XIX-A Clock Separation Before Coordinate Identification: Four Labelled Objects and Five Non-Identifications

We distinguish four time-related constructions: a finite-time harmonic relational scalar, an inflaton-clock construction with original slice-harmonic spatial rods and its relational lapse, a prescribed conical incidence parameter with a finite register, and a smooth ramp inserted into a coordinate density. The last object is a regulator, not a dynamical clock.

The distinctions are formulated as statements about domains, defining equations, auxiliary data, and functional dependence. They are supported by explicit calculations. A relational scalar value does not determine the scalar-clock lapse. Original and modified cosmological rods have different first variations, and an explicit compact perturbation makes their quadratic relational lapses different. A fixed density ramp has vanishing field variation and fails the harmonic pure-gauge response test in its transition region. A branch-normal deformation speed depends on the prescribed parametrization and does not determine a matter-defined lapse.

These are classical separation results on the stated kinematic domains. They neither identify the coordinate prescription of Landmark VI with that required by XVI-D nor establish a quantum Ward identity, regulator removal, a continuum evolution, or quantum evolution without an external time. The supplied Family 271 catalogue and its reported Lean scope remain documentary comparisons, not theorem premises.

Keywords: relational observables; harmonic coordinates; scalar clocks; relational lapse; spatial rods; prescribed slicing; density ramp.

The question is whether several constructions called a “clock”, “time”, or “lapse” can be used as the same functional. The answer requires more than a shared word. It requires compatible configuration spaces, prescriptions, normalizations, and dependence on the declared data.

The comparison is motivated by Cheung's Landmark VI, VIII-A, VIII-B, X-A, and Em02-E CheungVI,CheungVIIIA,CheungVIIIB,CheungXA,CheungEm02E. The supplied records separate their classical constructions from their remaining quantum and dynamical obligations. We do not use a documentary separation edge as a mathematical premise. Instead, the definitions needed for the present calculations are written below, and the separation statements are proved directly.

The general relational-coordinate framework is developed in BFHPR,Frob. Hyperbolic coordinate prescriptions and slice-elliptic prescriptions are different choices within that framework. Their equations and boundary data form part of the observable definition.

correction to an overstrong reading. “Not the same functional” does not mean that the numerical values must differ at every configuration or point. Nor does it mean that no future comparison map could relate the constructions. The claim proved here is that they are not interchangeable under the declarations in this paper. Where two expressions can be compared on a common domain, we give a detecting variation or a counterexample to the proposed identification.

correction to an underdetermined reading. The supplied VI excerpts state that a harmonic-coordinate Cauchy prescription is fixed, but do not reproduce all of its defining equations and initial data. Consequently, Section sec:harmonic declares an explicit finite-time harmonic prescription for the mathematical comparison. It does not silently authenticate every detail of the registered VI prescription. A VI instance can use the results only after its actual prescription has been matched to the displayed one. This qualification prevents a separation paper from becoming an unproved identification paper.

domain. Unless explicitly stated otherwise, perturbations below are off shell. They need not solve the coupled Einstein–scalar equations. A counterexample on this kinematic domain is not automatically a counterexample on a smaller solution space.

boundary. The paper does not establish quantum evolution without an external time. It constructs no interacting quantum product, state, field-theoretic instrument, or dynamical continuum limit.

A declared functional object consists of

where $\mathcal{D}$ is its domain, $\mathcal{V}$ its target, $F:\mathcal{D}\to \mathcal{V}$ its functional, $\mathcal{A}$ its auxiliary data, and $\mathcal{E}$ its defining equations and boundary or update conditions.

A literal functional identification requires the same declared domain and target, with equality of the functionals. An identification between different domains additionally requires specified comparison maps and an equality after those maps. Equality of one evaluation is not either identification.

The labels used throughout the paper are

These are four labelled entries in the time-related ledger. They are not four verified physical clocks.

Let $\pi :\mathcal{D}\to \mathcal{B}$ and $F:\mathcal{D}\to \mathcal{V}$. There exists a map $\overline{F}:\pi (\mathcal{D})\to \mathcal{V}$ such that $F=\overline{F}\circ \pi$ if and only if $F$ is constant on every fiber of $\pi$.

