathrm{eff}}(H_0)^2 h^{ij}\nabla_i\eta\,\nabla_j\eta
-
\frac12 U_J(\mathbf x)\eta^2
+
\nabla_i\mathcal B^i
+
O(\epsad\,\eta^2),

with an explicit lower-order coefficient $U_J$TeXU_J assembled from $U_{\mathrm{red}}''(H_0)$TeXU_{\mathrm{red}}''(H_0) and background-gradient terms of $M_{\mathrm{eff}}$TeXM_{\mathrm{eff}}. Hence, on this branch,

Z_t=M_{\mathrm{eff}}(H_0)^2,
\qquad
A^{ij}=M_{\mathrm{eff}}(H_0)^2 h^{ij},
\qquad
B^i=0.

proof. Substitute reference into reference to obtain reference and reference. Expand around $H_0$TeXH_0 to second order. With signature $(-,+,+,+)$TeX(-,+,+,+), the coefficient of $(\partial_t\eta)^2$TeX(\partial_t\eta)^2 is $M_{\mathrm{eff}}(H_0)^2$TeXM_{\mathrm{eff}}(H_0)^2, the coefficient of $h^{ij}\nabla_i\eta\nabla_j\eta$TeXh^{ij}\nabla_i\eta\nabla_j\eta is the same function with the opposite sign in the quadratic density, and all terms linear in $\nabla\eta$TeX\nabla\eta are removable into the divergence $\nabla_i\mathcal B^i$TeX\nabla_i\mathcal B^i and the lower-order coefficient $U_J$TeXU_J after one integration by parts.

General reduced Hessian coefficients

The worked example shows what must happen in any branch that genuinely descends from a single reduced action. This is the reduced-action version of the Hessian/Jacobi route familiar in geometric field theory [citation].

proposition: Origin of the reduced quadratic coefficients. Let $\HH_0$TeX\HH_0 be a stationary coherent background and write $\HH=\HH_0+\eta$TeX\HH=\HH_0+\eta. Then the reduced quadratic density can be written, up to a spatial divergence, as

\Lred^{(2)}
=
\frac12 Z_t(\mathbf x)(\partial_t\eta)^2
-
B^i(\mathbf x)\,\partial_t\eta\,\nabla_i\eta
-
\frac12 A^{ij}(\mathbf x)\,\nabla_i\eta\,\nabla_j\eta
-
\frac12 U_2(\mathbf x)\eta^2,

where

Z_t
=
\left.\frac{\partial^2 \Lred}{\partial(\partial_t\HH)^2}\right|_{\HH_0},

[0.4em]
A^{ij}
=
-\left.\frac{\partial^2 \Lred}{\partial(\nabla_i\HH)\,\partial(\nabla_j\HH)}\right|_{\HH_0},

[0.4em]
B^i
=
-\left.\frac{\partial^2 \Lred}{\partial(\partial_t\HH)\,\partial(\nabla_i\HH)}\right|_{\HH_0},

and $U_2$TeXU_2 is the second $\HH$TeX\HH-derivative corrected by the background-derivative terms generated when first-order pieces are integrated by parts. All four coefficients are therefore fixed by the same reduced branch.

proof. Expand $\Lred(\HH_0+\eta)$TeX\Lred(\HH_0+\eta) to quadratic order. Stationarity eliminates explicit time dependence in the coefficients. All terms linear in derivatives of $\eta$TeX\eta are removed into the coefficient functions and a spatial divergence by integration by parts.

