XX-A Original-Rod Lapse: Classical Freeze and Quantum Admission Contract The First Artifact of Track A on a Fixed Cosmological Background

Track A seeks a renormalized relational observable on a fixed external-time background. This first artifact does not announce that quantum construction. It specifies an independent original-rod lapse prescription on an explicit on-shell power-law Einstein–scalar background and proves its complete classical expansion through second perturbative order.

The clock calibration, positive lapse branch, slice-harmonic rod equations, spatial boundary condition, perturbation convention, and gauge group are fixed before calculation. The second spatial rod is constructed, and its absence from the quadratic lapse is proved. An explicit compact perturbation distinguishes the original rods from lapse-weighted modified rods. The complete linear observable and quadratic Hessian satisfy a coupled classical gauge identity; the Hessian alone is not declared linearly gauge invariant.

We also isolate the instantaneous kernel and spatial-tail issues that a quantum construction must address. A regularized polynomial has a conditional finite-order Wick realization once contraction data are supplied, but this fact does not construct the unsmoothed lapse. An exact affine Ward-repair criterion is proved without assuming that its target breaking or counterterm images have been calculated.

The native quantum milestone remains unresolved at Q1: a complete coupled gauge-fixed realization, Green distributions, and normalized contraction data have not been supplied. No retained-order quantum Ward identity, regulator removal, inherited characteristic quantum coefficient, external-time-free evolution, or compilation is asserted. All predecessor comparisons retain declared path null, and the standing insertion, route, and admission boundaries are unchanged.

Keywords: relational lapse; harmonic spatial rods; Einstein–scalar background; classical gauge identity; quantum admission; nonlocal composite; Ward repair.

The proposed Track A milestone is a well-defined quantum relational observable whose rod contributions, product domains, counterterm images, and retained-order gauge identity are all supplied. A plan naming these objects does not construct them.

XX-A is the first classical-prescription and admission-contract artifact of that track. Its mathematical release is the printed classical construction and the comparison and repair theorems below. Its quantum ending is explicitly unresolved.

The motivating target is Cheung's original-rod lapse in Em02-E CheungEm02E. The supplied record distinguishes:

It also distinguishes those rods from modified rods.

The available excerpts do not reproduce every native background constant, rod formula, domain, and normalization needed for a verbatim reconstruction of Em02-E. Consequently, this paper freezes an explicit original-rod variant in Sections sec:background–sec:lapse. Every equation used in its proofs is displayed here. An exact identification with Em02-E would require a separate comparison of all defining data; it is not asserted.

The variant retains the geometric original-rod rule: harmonic spatial coordinates on the actual matter-clock slices, with asymptotically fixed spatial labels. It does not insert a lapse weight into that rule.

The five quantum audit slots of XIX-D are used as a checklist XIXD. No unprinted predecessor proof, catalogue claim, reported artifact, or rebuild result is a theorem premise. Published mathematical frameworks are cited for context and for the precise analytic results identified where used.

XIX-B's reported forcing residual

excludes one direct identification between two specified coordinate prescriptions XIXB,CheungXVID. It does not show that every independently frozen VI harmonic prescription is defective. Choosing the original-rod lapse avoids importing that particular identification; it does not establish that the lapse is easier to quantize.

Likewise, Em02-E's supplied record already specifies a classical observable and classical covariance. Its incomplete quantum audit must not be renamed as failure of all its classical results CheungEm02E. The classical construction below is proved independently on a narrower, explicitly stated domain.

Pure-trace kernel non-determination is the relevant shape of XVIII-A CheungXVIIIA. XVIII-B supplies a different massive stress-response benchmark CheungXVIIIB; it is not silently renamed as that kernel theorem.

Throughout,

This is a relationship convention: the comparisons do not supply a cross-programme identification or import unproved source theorems. It does not mean that a displayed mathematical map is zero.

The work remains on a fixed cosmological background and uses an external cosmic-time coordinate. Field-dependent labels do not remove that background from the quantum construction.

Use signature $(-,+,+,+)$, a reduced Planck mass $M>0$, and the minimally coupled action density

On noncompact spatial slices the background action need not be finite. We use its local Euler equations and compactly supported variations, not an asserted finite total background action.