Proof. A composition with $\pi$ is constant on each fiber. Conversely, if $F$ is constant on fibers, define $\overline{F}(\pi (d))=F(d)$. Fiber constancy makes this definition independent of the representative.

The lemma will distinguish a scalar value from the lapse depending on its gradient, and a geometric schedule from a lapse requiring additional matter data. It is an elementary criterion, not a transfer of a companion theorem.

Let

All fields extend smoothly to a neighborhood of the closed slab. We use a formal parameter $e$ and work modulo $e^3$:

The coefficients $h,k$ are real smooth symmetric tensors with spatial supports contained in one fixed compact set over the slab. Indices in this subsection are raised with $\eta$.

Write the inverse metric density as

Put $h_{\mathrm{tr}}={\eta }^{\mu \nu }{h}_{\mu \nu }$ and $k_{\mathrm{tr}}={\eta }^{\mu \nu }{k}_{\mu \nu }$. Direct matrix expansion gives

The formal coordinate map

is defined by

and the ordinary coordinate-time Cauchy conditions

A metric-dependent unit-normal derivative is not substituted for ${\partial }_t$ in these conditions.

Expanding (eq:harmonic) gives

Equations (eq:W1)–(eq:W2) with (eq:harmonic-data) have unique smooth coefficient fields. Uniqueness does not require spatial decay of the compared smooth solutions.

Proof. For a smooth spatially compact forcing $F$, the solution of ${\square }_{\eta }u=F$ with zero coordinate-time data is

The quotient has its continuous value $t-s$ at zero frequency. Spatial Fourier transformation verifies the equation and data. Smoothness follows by differentiating the Fourier integral, or by the standard local wave estimates. Apply this construction first to (eq:W1), and then to (eq:W2). The second forcing is spatially compact because $A_1,A_2$ are.

For uniqueness, let $u$ be a smooth homogeneous solution with zero data. On a shrinking ball of radius $R-(t-t_0)$, its wave energy

has nonpositive derivative. The boundary flux is

Since $E(t_0)=0$, the solution vanishes on the corresponding interior cone. Taking $R$ large enough treats any chosen point. The argument is local and imposes no condition at spatial infinity. Apply it successively to the differences of the first and second coefficient solutions.

This calculation is the flat finite-time version of the Cauchy construction discussed in BGP. It does not construct an interacting gravitational solution.

For a formal scalar

define

The inverse is formal, so no global numerical convergence of a perturbation series is asserted.

The coefficients of (eq:rel-scalar) are

Proof. Solving ${\mathsf{H}}_{q_e}\circ {{\mathsf{H}}_{{\mathsf{q}}_{\mathsf{e}}}}^{-1}=\mathrm{id}$ coefficientwise gives

Taylor expansion of $T_e$ at this inverse map gives the formula.

A harmonic coordinate component ${{\mathsf{H}}_{\mathsf{q}}}^0$ and the scalar $T\circ {{\mathsf{H}}_{\mathsf{q}}}^{-1}$ are different objects. Moreover, a general relational scalar need not have a timelike gradient and therefore need not be a scalar clock.

Let $F_e$ be a formal diffeomorphism equal to the identity on a neighborhood of the initial surface, with compactly supported positive-order generators. Then

Proof. The scalar wave equation is covariant under pullback, so ${\mathsf{H}}_{q_e}\circ F_e$ solves the coordinate equation for $F_e^{\ast }{q}_e$. The initial data agree because $F_e$ is the identity in an initial collar. Coefficientwise uniqueness proves the first equality. Since $F_e^{\ast }{T}_e=T_e\circ F_e$, the inverse-map identity proves the second.

For example, if $F_e=\exp (e\xi +\frac{1}{2}{e}^2\zeta )$, the factorial convention gives

A second-order coordinate expansion is not combined with only a first-order metric transformation. The covariance statement is classical and includes the complete retained jet.

Let $q$ have signature $(-,+,+,+)$, and let $T$ be a smooth scalar with

The clock orientation is chosen so that $T$ increases to the future. Its future unit normal and raw lapse are

Thus $n_T(T)=\sqrt{S}$ and $N{n}_T(T)=1$.

The lapse is neither the scalar value $T$ nor its normal derivative. It is a functional of the metric and the first derivatives of $T$.