The quadratic action determines both an interior Jacobi operator and a boundary form.

proposition: Jacobi operator and boundary concomitant. There exist coefficient functions $C^i_J(\mathbf x)$TeXC^i_J(\mathbf x) and $U_J(\mathbf x)$TeXU_J(\mathbf x), built from $A^{ij},B^i,U_2$TeXA^{ij},B^i,U_2 and their first derivatives, such that

\delta^2\Sred[\HH_0](\eta,\eta)
=
\frac12
\int dt\int_{\Sig}d\mu_h\,
\Bigl(
Z_t(\partial_t\eta)^2
-
\eta\,\Jcore\eta
\Bigr)
+
\frac12
\int dt\int_{\partial\Sig}d\mu_\gamma\,
\omega_{\partial\Sig}(\eta,\eta),

with

\Jcore
=
-\nabla_i(A^{ij}\nabla_j)
+
C^i_J\nabla_i
+
U_J,

and Green boundary concomitant

\omega_{\partial\Sig}(u,v)
=
n_iA^{ij}\bigl(u\,\nabla_j v-v\,\nabla_j u\bigr)
+
n_i\Xi^i(u,v),

where $n_i$TeXn_i is the outward unit normal and $\Xi^i$TeX\Xi^i collects lower-order contributions coming from the mixed term and coefficient gradients.

proof. Insert reference into the action and integrate by parts in the spatial variables once. The second-order spatial part gives the symmetric term $-\nabla_i(A^{ij}\nabla_j)$TeX-\nabla_i(A^{ij}\nabla_j). All first-order terms are absorbed into $C^i_J$TeXC^i_J and $U_J$TeXU_J, while the boundary pieces assemble into reference.

Drift splitting and the drift-corrected elliptic core

The mixed time--space coefficient does not supply a new benchmark-specific Hamiltonian. It is part of the same reduced Hessian and can be isolated by a completion of the square.

definition: Drift-corrected principal tensor. Define

\beta^i=Z_t^{-1}B^i,
\qquad
\widehat A^{ij}=A^{ij}-Z_t\,\beta^i\beta^j
=
A^{ij}-Z_t^{-1}B^iB^j.

lemma: Completion of the square on the declared stationary coherent window. Let

\Qchc
=
-Z_t\partial_t^2
-
2B^i\partial_t\nabla_i
-
\nabla_i(A^{ij}\nabla_j)
+
U_2

be the quadratic operator associated with reference. Then

\Qchc
=
-Z_t\Dcore^2
-
\nabla_i(\widehat A^{ij}\nabla_j)
+
\mathcal R_{1}
+
U_2,
\qquad
\Dcore:=\partial_t+\beta^i\nabla_i,

where $\mathcal R_{1}$TeX\mathcal R_{1} is a first-order spatial operator whose coefficients depend algebraically on $Z_t$TeXZ_t, $A^{ij}$TeXA^{ij}, $B^i$TeXB^i and their first derivatives.

proof. Expand $-Z_t(\partial_t+\beta^i\nabla_i)^2$TeX-Z_t(\partial_t+\beta^i\nabla_i)^2. The quadratic spatial contribution generated by the square equals $-Z_t\beta^i\beta^j\nabla_i\nabla_j$TeX-Z_t\beta^i\beta^j\nabla_i\nabla_j and therefore shifts the principal tensor from $A^{ij}$TeXA^{ij} to $\widehat A^{ij}$TeX\widehat A^{ij}. The terms containing derivatives of $\beta^i$TeX\beta^i are at most first order and are collected in $\mathcal R_{1}$TeX\mathcal R_{1}.

theorem: Common propagation class on the declared stationary coherent window at fixed carrier scale. Fix the family-fixed carrier scale $\Ombar$TeX\Ombar of reference on the declared stationary coherent window, and suppose that there exist constants $0<z_0\le z_1<\infty$TeX0<z_0\le z_1<\infty and $0<\lambda_0\le\lambda_1<\infty$TeX0<\lambda_0\le\lambda_1<\infty such that

z_0\le Z_t(\mathbf x)\le z_1,
\qquad
\lambda_0 |\xi|_h^2
\le
\widehat A^{ij}(\mathbf x)\xi_i\xi_j
\le
\lambda_1 |\xi|_h^2

for all $\mathbf x\in\Sig$TeX\mathbf x\in\Sig and all covectors $\xi$TeX\xi. Assume moreover that the first-order remainder obeys a graph-norm bound