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

Work on

No limit toward the cosmological singularity $t=0$ is taken.

The Euler equations are

The data (eq:background-data)–(eq:potential) solve (eq:einstein)–(eq:scalar). The gradient of $\bar{\varphi }$ is timelike and future-increasing.

Proof. The Hubble function and scalar derivative are

The nonzero background curvature expressions give

On the displayed scalar,

Thus the energy density is $3M^2{p}^2/t^2$, and $-2M^2\dot{H}={\dot{\bar{\varphi }}}^2$. These verify both independent Einstein equations.

The scalar equation is equivalent to $\ddot{\bar{\varphi }}+3H\dot{\bar{\varphi }}+V^{\prime }(\bar{\varphi })=0$. Its left side is

because $A^2=2p{M}^2$. Finally, ${\bar{g}}^{-1}(\mathrm{d}\bar{\varphi },\mathrm{d}\bar{\varphi })=-A^2/t^2<0$ and $\dot{\bar{\varphi }}>0$.

Let $s$ be a dimensionless formal perturbation parameter. It is not automatically an independently normalized gravitational coupling. Freeze the factorial convention

All tensor components are retained.

The classical coefficients are smooth and, on each compact time interval used in the calculation, have a common compact spatial support. They are not required to solve the perturbed Euler equations. This is an off-shell classical observable prescription on an on-shell background.

Choose a compact label-time interval $J⋐I$, and perform the construction on a slightly larger interval $J_{+}⋐I$. This leaves room for the formal time displacement.

The external label coordinates are $(\tau ,\mathbf{x})$. The clock calibration is fixed by

The raw lapse is calibrated to the matter scalar ${\varphi }_s$, not to a Euclidean norm and not to a separately rescaled clock:

The positive timelike branch is used.

Formal square roots and inverses are well defined around the positive background coefficient. No assertion of a nonlinear global coordinate chart for arbitrary large perturbations is needed.

The proposed quantum target is the coefficient range

A formal truncation is not a numerical error estimate. The classical local expressions below use at most second derivatives of the elementary fields before the displayed spatial inverses. A proposed local counterterm search may be restricted to four derivatives, but such a restriction is not declared exhaustive for this nonlocal observable.

Write

where $T_s({\theta }_s(\tau ,\mathbf{x}),\mathbf{x})=\tau$. Taylor expansion gives

All coefficients on the right of the inverse formula are evaluated at $t=\tau$.

Proof. Substitution yields the first coefficient $u_1+\chi =0$. The second coefficient is $u_2+2u_1\dot{\chi }+\zeta =0$, which gives (eq:time-inverse). Expansion of the exponential in (eq:clock) gives (eq:clock-coeff).

The inverse ${\theta }_s$ is algebraic and local in time to this order. It is not a retarded Green operator.

Embed a clock slice by

Its induced spatial metric is

Write

Direct expansion gives

In particular, scalar gradients, mixed metric components, and the perturbation of the embedding have not been omitted.

On each clock slice, the original rods satisfy

with

The condition is imposed coefficientwise through $s^2$. There is no lapse weight in (eq:rod-exact).

This is a slice-elliptic prescription, not a hyperbolic coordinate Cauchy problem. There are no rod position and velocity data on an initial time surface to replace its spatial boundary condition.

Normalize its density by the background spatial scale:

Let

with spatial indices raised by $\delta$. Then

These formulas follow by multiplying the second-order expansions of the determinant square root and matrix inverse.

Use the Newton inverse

For

the complete coefficient equations and solutions are

Equations (eq:Y1)–(eq:Y2) construct unique smooth decaying rod coefficients on the frozen classical domain. On compact time intervals their spatial decay is $O(|\mathbf{x}{|}^{-2})$. The coefficient maps are continuous in the smooth topology on fixed compact spatial supports.

Proof. The tensors $B_1,B_2$ are smooth and compactly supported in space. The first source is a divergence of a compact smooth vector. Convolution with the locally integrable fundamental solution (eq:newton) is smooth; derivatives can be transferred to the compact source.