Let ${\Delta }_{q,T}$ be the Laplace–Beltrami operator of the Riemannian metric induced on a $T$-level set. The original rods satisfy

These are slice-elliptic equations, not spacetime wave equations. The original and modified choices in BFHPR (arXiv equations (28)–(35)) must be distinguished. The supplied Em02-E record explicitly selects the original rods CheungEm02E.

For a fixed background clock $T_0(t)$ with ${\dot{T}}_0>0$, let

The relational lapse is

The fixed clock calibration in (eq:relmap) is part of this definition.

Use the displayed background of the supplied Em02-E record:

Work on a fixed closed subinterval of $(t_{\ast }/2,2t_{\ast })$. There is no limit toward $t=0$. The background lapse is

Only the displayed geometry and nondegenerate clock are needed below. We do not import an Einstein-equation theorem as a premise of a separation calculation.

On (eq:background),

In particular, the inflaton clock is not a spacetime-harmonic coordinate component on this background.

Proof. For a homogeneous scalar, ${\square }_{q_0}u=-\ddot{u}-3H\dot{u}$. Substitution of (eq:background) gives the formula.

This lemma concerns the clock field itself. It does not identify or compare an independently dressed matter scalar with a coordinate component.

All perturbations use the factorial convention

Write

Matrix inversion gives

Consequently,

No vector or tensor metric components have been discarded.

The raw lapse coefficients are

Write

Define

The coefficients of ${\mathcal{N}}_{\mathrm{E}}=N\circ \alpha$ are

In particular, $Y_2$ does not occur in $n_2$.

Proof. Let

The equations $T_e\circ {\alpha }_e=T_0$ and ${Y_e}^i\circ {\alpha }_e=x^i$ imply

At second order in the clock equation,

Since ${\partial }_i{u}_1=-{\partial }_i{\theta }_1/v$, this gives

Taylor expansion yields

Because $N_0$ depends only on $t$, this becomes

Substitute $z_1=-Y_1$. The second spatial displacement multiplies ${\partial }_i{N}_0=0$, so $Y_2$ is absent. The first-order formula follows from the same Taylor expansion.

The result agrees with the composition structure recorded in Em02-E CheungEm02E, but its proof here uses only the displayed definitions. It is not a quantum gauge identity.

The harmonic relational scalar ${\mathcal{R}}_{\mathrm{H}}$ and the scalar-clock relational lapse ${\mathcal{N}}_{\mathrm{E}}$ are different functional prescriptions.

More concretely, on a common flat, homogeneous comparison domain:

1. ${\mathcal{R}}_{\mathrm{H}}$ and ${\mathcal{N}}_{\mathrm{E}}$ are not equal as functionals;

2. at a fixed point, ${\mathcal{N}}_{\mathrm{E}}$ does not factor through the scalar value ${\mathcal{R}}_{\mathrm{H}}$ alone;

3. original spatial rods are not the relational scalar.

These statements do not exclude identities on specially restricted families or after additional, explicitly specified constructions.

Proof. Take $q=\eta$. The harmonic map of Definition def:harmonic is ${\mathsf{H}}_q=\mathrm{id}$. For any homogeneous clock $T(t)$ with $\dot{T}>0$, the original spatial rods are ${Y_T}^i=x^i$. Choose the clock itself as the background reference, so the relational map is the identity. Then

For $T(t)=ct+b$ with $c>0$, these are $ct+b$ and $1/c$. They are not the same functional.

For the factorization claim, fix a background clock $T_0(t)=ct+b$, a point $p=(t_c,{\mathbf{x}}_c)$, and a smooth time bump $\chi$ equal to one near $t_c$. Compare

For sufficiently small $e$, these are timelike clocks. At $p$ they all have the same scalar value,

The relational inverse sends $p$ to $p$ in time because the clock values agree there. Spatial rods remain $x^i$. Thus

The lapse varies with $a$ on a fiber of the scalar-value map. Lemma lem:factor excludes factorization through that value.

Finally, on the homogeneous flat family the scalar is independent of $\mathbf{x}$, whereas ${Y_T}^i=x^i$ is not. Even componentwise, they cannot be equal as fields on an open spatial set.

The homogeneous flat comparison is not the power-law Einstein–scalar solution of Em02-E and is not an assertion about a restricted VI solution jet. It detects nonidentity of the displayed kinematic prescriptions. The native manuscript objects also retain different domains and auxiliary data.