\|\mathcal R_{1}\varphi\|_{L^2(\Sig)}
\le
C\epsad\bigl(
\|\nabla_h^2\varphi\|_{L^2}
+
\|\nabla_h\varphi\|_{L^2}
+
\|\varphi\|_{L^2}
\bigr)

on the declared domain. Then each admissible benchmark $b\in\Bfam$TeXb\in\Bfam can be written as

\Dcore^2\eta
+
(\Ombar^2+\Lchc^{(b)})\eta
=
\Rad^{(b)}[\eta],

with

\Lchc^{(b)}
=
-Z_t^{-1}\nabla_i(\widehat A^{ij}\nabla_j)
+
V_{\mathrm{geom}}
+
V_{\mathrm{conf}}^{(b)},

where $V_{\mathrm{geom}}$TeXV_{\mathrm{geom}} is fixed by the branch data and coefficient normalization, $V_{\mathrm{conf}}^{(b)}$TeXV_{\mathrm{conf}}^{(b)} is the declared benchmark-dependent lower-order confinement profile, and

\|\Rad^{(b)}[\varphi]\|_{L^2}
\le
C_b\epsad\bigl(
\|\nabla_h^2\varphi\|_{L^2}
+
\|\nabla_h\varphi\|_{L^2}
+
\|\varphi\|_{L^2}
\bigr).

The principal symbol of $\Lchc^{(b)}$TeX\Lchc^{(b)} is therefore common across the family.

proof. Apply reference and divide by $Z_t$TeXZ_t. The zeroth-order terms that do not vary across the benchmark family are grouped into $V_{\mathrm{geom}}$TeXV_{\mathrm{geom}}. All family dependence not already contained in the common branch data is, by assumption, restricted to the declared lower-order confinement profile and the realization data. The graph-norm estimate follows directly from reference.

corollary: Principal-symbol replacement excludes common-family membership. If a claimed benchmark requires a change of the principal tensor $\widehat A^{ij}$TeX\widehat A^{ij} rather than a change of domain, boundary realization, or lower-order confinement profile, then that benchmark is not generated by the same coherent CHC branch.

proof. A change of $\widehat A^{ij}$TeX\widehat A^{ij} changes the principal symbol of reference. By reference that symbol is fixed by the same reduced Hessian.

Self-adjoint realizations and benchmark transfer

definition: Admissible benchmark family. A benchmark family $\Bfam$TeX\Bfam is admissible if every member is described by the same principal operator reference and differs from the others only through

- the spatial domain $\Sig$TeX\Sig; - the self-adjoint realization selected through the boundary concomitant reference; - the lower-order confinement profile $V_{\mathrm{conf}}^{(b)}$TeXV_{\mathrm{conf}}^{(b)}.

theorem: Admissible self-adjoint realizations under declared compact-boundary or coercive-confinement hypotheses. Let $b\in\Bfam$TeXb\in\Bfam. Assume either

- $\Sig$TeX\Sig is compact and the declared realization data define a regular self-adjoint boundary condition compatible with reference; or - $\Sig$TeX\Sig is noncompact and the admitted benchmark data include a coercive-confinement hypothesis for which the closed quadratic form associated with $\Lchc^{(b)}$TeX\Lchc^{(b)} is semibounded and has compact resolvent.