The vector inside the second divergence is also compact: the possibly noncompact $\partial Y_1$ is multiplied by $B_1$. The same argument constructs $Y_2$.

For a compact smooth vector $b^i$,

Outside a sufficiently large ball the derivative of $1/|\mathbf{x}-\mathbf{y}|$ is $O(|\mathbf{x}{|}^{-2})$, uniformly for $\mathbf{y}$ in the source support. This proves the decay and its compact-time uniformity.

A difference of two first-order solutions is an entire harmonic function tending to zero at infinity. The maximum principle on large balls gives zero. Once $Y_1$ is fixed, the same argument proves uniqueness of $Y_2$.

On compact spatial sets, convolution and transferred derivatives are bounded by finitely many source seminorms. Outside such sets the explicit kernel estimates give the stated uniform bounds. The sources are finite smooth algebraic expressions in the input coefficients and the previously constructed $Y_1$. This proves continuity and smooth time dependence.

Without decay, constants or other harmonic functions can be added to a homogeneous rod coefficient. Smooth uniqueness for a zero-data hyperbolic problem without spatial decay is therefore not a uniqueness theorem for (eq:rod-exact). No such transfer from XVI-D is made.

Raise four-dimensional indices using $\bar{g}$, and put $v_{\mu }={\partial }_{\mu }\bar{\varphi }$. The inverse metric coefficients are

Let

Its coefficients are

For $N_s=N_0+s{N}_1+s^2{N}_2/2$,

These follow from two derivatives of $w^{-1/2}$.

The cosmic-clock raw lapse is a different normalization:

Since $F^{\prime }({\varphi }_s)=T_s/A$,

Neither normalization is silently substituted for the other.

The forward label map is

Let its formal inverse be

First invert $R_s$ at fixed $\tau$, then insert the resulting spatial argument into ${\theta }_s$. This gives

The observable scalar is

For $f\in {C_c}^{\infty }(J\times {\mathbb{R}}^3)$, fixed in label space, its smeared functional is

The smearing uses the fixed label density, not a fluctuating spacetime volume element.

The frozen relational lapse has the expansion

where

All terms are supplied by (eq:clock-coeff)–(eq:Y2) and (eq:W0)–(eq:N2). The second spatial rod $Y_2$ is constructed but does not enter $n_2$.

For this background,

Proof. Composition with a formal inverse gives

The background $N_0$ depends only on time. Substitution of (eq:alpha1)–(eq:alpha2) gives

which is (eq:n2). The coefficient ${{\alpha }_2}^i$, containing $Y_2$, multiplies ${\partial }_i{N}_0=0$.

Since $v_0=A/t$, ${\bar{g}}^{00}=-1$, and $h^{00}=h_{00}$,

Equations (eq:N1), (eq:time-inverse), and ${\dot{N}}_0=1/A$ give (eq:n1-explicit).

The familiar shape of (eq:n1)–(eq:n2) agrees with the supplied composition formula in Em02-E CheungEm02E. The agreement of that shape is not a proof that every native definition has been identified.

Admit spacetime diffeomorphisms with compact support in $\mathcal{M}$, preserving orientation and the perturbative timelike branch. They are the identity in the spatial asymptotic region and near the time ends of their support. They therefore preserve the fixed asymptotic rod labels.

The group acts by pullback on $g,\varphi$. The background is held as the reference for the perturbative expansion, not treated as a newly dynamical quantum variable.

The formal original-rod construction is gauge invariant through second order:

for the admitted formal diffeomorphisms. Consequently, every ${\mathcal{O}}_f$ is invariant to that order.

Proof. The scalar $T=F(\varphi )$ transforms by composition with $\rho$. The diffeomorphism maps its clock slices to the corresponding old slices. Pullback preserves the induced metric and the slice-harmonic scalar equation.

The pullback of an old rod is therefore a solution of the new rod equation with the same asymptotic labels. The uniqueness argument of Proposition prop:rods, applied coefficientwise, identifies it with the new rod through second order. Thus

The raw lapse is a scalar formed by contraction of a covector with the inverse metric, so $N^{\prime }=N\circ \rho$. The inverse coordinate map is ${\alpha }^{\prime }={\rho }^{-1}\circ \alpha$. Hence $N^{\prime }\circ {\alpha }^{\prime }=N\circ \alpha$. The fixed label-space smearing preserves the equality.