Classical covariance or descent of a functional does not imply existence of a regulator-removal limit, even when every positive-regulator member is smooth and covariant.

Proof. Let a gauge group act on any nonempty configuration space and let $C$ be the constant scalar field $1$ on a spacetime region. For $\delta >0$, set

Every $Z_{\delta }$ is smooth and gauge invariant. For a compact test $f$ with $\int f{\mathrm{d}}^{4}x\ne 0$,

has no finite limit. Hence covariance does not imply distributional regulator removal.

The example is a logical counterexample, not a model of the Em02-E quantum family. It justifies keeping the following statement as a lemma-level separation:

The supplied VI record leaves its quantum certificate RG-open; the supplied Em02-E record leaves its quantum audit incomplete CheungVI,CheungEm02E. Neither status is changed here.

Use

for $f\in {C_c}^{\infty }({\mathbb{R}}^3)$. Then $\Delta G_{\Delta }f=f$, and $G_{\Delta }f$ decays at infinity.

For an induced metric ${\gamma }_e=a^2(\delta +ec+\frac{1}{2}{e}^{2}d)$ with compactly supported coefficient perturbations, expand its Laplacian as

The rod equations recursively read

Their right-hand sides are spatially compact: the perturbation operators have compact coefficient support. Application of $G_{\Delta }$ gives smooth decaying coefficients. A smooth harmonic difference tending to zero at infinity vanishes by the maximum principle on expanding balls. This proves coefficientwise existence and uniqueness for the rod prescriptions used below.

Consider the scalar metric subclass

where $A,D$ are spatially compact smooth functions.

For the original rods, the clock slices are exactly $t=\text{constant}$. For the modified rods, it suffices to use the first clock jet

Define ${Z_e}^i$ to be harmonic on the ${\Theta }_e$-slices with the same asymptotic Cartesian boundary condition.

Equation (eq:modified-clock) has the first jet of the modified slicing used in BFHPR. No choice of its second-order coefficient is needed for the quadratic distinction proved below. In both relational observables, the {time coordinate remains $T_e$}; only the spatial rods are replaced.

On (eq:scalar-metric),

Thus original and modified rods are not identical functional prescriptions.

Proof. On an original clock slice,

Expanding ${\Delta }_{\gamma }f=(\det \gamma {)}^{-1/2}{\partial }_i((\det \gamma {)}^{1/2}{\gamma }^{ij}{\partial }_{j}f)$ gives

The first harmonic equation is $\Delta {Y_1}^i=-{\partial }_{i}D$, which proves the first formula.

A modified slice through background time $\tau$ has embedding

because $v={\dot{T}}_0$. Its induced spatial metric at first order is

The contribution involving ${\partial }_{i}t{\partial }_{j}t$ starts at order $e^2$; the lapse perturbation $A$ therefore does not alter this first-order calculation. The modified harmonic equation is $\Delta {Z_1}^i=0$ with decaying boundary data, so ${Z_1}^i=0$. For any $D$ such that ${\partial }_i{G}_{\Delta }D$ is nonzero, the prescriptions differ.

Let $\psi \in {C_c}^{\infty }({\mathbb{R}}^3;\mathbb{R})$ be nonconstant and let $b$ be a smooth time bump supported strictly inside the comparison interval. In (eq:scalar-metric), choose

Let $n_{2,Y}$ and $n_{2,Z}$ be the second coefficients of the relational lapse using original and modified rods, respectively, with the same clock $T_e=T_0$. Then

The difference is detected by a nonnegative compact spacetime test. Hence the quadratic relational lapse depends on the rod choice.

Proof. Since $G_{\Delta }\Delta \psi =\psi$ for compact $\psi$, Lemma lem:rod-first gives

There is no clock perturbation, so $u_1=u_2=0$. For (eq:scalar-metric), the raw lapse is

and therefore

The local raw second coefficient is the same for the two rod choices. Proposition prop:lapse-composition consequently gives

Choose a nonnegative test supported where $b\ne 0$ and $\nabla \psi \ne 0$, positive on a nonempty open subset. Its pairing with the displayed difference is strictly negative.

