XX-B Coupled Gauge-Fixed Free Input for the Original-Rod Lapse A Q1 Construction on the Frozen XX-A Background

We construct the coupled gauge-fixed free input for the original-rod lapse variant explicitly frozen in XX-A. The background, scalar calibration, positive lapse branch, slice-harmonic rods, decaying spatial boundary condition, and factorial perturbation convention are retained and reproduced.

The Einstein–scalar Euler linearization is converted to its correctly normalized action Hessian. A generalized de Donder gauge, with a specified auxiliary-field normalization, gives a reduced operator whose principal part is an invertible fiber matrix times the background wave operator. After fiber normalization it is normally hyperbolic. Retarded and advanced inverses are constructed on the open finite-time slab, and the complete auxiliary–ghost inverse is obtained by an exact block calculation. Causal intertwining identities and wavefront bounds are proved.

A nonempty class of normalized Hadamard contraction matrices is constructed using exact Feynman inverses for the resulting bundle operators. The construction is algebraic: positivity on the physical quotient, spatial-tail estimates for the lapse, and an exact BRST-invariant symmetric covariance are not claimed. The complete second-order classical field equations and the embedding, induced-metric, clock, and rod contributions to the lapse are displayed. The second spatial rod is constructed and its absence from the quadratic lapse is retained.

Q1 is closed for the compact-test, gauge-fixed free-input target defined in this paper. This does not close Q2–Q5 for the nonlocal relational lapse, identify the variant with every native Em02-E convention, or admit an inherited characteristic quantum coefficient. No quantum evolution without external time, regulator removal, bulk symmetry restoration, or compilation is asserted.

XX-A froze an explicit original-rod lapse variant on a power-law Einstein–scalar background XXA. This paper constructs its free gauge-fixed field input. It does not reconstruct an unavailable interacting calculation.

The target consists of:

1. the coupled metric–scalar action Hessian and a fixed gauge-fixing realization;

2. the auxiliary and ghost sectors, including their signs and normalization;

3. retarded and advanced inverses on compact spacetime tests;

4. a normalized Hadamard contraction matrix on that same free field system;

5. the classical principal-symbol comparison with the independently displayed original-rod lapse.

Here “normalized contraction” means a specified field-equation and graded canonical-commutation normalization with a stated microlocal singularity class. It does not mean a positive physical state or a chosen cosmological vacuum.

The original rods are derived coordinate functionals. They do not appear as additional propagating variables in the invariant bulk action. Their spatial inverse is supplied separately. Consequently, “original-rod Green distributions” must not mean that the Newton inverse is replaced by a retarded spacetime inverse.

We use two published analytic results as explicit mathematical inputs, after checking their hypotheses:

1. The Cauchy and Green-operator theorems for normally hyperbolic bundle operators on globally hyperbolic spacetimes: BGP (Theorem 3.2.11 and Corollary 3.4.3).

2. The bundle-valued Feynman-parametrix and exact Hadamard-bisolution construction: IS (Theorems 1.2 and 1.4). Only the existence and microlocal conclusions are used. No positivity conclusion for an indefinite bundle pairing is imported.

Hack's Einstein–Klein–Gordon construction and the BFHPR relational framework are comparison templates Hack,BFHPR. Their particular gauges, state choices, and observable definitions are not substituted for the definitions below.

The supplied Cheung records provide version-qualified comparisons and audit shapes CheungEm02E,CheungXVIIIA,CheungXIXC, CheungXIXD. Their unprinted proofs and documentary relationship edges are not premises.

For all predecessor-series comparisons,

This convention does not prohibit using the equations explicitly frozen and proved in the present manuscript. It prohibits treating a documentary relationship as a theorem or a prescription identification.

The construction remains on a fixed background with external cosmic time. No field-dependent coordinate map removes that external structure from the free quantum construction.

Use signature $(-,+,+,+)$, a reduced Planck mass $M>0$, and

On noncompact slices, this notation specifies the local action density and its compactly supported variations. No finite total background action is assumed.