Then $\Lchc^{(b)}$TeX\Lchc^{(b)} admits a self-adjoint realization on $L^2(\Sig,d\mu_h)$TeXL^2(\Sig,d\mu_h) with domain fixed by the declared realization data. Under the same admitted compact-resolvent hypotheses, its spectrum is discrete and bounded below [citation].

proof. Uniform ellipticity follows from reference. For compact $\Sig$TeX\Sig, the cited elliptic realization theory yields regular self-adjoint boundary realizations compatible with the Green boundary form. For noncompact $\Sig$TeX\Sig, self-adjointness and compact-resolvent control are asserted only on declared coercive-confinement families for which semibounded closed forms and compact-resolvent hypotheses are part of the admitted benchmark data. Under those hypotheses, the associated form defines the self-adjoint realization and the spectrum is discrete and bounded below.

proposition: Norm-resolvent transfer along a declared realization path. Let $\{b_s\}_{s\in[0,1]}$TeX\{b_s\}_{s\in[0,1]} be a declared path inside an admissible family $\Bfam$TeX\Bfam with fixed principal tensor $\widehat A^{ij}$TeX\widehat A^{ij}. Assume that the path varies only through domain, self-adjoint boundary realization, or lower-order confinement data, that the associated closed quadratic forms $q_s$TeXq_s vary continuously in the sense of Kato, and that the confinement terms remain uniformly form-bounded with respect to a common reference realization. Then the corresponding self-adjoint realizations $\Lchc^{(b_s)}$TeX\Lchc^{(b_s)} are norm-resolvent continuous along the declared path. Discrete eigenvalues therefore move continuously along the family except at the usual multiplicity changes or crossings [citation].

proof. The fixed principal tensor keeps the leading form unchanged. The additional assumptions place the path inside the standard closed-form perturbation framework of Kato. The declared realization data and uniformly form-bounded confinement terms therefore give norm-resolvent continuity of the associated self-adjoint operators.

No claim is made for undeclared topology changes or for paths leaving the common form domain.

Representative confinement family

For a local representative patch, flatten the common principal symbol on one spatial coordinate. The normal form below is local to that coherent patch and is not continued globally without additional hypotheses. The resulting normalized operator is written as

\Lchc^{(b)}=-\frac{d^2}{dx^2}+V^{(b)}(x).

It is the local normal form of the same drift-corrected elliptic core for the admitted benchmark family.

A. Finite interval with Dirichlet realization..

On $x\in[0,L]$TeXx\in[0,L] with $u(0)=u(L)=0$TeXu(0)=u(L)=0,

\omega_n^{(A)}=\frac{n^2\pi^2}{L^2},
\qquad
u_n^{(A)}(x)=\sqrt{\frac{2}{L}}\sin\!\frac{n\pi x}{L},
\qquad
n\in\mathbb N.

$A_R$TeXA_RA_R. Finite interval with Robin realization..

With

u'(0)=\alpha\,u(0),
\qquad
u'(L)=-\beta\,u(L),
\qquad
\alpha,\beta\in\mathbb R,

the eigenvalues satisfy

(\alpha\beta-k^2)\sin(kL)+(\alpha+\beta)k\cos(kL)=0,
\qquad
\omega_n^{(A_R)}=k_n^2.

The principal operator remains $-d^2/dx^2$TeX-d^2/dx^2; only the self-adjoint boundary realization changes.

B. Local quadratic confinement..

For

V^{(B)}(x)=\alpha(x-x_0)^2+V_0,
\qquad
\alpha>0,

the discrete spectrum is

\omega_n^{(B)}=(2n+1)\sqrt{\alpha}+V_0,
\qquad
n=0,1,2,\dots.

C. Soft-wall deformation..

For

V^{(C)}(x)=\alpha(x-x_0)^2+\beta(x-x_0)^4+V_0,
\qquad
\alpha>0,\ \beta>0,

coercivity preserves discreteness. At small $\beta$TeX\beta the leading perturbative shift is

\omega_n^{(C)}
=
\omega_n^{(B)}
+
\frac{3\beta}{4\alpha}\bigl(2n^2+2n+1\bigr)
+
O(\beta^2).

These four representatives exhibit the same pattern: the operator class is fixed first, while compactness, boundary response, and lower-order confinement determine the spectral realization.