Let $z=(h,\phi )$, $\ell =(k,\psi )$. Define

and polarize $F_{2,f}$ bilinearly. The complete expansion is

For a compact vector field $\xi$, put

For every admitted $\xi$ and every classical $z$,

The second equation does not assert $F_{2,f}(z,K\xi )=0$ separately.

Proof. Use the infinitesimal pullback generated by $s\xi$. The variation of the first jet is $K\xi$, and the variation of the second jet is $2{\mathcal{L}}_{\xi }z$. Differentiate (eq:observable-jet) and use Theorem thm:classical-gauge. The coefficients of $s$ and $s^2$ give the two equations.

As a direct first-order check,

Substitution in (eq:n1-explicit) gives zero.

This is an off-shell kinematic identity. It does not construct an interacting quantum BRST operation on the free solution quotient. In particular, tangency of the nonlinear gauge operation to that quotient is not inferred.

For comparison only, define lapse-weighted rods by

This is a different prescription.

There is a compact metric perturbation for which the quadratic relational lapses formed with (eq:rod-exact) and (eq:modified-rods) differ.

Proof. Take a real $w\in {C_c}^{\infty }(J\times {\mathbb{R}}^3)$, not identically zero, and choose

Then $T=t$, ${\gamma }_s=a^2\delta$, and the original rods are ${R_s}^a=x^a$ through every displayed order. The raw lapse is

The first coefficient of (eq:modified-rods) gives

The composition formula therefore gives

At each time, integration by parts and $\Delta \mathsf{G}w=w$ yield

The difference is supported where $\nabla w$ is supported. Choose a compact spatial smearing equal to one there, and a nonnegative time smearing detecting a time where $w\ne 0$. The smeared difference is strictly negative.

The witness is off shell. It is not asserted to be a solution of the perturbed Einstein–scalar equations. It independently verifies a prescription distinction rather than importing XIX-A's witness as a premise.

A future rod replacement $Y_1\mapsto Y_1+Z_1$ on the same clock slices changes the quadratic lapse, with other data fixed, by

A claim of identity must either prove this expression vanishes on the named domain or supply the compensating changes in the other coefficients. Equality of selected source contractions is not that proof.

Equation (eq:rod-residual) is the present residual-check template. The prepared source $S_{\lambda }$ and the Cauchy residual of XIX-B are documentary warnings, not cosmological rod inputs.

Let

and define $Pz=(\delta E_{\mu \nu },\delta E_{\varphi })$ at the displayed background. This defines the coupled linearized operator without discarding mixed metric–scalar terms.

For completeness, write

Here $\nabla$ is the background Levi-Civita derivative. The matter part is

The formulas follow by differentiating the metric inverse, connection, curvature, and scalar stress tensor.

The displayed operator satisfies $PK\xi =0$. It cannot therefore be used as an invertible gauge-fixed covariance operator.

Proof. The exact Euler equations transform tensorially under pullback. Differentiate $E[{\rho }^{\ast }\bar{g},{\rho }^{\ast }\bar{\varphi }]={\rho }^{\ast }E[\bar{g},\bar{\varphi }]=0$ along the flow of $\xi$. This gives $PK\xi =0$. Compact vector fields with $K\xi \ne 0$ exist, so the kernel is nontrivial.

The field content for the proposed construction is

The scalar fluctuation defining the clock and slices is retained. The ghosts and auxiliary variables are gauge-fixing fields, not additional physical clocks.

Let $S_2(z)=\frac{1}{2}{S^{(2)}}_{\bar{g},\bar{\varphi }}(z,z)$, and define the linear differential

At this linear level ${s_0}^2=0$. Freeze the gauge-fixing candidate

where $\beta >0$ is fixed. Then

with

The sign of the ghost term follows from the odd Leibniz rule. Equation (eq:ghost-operator) follows by applying $\mathcal{F}$ to $Kc$ and commuting background derivatives.