Fix $p>1$, $t_{\ast }>0$, ${\varphi }_{\ast }\in \mathbb{R}$, and

The spacetime is

There are no timelike or spacelike boundary conditions at the missing endpoints $t_{\pm }$. The interval is open.

Write

The background solves $E_{\mu \nu }=E_{\varphi }=0$:

The scalar equation follows from $A^2=2p{M}^2$. Also

Every surface $t=t_0$, $t_0\in I$, is a Cauchy surface of $(\mathcal{M},\bar{g})$.

Proof. Conformal time $\eta$, defined by $\mathrm{d}\eta =\mathrm{d}t/a(t)$, identifies the metric conformally with Minkowski space on an open time interval. Along a causal curve, $\text{abs}\mathrm{d}\mathbf{x}/\mathrm{d}t\le a(t{)}^{-1}$. If an inextendible causal curve had a time endpoint strictly inside $I$, this estimate would give a finite spatial endpoint, and the curve could be extended. Its time range therefore crosses every $t_0\in I$, exactly once for a future-directed causal curve after the usual harmless removal of constant parameter intervals.

Retain the factorial convention

The classical coefficient fields are smooth and have a common compact spatial support on each compact time interval used in the observable calculation. They need not solve the perturbed Euler equations.

The field-dependent cosmic label is

The raw matter-scalar lapse, on the positive branch, is

It is not silently replaced by the lapse calibrated to $T_s$. The two normalizations satisfy

Let $t={\theta }_s(\tau ,\mathbf{x})$ invert the clock at fixed spatial coordinate. Then

All inverse coefficients are evaluated at $t=\tau$.

On each clock slice, use the induced metric

The original rods are defined by

There is no lapse weight in this equation.

Write

The complete embedding contributions are

Define

Then

The spatial inverse is

For $R_s=x+s{Y}_1+s^2{Y}_2/2$,

Equations (eq:Y1)–(eq:Y2) give unique smooth decaying rod coefficients on the stated classical domain. They are $O(\text{abs}{\mathbf{x}}^{-2})$ on compact time intervals.

Proof. Both sources are divergences of compact smooth spatial vectors. For the second source, $B_1\partial Y_1$ is compact because $B_1$ is compact. Newton convolution gives smooth solutions. Integration by parts transfers one derivative to the Newton kernel and yields the $1/r^2$ bound.

The difference of two decaying solutions is entire harmonic and tends to zero at infinity. The maximum principle on successively larger balls makes it zero.

This elliptic realization is retained unchanged. No hyperbolic Green theorem removes its spatial boundary condition.

Let

Use $\bar{g}$ to identify vector and covector gauge descriptors. For $z=(h,\phi )$, $w=(j,\omega )$, and $a,b\in \mathcal{W}$, the integrated pairings are

These pairings are nondegenerate, not positive definite. Complexification uses their Hermitian versions.

Set $v_{\mu }={\partial }_{\mu }\bar{\varphi }$. The gauge generator is

where $\nabla$ is the background covariant derivative. Its adjoint is

The first curvature variations are

The matter variations are

All potentials in these expressions are evaluated at $\bar{\varphi }$.

The coupled action Hessian is the operator

The metric factor $-1/2$ is essential. It follows from

At an on-shell background, the variations of the volume and of the indices multiplying $E$ contribute no extra term to the second variation. Therefore

is the actual quadratic action.

The operator $\mathcal{H}$ is formally self-adjoint with respect to $(\cdot ,\cdot {)}_0$, and

Proof. Self-adjointness is symmetry of the second variation on compact variations. Covariance of the Euler equations and the on-shell background give $\mathcal{H}K=0$. Taking the formal adjoint gives the second identity.

The raw Euler map, the action Hessian, and its fiber-normalized wave operator are different maps. Their inverses must not be identified without the factors displayed below.

This section supplies a finite coefficient algorithm for the quadratic Einstein–scalar terms. It also fixes what “through quadratic order” means. No nonlinear differential operator is being assigned a linear Green inverse.