The witness is an off-shell compact perturbation of the cosmological background. It proves that the two prescriptions are different, not that every configuration distinguishes them. The proof uses only their first spatial jets, so it applies to any smooth completion of the modified prescription having (eq:modified-clock). It does not reconstruct or verify every higher-order formula in the external reference.

On the cosmological background, original slice-harmonic spatial rods are not interchangeable with spatial harmonic coordinates having fixed zero perturbative Cauchy data.

Proof. The background spatial functions $x^i$ are both slice harmonic and spacetime harmonic. This is a background coincidence. Choose the perturbation $D=-b(t)\Delta \psi$ with $b(t_0)\ne 0$. The original rods have

The perturbative spacetime-harmonic coordinates with fixed zero Cauchy positions have ${W_1}^i(t_0,\mathbf{x})=0$. They therefore differ at the initial surface. Existence of the corresponding hyperbolic coefficient problem follows from the Cauchy theorem for the smooth background BGP; only the fixed initial position is needed for the contradiction.

The lemma concerns the stated Cauchy prescription. Replacing its initial position by the elliptic rod value changes that prescription and requires a separate comparison.

The supplied Em02-A record distinguishes localization asymptotics in $ϵ$ from smoothing asymptotics in $\delta$ CheungEm02A. The supplied Em02-D record concerns one fixed distribution model with leading inverse-cubic smoothing behavior CheungEm02D. Neither result defines the quantum contraction of the lapse in this paper.

Let $t_{\delta }={\delta }^{-r}C+R_{\delta }$ in a topological vector space, where ${\delta }^r{R}_{\delta }\to 0$. If $B$ is a fixed continuous linear comparison map, then

Nonvanishing of $C$ does not imply nonvanishing of $BC$. Without a specified comparison map and matching regulated families, this expansion determines no leading coefficient for a different observable.

Proof. Apply continuity of $B$ to ${\delta }^r{t}_{\delta }=C+{\delta }^r{R}_{\delta }$. A nonzero $C$ may belong to $\ker B$, including when $B$ is a nonzero projection. Thus the source coefficient need not survive the target operation. An expansion in the source space, by itself, contains no equality defining a distinct target family.

Even the parameters cannot be identified by notation alone:

is constant along $ϵ={\delta }^{3/2}$, divergent along $ϵ=\delta$, and tends to zero along $ϵ={\delta }^2$. A joint limit requires its own hypotheses.

Accordingly, neither Em02-A's localization powers nor Em02-D's inverse-cubic coefficient is assigned to the relational-lapse kernel. The rod distinction of Theorem thm:rod-detection is already enough to rule out replacing the original rods by a modified formula without recalculation.

Choose a fixed $\chi \in C^{\infty }(\mathbb{R})$ with

and a nonempty transition region. Put

The X-A-type regulated density is

It defines a coordinate equation by replacing ${\mathfrak{a}}_e$ with ${\mathfrak{b}}_{\epsilon ,e}$ in (eq:harmonic), with the same zero coefficient Cauchy data.

This is the density-ramp placement described in the supplied X-A record CheungXA. It is not field-argument convolution, and it is not a scalar-clock equation.

In this paper, a dynamical scalar clock is a scalar functional of the dynamical configurations whose gradient is timelike and nonzero on its declared clock region, with a fixed calibration. A prescribed external profile is not a dynamical clock merely because it is a function of coordinate time.

With ${\sigma }_{\epsilon }$ held fixed:

1. its variation with respect to $q$ and $T$ is zero;

2. $\mathrm{d}{\sigma }_{\epsilon }=0$ on both plateau regions, so it fails the nondegeneracy criterion there;

3. the same ramp is compatible with different scalar-clock lapses;

4. it is not the harmonic relational scalar or the incidence register update.

Proof. The first claim follows from (eq:ramp), which contains no dynamical field. The second follows from the constant values of $\chi$.

For the third, keep $q=\eta$ and the same ramp, and use ${T^{(a)}}_e$ from Lemma lem:scalar-lapse. The ramp is unchanged while the lapse at the detecting point is $1/(c+ea)$. Lemma lem:factor excludes determination of that lapse by the ramp alone.

For the fourth, changing $T$ changes ${\mathcal{R}}_{\mathrm{H}}$ even at $q=\eta$, whereas it does not change the ramp. The register is a map on a finite code set with a declared update; a fixed smooth function on a time slab is not that map or its update.