These formulas specify a quadratic candidate and its normalization. They do not prove that its complete reduced metric–scalar realization has the intended Green properties, nor select a state. Eliminating $B$, changing field variables, or using another gauge requires a checked normalization and comparison, not an implicit substitution.

Hack's coupled linearized Einstein–Klein–Gordon construction is relevant framework Hack. It is not used as a premise that the particular $\mathcal{F}$, covariance, rods, spatial tails, or composite in this paper have been constructed. Relational quantum-observable and perturbative BV frameworks likewise do not supply this particular realization by name BFHPR,Frob,BFR.

A Q1 completion must supply a specified realization $\tilde{P}$ of the coupled gauge-fixed operator, its domain, and distributions

with the actual left and right inverse, support, normalization, and field-equation identities. The graded ghost and auxiliary blocks must be included.

A Q2 completion must additionally specify a compatible two-point matrix or subtraction parametrix. In a bosonic physical sector, a state-dependent two-point function $W$ must satisfy the appropriate field equations, Hermiticity, positivity, and

under its declared component pairing; factors of $\hslash$ are then restored in the field expectation.

The Hadamard wavefront condition uses the positive-frequency null relation

Its scalar characterization and appropriate vector-valued extensions have precise hypotheses Radzikowski,SV. They are not claims of positivity for the entire ghost-extended field space.

Suppose a bosonic physical two-point function $W$ has been constructed, and $v$ is a smooth real solution of its linear field equation. For every $c\ge 0$,

has the same antisymmetric part and singular wavefront data, and is positive whenever $W$ is positive. Distinct $c$'s can therefore give different states with the same stated local singularity condition.

Proof. The addition is a smooth symmetric bisolution. Its pairing with a conjugate test pair is $c|⟨v,f⟩{|}^2\ge 0$. It changes neither the antisymmetric part nor the wavefront set modulo smooth terms.

The lemma assumes a physical $W$; it does not construct one for the present coupled system. Nor does local Hadamard behavior impose the weighted spatial-tail bounds needed for the rods.

The first unclosed quantum gate is Q1. It is not bypassed by calling a quadratic action a covariance.

The first-order rod operator is instantaneous in clock time. Its scalar Newton kernel has the spacetime form

The actual rod kernels contain its derivatives and the explicitly specified coefficient matrices.

For compact $b$ of nonzero spatial integral,

Thus the first rod can have a nonzero $1/r^2$ tail.

There are compact perturbations for which $Y_1$ is not compactly supported. Consequently, the rod part of the Hessian of a compactly label-smeared lapse is not automatically a compact joint field kernel.

Proof. Take $\phi =h_{00}=h_{0i}=0$ and $h_{ij}=2a^{2}b{\delta }_{ij}$, where $b$ is compact and has nonzero spatial integral at some time. Then $C=2bI$, $B_1=bI$, and ${Y_1}^i=-\mathsf{G}{\partial }_{i}b$. The displayed asymptotic shows its noncompact tail.

Polarize the rod term $-2{Y_1}^i{\partial }_i{n}_1$ against a second perturbation having $h_{00}\ne 0$ near the label-smearing support. Its dependence on the first perturbation is through the Newton kernel even at distant spatial points. No compact support in those first-input variables follows from compact support of the output test.

The classical integrals are defined on the frozen spatially compact input class. Quantum fields are distributions, not compactly supported smooth configurations. Extension to their actual smearing spaces needs its own tail theorem.

On $\mathbf{x}\ne \mathbf{y}$,

At the full diagonal there are, in addition, conormal directions from the spatial Newton singularity. In particular, the kernel cannot be treated as smooth merely because its spatial endpoints are distinct.

Proof. Away from the spatial diagonal the Newton coefficient is smooth and nonzero. Multiplication by it preserves the wavefront set of $\delta (\tau -{\tau }^{\prime })$, which is the conormal set displayed in (eq:instantWF).

At coincidence, the relative-coordinate kernel is the tensor product of a temporal delta and the spatial Newton distribution. The latter is singular at its spatial origin. Its Fourier transform has inverse-quadratic spatial behavior rather than rapid decay; the temporal delta supplies arbitrary temporal frequencies. These produce conormal directions at the full diagonal.