Define

For the jet (eq:jet), the inverse metric coefficients are

The connection difference coefficients are

Thus

Put $a_{\mu }={\partial }_{\mu }\phi$, $b_{\mu }={\partial }_{\mu }\psi$. For $U_s={g_s}^{-1}(\mathrm{d}{\varphi }_s,\mathrm{d}{\varphi }_s)$,

Write $V_0=V(\bar{\varphi })$, $V_1=V^{\prime }\phi$, and $V_2=V^{\prime }\psi +V^{\prime \prime }{\phi }^2$. Then

For the scalar equation, define

Its complete second coefficient is

Equations (eq:J)–(eq:scalar2), together with (eq:Gamma1)–(eq:scalar1), give every coefficient of the coupled Euler equations through $s^2$. The second coefficient separates uniquely into the linear operator applied to $(k,\psi )$ and a homogeneous quadratic source in $(h,\phi )$.

Proof. The exact connection-difference formula is

Its Taylor expansion gives (eq:C). The exact curvature-difference formula is its covariant derivative plus the two connection products in (eq:R2). Contracting with the inverse metric gives the scalar curvature coefficients.

Expanding the stress tensor gives (eq:U) and (eq:T2). Finally,

gives (eq:A) and (eq:scalar2). Every occurrence of $k,\psi$ is linear; setting them to zero leaves the homogeneous quadratic source.

Rods do not add terms to these bulk equations. Their complete contributions belong to the insertion, and are retained in Sections sec:freeze and sec:lapse.

Retain XX-A's gauge-condition candidate:

Fix

Thus a previously unspecified positive gauge parameter is now fixed; the rod prescription is not changed.

The free graded fields are

with $z,B$ even and $c,\bar{c}$ odd. Set

The gauge fermion and quadratic action are

The odd Leibniz rule gives the ghost sign in (eq:Sgf). Since $\mathcal{H}K=0$, $s_0{S}_{2,\mathrm{gf}}=0$ and ${s_0}^2=0$ on the off-shell free multiplet.

Let

Its adjoint and the adjoint of $\mathcal{F}$ are

The ghost operator is explicitly

It is normally hyperbolic and formally self-adjoint for $(\cdot ,\cdot {)}_1$.

The auxiliary equation is $B=-r\mathcal{F}z$. Eliminating $B$ gives

Its operator is

Define the four-dimensional trace reversal

and the parallel invertible fiber map

The operator

has differential principal part ${\square }_{\bar{g}}{\mathrm{id}}_{\mathcal{V}}$. All its lower-order coefficients are smooth on $\mathcal{M}$. It is formally self-adjoint for the nondegenerate pairing

Proof. The second-derivative part of the linearized Einstein tensor is

Therefore the metric block of $\mathcal{H}$ has second-derivative part

The second-derivative part of $r{\mathcal{F}}^{†}\mathcal{F}$ is $r{D}^{†}D$. Equation (eq:beta) cancels it exactly.

The scalar block has principal part ${\square }_{\bar{g}}$. Mixed metric–scalar entries have at most one derivative: the background scalar gradient and its derivatives are fixed coefficients. The scalar term in $\mathcal{F}$ likewise creates only first- and zeroth-order mixed entries.

Thus $\mathcal{B}=\mathcal{A}{\square }_{\bar{g}}+\text{lower differential order}$. All coefficients are smooth because $t$ stays away from zero. Finally, $\mathcal{B}$ is self-adjoint for $(\cdot ,\cdot {)}_0$; using $\mathcal{A}\mathcal{L}=\mathcal{B}$ proves the asserted weighted self-adjointness.

This proves hyperbolicity for the actual displayed gauge and normalization. It does not rely on identifying this operator with Hack's differently presented operator.

Within XX-A's compactly supported spacetime gauge group, the linear residual gauge condition is

and its only solution is $\xi =0$.

Proof. A compact spacetime support lies strictly to the future of some Cauchy surface in the slab. Both Cauchy data of $\xi$ vanish there. Uniqueness for the normally hyperbolic operator $Q$ gives zero.