The lemma does not deny that a strictly monotone segment of an external profile can be used to reparametrize an interval. That operation remains an externally prescribed parametrization; it does not turn (eq:ramp) into the matter-defined clock of Section sec:inflaton.

Let $\xi$ be a compact smooth vector field vanishing in an initial collar, and set

For the first metric-density coefficient,

Let $G_{\mathrm{C}}$ denote the zero-Cauchy solution operator for ${\square }_{\eta }$ on the slab.

The unregulated first coordinate correction satisfies $W_1[K\xi ]=\xi$. The ramped first correction satisfies

It returns $\xi$ when its support lies strictly inside the ${\sigma }_{\epsilon }=1$ plateau. There is a compact pure-gauge input in the transition region for which it does not return $\xi$.

Proof. For $h=K\xi$,

Taking the divergence proves (eq:pure-gauge-density). The unregulated equation is ${\square }_{\eta }{W}_1={\square }_{\eta }\xi$ with the same zero data as $\xi$; uniqueness proves $W_1=\xi$.

The ramped equation is

Subtract the equation for $\xi$ and apply $G_{\mathrm{C}}$. On the plateau the forcing in (eq:ramp-defect) vanishes.

For a detecting input, choose a transition time $t_c$ with ${\sigma }_{\epsilon }(t_c)<1$, a compact spatial bump $\psi$ equal to one near ${\mathbf{x}}_c$, and a compact time bump $a$ in the transition region such that $a^{\prime }(t_c)=0$ and $a^{\prime \prime }(t_c)\ne 0$. Set

Near the detecting point,

The forcing for ${W_{1,\epsilon }}^0-{\xi }^0$ there equals

If the difference field vanished identically, its wave operator would vanish identically, contradicting this forcing.

A correction beginning at order $e^2$ cannot remove the nonzero order-$e$ defect of Proposition prop:ramp-defect while the ramp and first-order prescription remain fixed.

Proof. Equality of formal series modulo $e^3$ requires equality of their order-$e$ coefficients. Adding an order-$e^2$ term does not change that coefficient.

This is a calculation for the displayed density ramp. It is not an identification with VI's separately reported compact-switch defect. The two defects retain their own prescriptions and domains.

If the external ramp is also pulled back under a diffeomorphism, one instead studies a construction depending covariantly on the enlarged data $(q,T,{\sigma }_{\epsilon })$. Treating a profile as an additional transforming background does not make it a dynamical matter clock.

Let $C_{\mu }$ be a spinless spatial cone with a supplied deficit $\mu$, and use the regular-part ultrastatic metric

The cone and its metric are declared inputs.

A slicing family is a graph

on each regular branch. The incidence parameter $s$ is supplied by the controller, not solved from a scalar-field equation.

A finite implementation consists of a finite code set $\mathcal{J}$, an installed parameter table

and a declared update $U_{\mathcal{J}}:\mathcal{J}\to \mathcal{J}$. When a reversible realization is intended, the corresponding complete finite update is a permutation of its declared code space. These definitions do not acquire a matter expectation or define a Hamiltonian.

This separates the geometric parameter from its finite code, as required by the supplied VIII-A/B records CheungVIIIA,CheungVIIIB. The clock code is denoted by $j$; it is not the finite Hermitian source matrix also written $j[f]$ in that track.

On a regular branch of (eq:graph), the future unit normal is

For the deformation $V_s=({\partial }_{s}F){\partial }_U$, its positive normal component is

Proof. A tangent vector to the graph is $w+({\mathrm{d}}_{x}F)(w){\partial }_U$. Its inner product with ${\partial }_U+\nabla F$ is zero. The squared norm of the latter is $-1+{\left|\nabla F\right|}^2$. Normalization and contraction with $V_s$ give the formulas.

For the local branch data reported in VIII-B, the calculation is

The arithmetic follows directly from (eq:normal-speed). The geometric construction establishing those branch data belongs to VIII-B CheungVIIIB; it is not reconstructed by the table.

In oblique cone coordinates, the gradient norm must be computed with the inverse physical Gram matrix, not with the Euclidean norm of the coordinate covector. Formula (eq:normal-speed) is coordinate independent.

Let $r=f(s)$ with $f^{\prime }>0$. Keeping the same unparametrized graph family,