Drift frame and exact core envelope

The drift field generated by $B^i$TeXB^i is part of the same reduced Hessian. On each simply connected patch it defines a natural propagation time.

lemma: Drift frame. Let $\Dcore=\partial_t+\beta^i(\mathbf x)\nabla_i$TeX\Dcore=\partial_t+\beta^i(\mathbf x)\nabla_i with smooth time-independent $\beta^i$TeX\beta^i. On each simply connected coherent patch there exist local coordinates $(\tau,\mathbf y)$TeX(\tau,\mathbf y) such that

\Dcore=\partial_\tau.

In these coordinates the remainder-free core of reference becomes

\partial_\tau^2\eta
+
(\Ombar^2+\Lchc^{(b)})\eta
=
0.

proof. The flow-box theorem applied to the smooth vector field $\partial_t+\beta^i\nabla_i$TeX\partial_t+\beta^i\nabla_i supplies local coordinates along its integral curves.

definition: Shifted-positive stable realization. A self-adjoint realization $\Lchc^{(b)}$TeX\Lchc^{(b)} on $L^2(\Sig,d\mu_h)$TeXL^2(\Sig,d\mu_h) is shifted-positive at the family-fixed carrier scale $\Ombar$TeX\Ombar if there exists a constant $c_{\ast}>0$TeXc_{\ast}>0 such that

\Ombar^2+\Lchc^{(b)}\ge c_{\ast} I

as a quadratic-form inequality on its natural domain.

theorem: Exact positive-frequency envelope of the drift-corrected core. Let $\Lchc^{(b)}$TeX\Lchc^{(b)} be a shifted-positive stable realization in the sense of reference. Then

\Omega_{+}^{(b)}=\bigl(\Ombar^2+\Lchc^{(b)}\bigr)^{1/2}

is a self-adjoint strictly positive operator by functional calculus. For any initial datum $\psi_0$TeX\psi_0 in the domain of $\Omega_{+}^{(b)}$TeX\Omega_{+}^{(b)}, the core solution

\eta_{\mathrm{core}}(\tau)
=
\Re\!\left(e^{-i\Omega_{+}^{(b)}\tau}\psi_0\right)

solves reference. Equivalently, the complex envelope

\psi(\tau)=e^{-iH_{+}^{(b)}\tau}\psi_0,
\qquad
H_{+}^{(b)}:=\Omega_{+}^{(b)}-\Ombar,

obeys the exact first-order law

i\partial_\tau \psi = H_{+}^{(b)}\psi.

proof. By reference, the shifted operator $\Ombar^2+\Lchc^{(b)}$TeX\Ombar^2+\Lchc^{(b)} is self-adjoint and bounded below by a strictly positive constant. The spectral theorem therefore defines the self-adjoint square root reference. Substituting reference into reference gives the solution formula, and reference is the corresponding first-order representation of the same evolution.

proposition: Control of the full coherent family. Assume the hypotheses of reference and reference and let $\eta$TeX\eta solve the full equation reference on a bounded drift-time interval $|\tau|\le T$TeX|\tau|\le T with the same initial data as $\eta_{\mathrm{core}}$TeX\eta_{\mathrm{core}}. Then there exists $C_T>0$TeXC_T>0 such that

\sup_{|\tau|\le T}
\Bigl(
\|\eta(\tau)-\eta_{\mathrm{core}}(\tau)\|_{H^1(\Sig)}
+
\|\partial_\tau\eta(\tau)-\partial_\tau\eta_{\mathrm{core}}(\tau)\|_{L^2(\Sig)}
\Bigr)
\le
C_T\epsad\,
\|(1+\Lchc^{(b)})\psi_0\|_{L^2(\Sig)}.

proof. Subtract reference from reference and apply the standard energy estimate for a forced second-order equation with symmetric principal part. The forcing term is bounded by reference, and Gr\"onwall on the bounded interval yields reference.