There are nonzero homogeneous $Q$-solutions with general spacelike-compact Cauchy data. They are not compactly supported spacetime transformations of the frozen group.

Likewise, the retarded solution of $Q\xi =-\mathcal{F}z$ need not be compact in time. It does not prove that every configuration has a gauge-fixed representative inside XX-A's compact spacetime group. The gauge-fixed action is defined independently of that unproved reachability claim.

Compact admitted diffeomorphisms preserve the asymptotic rod labels. Their pullbacks preserve the timelike character of a classical scalar gradient and its chosen branch locally around the background. The gauge condition does not itself enforce a pointwise positive quantum lapse.

Write ${G_{\mathcal{L}}}^{\pm }$, ${G_Q}^{\pm }$ for the retarded and advanced Green operators, respectively. The superscripts $+$ and $-$ mean retarded and advanced throughout.

There are unique continuous operators

with the left and right inverse identities and

The correctly normalized reduced action inverse is

Moreover,

Proof. Lemma lem:GH verifies global hyperbolicity. The bundles have finite rank and smooth coefficients. Theorem thm:NH and (eq:Q) verify normal hyperbolicity. No boundary or positivity assumption is needed. The Green theorem BGP (Corollary 3.4.3) therefore applies.

The inverse identities in (eq:GB) follow from $\mathcal{B}=\mathcal{A}\mathcal{L}$. The fiber map is local and invertible, so it changes neither support nor wavefront relations. Formal self-adjointness, integration by parts, and uniqueness give the adjoint identities.

The corresponding distribution kernels use the background volume normalization:

In the displayed coordinates,

Choose the global coframe $\mathrm{d}t,a(t)\mathrm{d}{x}^i$. Spatial homogeneity writes either normalized operator as

The matrices are obtained directly from (eq:H), (eq:Fadj), (eq:Bbulk), (eq:Amap), and (eq:Q); this is an exact coordinate representation, not a new operator.

For each $\mathbf{q}\in {\mathbb{R}}^3$, let $S_P(t,t^{\prime };\mathbf{q})$ be the matrix solution of

Then

The integrals are distributional Fourier integrals.

Proof. The time-derivative jump is $-\mathrm{id}$, so the leading $-{{\partial }_t}^2$ produces the correctly normalized delta. Ordinary matrix ODE uniqueness supplies the mode functions. On compact time intervals, the usual wave-equation energy estimate bounds the Fourier-mode solutions and their fixed derivatives by polynomial momentum weights. Thus the formulas define distributions on compact tests. Their time support, inverse equations, and Green uniqueness identify them with Theorem thm:greens.

This is an exact continuous-momentum representation. No finite-mode approximation or finite-register continuum limit is used.

Define

On compact tests,

The same identities hold for the causal differences.

Proof. The first line follows from $\mathcal{H}K=0$, $\mathcal{F}K=Q$, and self-adjointness. For a compact test $a$, both ${G_{\mathcal{B}}}^{\pm }{\mathcal{F}}^{†}a$ and $-r^{-1}K{G}_Q^{\pm }a$ solve the same inhomogeneous $\mathcal{B}$-equation and have the same retarded or advanced support. Uniqueness gives the first Green identity. Taking adjoints gives the second.

If $\text{calBz}=j$, then

In particular, invariant sources $K^{†}j=0$ and zero subsidiary Cauchy data preserve $\text{calFz}=0$. This is a free subsidiary-equation statement, not an interacting lapse Ward identity.

The even quadratic operator on $(z,B)$ is

With left/right Grassmann derivatives fixed by $-(\bar{c},Qc{)}_1$, the odd matrix on $(c,\bar{c})$ is

The complete retarded and advanced inverses are

\mathbb G_{\mathrm{ev}}^\pm
 &=\begin{pmatrix}
       G_{\mathcal B}^\pm&K G_Q^\pm\\
       G_Q^\pm K^\dagger&0
   \end{pmatrix},
 \label{eq:even-green}\\
 \mathbb G_{\mathrm{odd}}^\pm
 &=\begin{pmatrix}
       0&-G_Q^\pm\\
       G_Q^\pm&0
   \end{pmatrix}.
 \label{eq:odd-green}

They are left and right inverses on compact tests, with causal support in every component.