These intermediate kernel directions are not, by themselves, a proof that every final observable fails a microcausal domain test. That test concerns the covectors in the final field slots after the actual operations and smearing.

For any proposed kernel operations, Q3 must check:

1. Products. The component factors $u,v$ must satisfy the appropriate product condition; a standard sufficient condition excludes $(x,k)\in \mathrm{WF}(u)$ and $(x,-k)\in \mathrm{WF}(v)$ simultaneously Hormander (Theorem 8.2.10).

2. Pullbacks. A proposed restriction or composition $F^{\ast }u$ must avoid the normal set

   $$N_F=\{(F(x),\eta ):(\mathrm{d}{F}_x{)}^{\mathrm{T}}\eta =0\}$$

   in its input wavefront set Hormander (Theorem 8.2.4).

3. Pushforwards and tails. Properness on the support, or a separately proved nonproper convergence estimate, is required. Compact external smearing alone does not supply it.

4. Extensions. The actual extension locus, reference distribution, scaling bounds, normalization, and allowed corrections must be specified before an extension theorem is applied.

Local Wick and time-ordered-product existence results have their own local field hypotheses HWlocal,HWexistence. Their conclusions are not an automatic extension theorem for (eq:instantaneous) composed with the complete nonlocal lapse Hessian. Brunetti–Fredenhagen's finite-scaling-degree extension criterion concerns an already defined reference distribution and an actual scaling bound BF (Theorem 5.2); it does not create the absent reference product.

In particular, the phrase “extend across the light cone” is not one universal operation. Some positive-frequency products already exist by the wavefront criterion, whereas time-ordered or nonlocal characteristic compositions may need extensions. The locus and the operation must be identified separately.

Failure of a sufficient ordinary-product criterion is not a theorem that every renormalized or generalized realization fails.

Let $P_{ϵ,R}$ be a fixed, properly supported smoothing operator whose kernel is smooth and whose relevant input and output supports are compact. A radius $R$ and a smoothing width $ϵ>0$ are independent external parameters.

Suppose normalized free contraction data $W,E$ on the required graded test space have been supplied. If a linear and quadratic observable have smooth compact coefficient kernels $l,k$, then all finite Wick contractions needed for that regular polynomial are defined.

For

one bosonic Wick convention is

Its finite polynomial star products are defined by the corresponding finite contraction sums.

Proof. Every contraction pairs a distribution or a tensor product of distributions with a smooth compact joint test kernel. The tensor products exist, and the pairings are finite by distributional continuity. A polynomial has only finitely many nonzero functional derivatives, so the contraction sums terminate. The displayed subtraction removes the unique self-contraction of the quadratic term in that convention.

Applied to a sufficiently smoothed and spatially cut off $F_{1,f},F_{2,f}$, this lemma gives a regular comparison polynomial conditional on Q1–Q2 data. It does not prove a realization of the original noncompact Hessian, a regulator-removal limit, or a quantum gauge identity.

A fixed smoothing operator generally changes the classical prescription. Under a linear gauge transformation, the regularized linear functional has variation

using (eq:linear-ward). There is no reason for this residual to vanish unless the particular smoothing comparison is proved compatible.

Thus ordinary field-independent smoothing cannot be advertised simultaneously as harmless classical identification and as proved gauge-preserving regularization.

A prospective counterterm calculation must distinguish:

A correction supported on an internal diagonal can acquire a spatial tail after a Newton inverse. No final support claim is made before that image is calculated.

The suggested four-derivative local search is only a declared search class. If the computed target requires other structures, one must report that fact or explicitly revise the class. A finite list copied from a different bulk two-point problem does not establish exhaustive coverage.

The complete classical insertion is (eq:observable-jet). Its homogeneous nonlinear part includes all of (eq:n2), not only $N_2$.

A possible renormalized coefficient would have the form

where $J_{\mathcal{R}}$, its domain, and $C_f$ must actually be constructed. Equation (eq:quantum-template) is a specification of the target, not its definition by fiat.

If interactions are included with the amplitude normalization (eq:jet), their action expansion must also be retained. For compact variational arguments around the on-shell background,