Shifted-positive low-spectral Schr\"odinger regime

The exact envelope law remains first-order but non-polynomial in $\Lchc^{(b)}$TeX\Lchc^{(b)}. The standard Schr\"odinger form appears only on declared spectral support for which both absolute spectral smallness and shifted positivity hold.

Let $I_{\mathrm{low}}\subset\mathbb R$TeXI_{\mathrm{low}}\subset\mathbb R be a declared Borel interval and let

P_{\mathrm{low}}=\mathbf 1_{I_{\mathrm{low}}}\!\bigl(\Lchc^{(b)}\bigr)

be the associated spectral projector. Assume

\etasp
=
\sup_{\lambda\in I_{\mathrm{low}}}
\frac{|\lambda|}{\Ombar^2},
\qquad
0<\etasp\ll 1,

and

\inf_{\lambda\in I_{\mathrm{low}}}\bigl(\Ombar^2+\lambda\bigr)\ge c_{\ast}>0.

The second condition keeps the declared interval inside the shifted-positive sector of reference. Because $\Ombar$TeX\Ombar is fixed across the admitted family, $\etasp$TeX\etasp measures spectral smallness on one common carrier scale rather than a benchmark-wise retuning.

For states with $P_{\mathrm{low}}\psi=\psi$TeXP_{\mathrm{low}}\psi=\psi, functional calculus gives

\Omega_{+}^{(b)}
=
\Ombar
+
\frac{1}{2\Ombar}\Lchc^{(b)}
-
\frac{1}{8\Ombar^3}\bigl(\Lchc^{(b)}\bigr)^2
+
O\!\left(\frac{|\Lchc^{(b)}|^3}{\Ombar^5}\right)

on the declared interval, and hence

H_{+}^{(b)}
=
\frac{1}{2\Ombar}\Lchc^{(b)}
-
\frac{1}{8\Ombar^3}\bigl(\Lchc^{(b)}\bigr)^2
+
O\!\left(\frac{|\Lchc^{(b)}|^3}{\Ombar^5}\right).

Define the Schr\"odinger-type effective generator

H_{\mathrm{eff}}^{(b)}=\frac{1}{2\Ombar}\Lchc^{(b)}.

proposition: Controlled Schr\"odinger regime. Let $\psi$TeX\psi satisfy $P_{\mathrm{low}}\psi=\psi$TeXP_{\mathrm{low}}\psi=\psi on an interval $I_{\mathrm{low}}$TeXI_{\mathrm{low}} obeying reference and reference. Then the exact core envelope law reference reduces to

i\partial_\tau\psi
=
H_{\mathrm{eff}}^{(b)}\psi
+
\Renv[\psi],

with remainder bound

\|\Renv[\psi]\|_{L^2}
\le
\frac18 \etasp^2\,\Ombar\,\|\psi\|_{L^2}
+
O\!\left(\etasp^3\Ombar\,\|\psi\|_{L^2}\right).

If no interval satisfying reference and reference exists for a claimed benchmark family, the Schr\"odinger-type reduction fails for that family.

proof. Apply the expansion reference on the spectral support of $\psi$TeX\psi. The leading term gives reference, and the next term is bounded by

\frac18 \frac{|\lambda|^2}{\Ombar^3}\le \frac18 \etasp^2\,\Ombar

for all $\lambda\in I_{\mathrm{low}}$TeX\lambda\in I_{\mathrm{low}}. The positivity condition reference guarantees that the branch remains inside the shifted-positive sector on which the square-root functional calculus is admissible.