Proof. The Schur complement of the lower-right even block is $\mathcal{B}=\mathcal{H}-r{\mathcal{F}}^{†}\mathcal{F}$. Its formal block inverse is

Proposition prop:intertwining converts the off-diagonal blocks to those in (eq:even-green). The lower-right block equals

Direct multiplication verifies both inverse identities. The odd inverse is verified by matrix multiplication. Differential operators are local, so all blocks retain the corresponding causal support.

The vanishing $BB$ Green block is a result of the exact calculation. It is not an omission of an auxiliary contact term.

Let ${\mathrm{WF}}^{\prime }$ denote the kernel wavefront relation with the second covector reversed. Define

where $\sim$ means transport along a null bicharacteristic. Let ${\mathcal{N}}^{\pm }$ restrict $x$ to the causal future/past of $y$, and let

For $P=\mathcal{L},\mathcal{B},Q$,

Every component of the full Green matrices has the same corresponding upper bound.

Proof. Away from coincidence, the inverse equation is homogeneous. The normally hyperbolic principal symbol gives null propagation. The diagonal inverse singularity is conormal to the diagonal; the retarded or advanced support chooses the corresponding directed null relation. This is the bundle-valued distinguished-parametrix construction IS.

In the causal difference the inhomogeneous diagonal source cancels. Elliptic regularity excludes non-null covectors. Multiplication by ${\mathcal{A}}^{-1}$, and application of the differential operators in the full matrices, do not enlarge the wavefront set.

A retarded Green distribution can have a smooth tail inside the light cone. The wavefront bound does not assert support only on the null cone.

By contrast, the spatial rod kernel is

At distinct spatial endpoints,

Indeed the nonzero smooth spatial factor preserves the wavefront set of the time delta.

The original rod inverse has instantaneous off-diagonal spatial support and temporal conormal singularities. It is not one of the causal inverses in Theorem thm:full-green.

Proof. Equation (eq:spkernel) is nonzero for arbitrary distinct spatial endpoints on the same time slice, whereas distinct same-time spatial points are spacelike for $\bar{g}$. A retarded or advanced bulk kernel vanishes there.

Thus Q1 supplies the propagating field input and retains the actual classical elliptic coordinate construction. It does not falsify that construction by imposing causal support on its spatial inverse.

The pairings used for $\mathcal{L}$ and $Q$ are indefinite. An algebraic Hadamard contraction can nevertheless be constructed. Positivity on a physical quotient is a different problem.

For a real self-adjoint action operator $P=\mathcal{B},Q$, choose an exact Feynman inverse $F_P$ such that

on compact tests. Choose the Feynman orientation for which $i(F_P-G_P^{-})$ has positive-frequency Hadamard wavefront. This is the convention appropriate to the action operator whose scalar principal differential part is ${\square }_{\bar{g}}$, with signature $(-,+,+,+)$. Under the opposite operator-sign convention, the familiar formula has the opposite prefactor.

The choices above exist. For $\mathcal{B}$, they are obtained from the normally hyperbolic operator $\mathcal{L}$ by

They can be symmetrized under the real formal transpose without changing their exact inverse equations or their Feynman wavefront class.

Proof. The operators are smooth, normally hyperbolic after the displayed fiber normalization, and formally self-adjoint for nondegenerate Hermitian bundle pairings. The spacetime is globally hyperbolic. The exact Feynman construction in IS (Theorems 1.2 and 1.4) therefore applies. The required operator-sign convention is obtained by multiplying the operator and its inverses by $-1$ if needed.

The formula for $F_{\mathcal{B}}$ follows from $\mathcal{B}=\mathcal{A}\mathcal{L}$. Its transpose is also a left and right inverse of $\mathcal{B}$. The Feynman relation is transpose-invariant, so averaging the inverse and its transpose preserves the required microlocal class.