Induced Hilbert-space representation

Because $\Lchc^{(b)}$TeX\Lchc^{(b)} is self-adjoint and shifted-positive, $H_{+}^{(b)}$TeXH_{+}^{(b)} is self-adjoint on its natural domain $D(H_{+}^{(b)})$TeXD(H_{+}^{(b)}) by spectral calculus, and $H_{\mathrm{eff}}^{(b)}$TeXH_{\mathrm{eff}}^{(b)} is self-adjoint on $D(\Lchc^{(b)})$TeXD(\Lchc^{(b)}). All comparisons between the exact and Schr\"odinger-type generators are asserted only on $P_{\mathrm{low}}\Khil\subset D(H_{+}^{(b)})\cap D(\Lchc^{(b)})$TeXP_{\mathrm{low}}\Khil\subset D(H_{+}^{(b)})\cap D(\Lchc^{(b)}), where both reference and the shifted-positivity condition remain controlled. The natural Hilbert space of the envelope is therefore

\Khil=L^2(\Sig,d\mu_h),
\qquad
\langle \psi_1,\psi_2\rangle_{\Sig}
=
\int_{\Sig}d\mu_h\,\psi_1^\ast(\mathbf x)\psi_2(\mathbf x).

Stone's theorem then gives strongly continuous unitary groups

\psi(\tau)=e^{-iH_{+}^{(b)}\tau}\psi_0,
\qquad
\psi_{\mathrm{eff}}(\tau)=e^{-iH_{\mathrm{eff}}^{(b)}\tau}\psi_0,

on their natural domains, while the comparison between them remains restricted to the declared low-spectral support [citation].

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

The propagation-side claim is rejected under any of the following conditions.

F0. No reduced-action Jacobi route..

If the coefficients $(Z_t,A^{ij},B^i,U_2)$TeX(Z_t,A^{ij},B^i,U_2) cannot be obtained as the quadratic Hessian data of one and the same reduced CHC branch on the declared coherent window, the construction fails before spectral analysis begins.

F1. Principal-symbol drift..

If different benchmarks can be matched only by changing $\widehat A^{ij}$TeX\widehat A^{ij} rather than by changing domain, boundary realization, or lower-order confinement, the common propagation class does not exist.

F2. Boundary-family failure..

If compact-domain realizations such as the Dirichlet-to-Robin deformation of reference and reference require replacement of the kinetic operator rather than a change of self-adjoint realization generated by the same quadratic core, the benchmark-family claim fails at the boundary level.

F3. Transfer failure..

If a specified confinement family does not admit continuous spectral transfer with one fixed principal symbol, the family claim fails for that family.

F4. No shifted-positive core..

If

\inf\sigma\!\bigl(\Ombar^2+\Lchc^{(b)}\bigr)\le 0,

the positive-frequency branch is not admissible on that realization.

F5. No controlled low-spectral window..

If there is no declared interval $I_{\mathrm{low}}$TeXI_{\mathrm{low}} satisfying both reference and reference, the Schr\"odinger-type reduction is not admissible for that family.

F6. Benchmark-specific operator replacement..

If discrete spectra or complex envelope laws appear only after benchmark-specific operators are inserted by choice rather than inherited from the same reduced Hessian and its shifted-positive branch, the propagation claim reduces to system-specific modeling.

Within a stationary coherent window, the propagation sector of CHC fixes a reduced quadratic Hessian, a drift-corrected elliptic core, and a positive-frequency propagation-envelope sector. The two-field coherent reduction only exhibits the descent of the quadratic coefficients from one coherent branch.

The resulting self-adjoint operator class is common across the admitted family. Finite-domain, Robin, local-quadratic, and soft-wall realizations differ only through domain, boundary response, and lower-order confinement. The drift-corrected core admits an exact first-order envelope only on shifted-positive stable realizations at the family-fixed carrier scale $\Ombar$TeX\Ombar, and the full coherent family remains $O(\epsad)$TeXO(\epsad)-close to that core on bounded drift-time intervals. The Schr\"odinger-type law is a controlled reduction on declared shifted-positive low-spectral support, not a universal consequence of self-adjointness alone.

No accessible-event probability rule, Born-type assignment, detector threshold, readout law, objectivity criterion, or record semantics is derived in the present propagation analysis.

Funding and competing interests..

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