XX-F Native Ward Admission for the Original-Rod Lapse Filtered Quantum-Jet Tests, Weighted Newton Domains, and the Exact Conditional Q4–Q5 Repair Problem

The proposed XX-F calculation assumes an admitted native Q3 insertion and a quantum Ward operation on its domain. Those objects are not supplied by XX-A–XX-E. An unconditional native breaking term, counterterm cancellation, or interacting identity therefore cannot be released from the attached package.

We nevertheless sharpen the missing interfaces by proving new results on the identical background and free operator. The coupled classical jet transformation has a calculable first-order Leibniz defect on the centrally extended free algebra. A same-operator Cauchy witness proves that its antisymmetric part can be nonzero, excluding a first-order quantum derivation with that leading action on the unfiltered algebra. We then correct the order interpretation: the central-second-jet defect starts at total perturbative order three on a specified weighted regular jet algebra. It is invisible through total order two, while the free-equation quotient obstruction persists at order one.

A weighted Newton theorem replaces unnecessarily strong Schwartz assumptions by explicit decreasing-source bounds, with cutoff-error and output-decay estimates. A finite spatial contact module is reduced to a unique normal form before its Newton image is taken. This prevents presentation-dependent contact counting and illegitimate diagonal readout.

Finally, we derive the exact conditional breaking and repair equations when both the insertion and Ward operation may receive quantum corrections. Linear image membership is separated from nonlinear admissibility restrictions such as positivity. None of these interface theorems verifies the missing native covariance, infrared estimates, quantum operation, or interacting contribution inventory. Native Q3 remains open; Q4 and Q5 remain unentered and unclosed. No universal obstruction to a future native completion is asserted.

The requested deliverable is a native quantum breaking term, a calculated admissible counterterm image, cancellation, and a retained-order quantum identity. The plan explicitly requires a completed native prerequisite package before those calculations begin.

XX-A supplies the classical freeze. XX-B supplies its stated compact-test free input. XX-C retains state and infrared choices. XX-D's native one-contraction construction is conditional. XX-E supplies domain tests and comparison images, not the missing quantum operation. The full texts of these five drafts are available in the present exchange XXA,XXB,XXC,XXD,XXE.

Accordingly, this manuscript is a complete proof of its stated interface results, not a purported completion of the absent native quantum calculation.

numbering. The original inventory assigns complete quantum remainder control to Q4 and the retained quantum identity to Q5. The proposed XX-F plan organizes the breaking calculation and its repair jointly under Q4–Q5. We distinguish these usages rather than declare the original remainder obligation closed by a renamed algebraic test.

discipline. Cheung's records provide comparison shapes: same-target admission, actual-image repair, and nontransfer CheungXVIIIA,CheungXIXD,CheungXIXE. Their documentary status does not supply an unprinted covariance, anomaly, or counterterm coefficient. Theorems proved here use the displayed equations.

Entry into the requested native Q4–Q5 calculation requires:

1. a specified normalized covariance representative;

2. verified products and infrared estimates for the insertion and every operation used by its Ward map;

3. an admitted native insertion, its output space, normalization, and allowed Newton-contact images;

4. a consistent treatment of second jets;

5. a quantum Ward operation on the same domain, including its action on field relations, products, and normalization data;

6. the complete contribution inventory at the orders claimed.

Any operation with two noncompact rod legs also requires its actual two-endpoint covariance domain.

No item is fulfilled merely by introducing its name. In particular, an undefined native breaking is not zero.

frameworks. Regular functional products provide the compact comparison algebra used below FR. Normally hyperbolic Cauchy theory supplies the same-operator witness BGP. Distribution-extension and quantum BV frameworks have separate hypotheses BF,FRBV,BDFrR. They are not invoked as automatic constructions of the nonlocal lapse.

Use signature $(-,+,+,+)$ and

The notation means the local action density with compact variations; no finite total background action is asserted.

Fix $M>0$, $p>1$, $t_{\ast }>0$, and

Work on $I\times {\mathbb{R}}^3$, where $I=(t_{-},t_{+})$ and $0<t_{-}<t_{+}<\infty$. The identities

and $A^2=2p{M}^2$ verify the background equations.

The factorial jet is

The clock label and raw matter-scalar lapse remain

The positive classical timelike branch is used.

All observable smearings below use the fixed label density $\mathrm{d}t{\mathrm{d}}^{3}x$. For comparison with XX-D's background-volume convention,

This identification changes test coordinates, not the geometric prescription.

The rods are harmonic on the actual matter-clock slices:

There is no lapse weight.

The first inverse clock displacement is $u_1=-t\phi /A$. The first induced metric is

Its original density-source map is

Retain

The linear lapse coefficient is

The full second coefficient is

where

For a compact label test $f$, let

If $b$ is the symmetric polarization of $q$, set

Then

In particular,

The second spatial rod is obtained from

It does not occur in $n_2$, because its second inverse-map coefficient multiplies ${\partial }_i{N}_0=0$, with $N_0=t/A$. Nothing in this manuscript discards its construction.

Let $\mathcal{V}=S^2{T}^{\ast }\mathcal{M}\oplus \underline{\mathbb{R}}$. The integrated field pairing is

It is nondegenerate, not positive definite.

At the background define

where $E_{\mu \nu }=M^2{G}_{\mu \nu }-T_{\mu \nu }$ and $E_{\varphi }={\square }_g\varphi -V^{\prime }(\varphi )$. Thus $S_2(z)=\frac{1}{2}(z,\mathcal{H}z{)}_0$.

Retain

Here $\mathsf{T}$ is four-dimensional trace reversal.

The second-derivative metric part of $\mathcal{H}$ is

The $r{\mathcal{F}}^{†}\mathcal{F}$ term cancels the second summand. Mixed metric–scalar entries have at most one derivative. Consequently

Also

The slab is globally hyperbolic: causal curves obey $|\mathrm{d}\mathbf{x}/\mathrm{d}t|\le a(t{)}^{-1}$, so an inextendible curve cannot terminate at an interior time. Normally hyperbolic Cauchy and Green theory therefore applies BGP. The causal action-normalized kernel is

When used in a comparison, $W$ denotes a bidistribution satisfying

No selected physical state or native infrared estimate is inferred from these relations.

For compact spacetime $\xi$, the infinitesimal pullback generated by $s\xi$ gives

Define on classical polynomial functionals

For compact $\xi$,

Therefore ${\mathcal{V}}_{\xi }{O}_f=0(\mathrm{mod}{s}^3)$.

Proof. The gauge fields satisfy

which give $\ell (K\xi )=0$. The time displacement cancels the background spatial scale variation, giving

Its divergence is $-\Delta {\xi }^a$. The decaying Newton inverse gives $\mathcal{R}(K\xi {)}^a={\xi }^a$.

Pullback maps clock slices, their induced metrics, and their harmonic equations to the corresponding new data. Uniqueness of the decaying rod coefficients identifies the transformed rods with the old rods pulled back. The raw lapse is a scalar. Hence its composition with the inverse clock-and-rod map is unchanged. Expanding this equality through second order gives the last identity.

Explicitly, the polarized rod contribution is

It is part of the identity, not a separately invariant piece. Substitution into (eq:O) proves the final statement.

Let ${\mathcal{P}}_{\mathrm{reg}}$ be the algebra of finite polynomials in compactly smeared linear variables

Declare $M_a$ central and use, through first order,

All pairings exist on this regular algebra. This is a bosonic comparison subalgebra; it neither discards interacting ghosts nor represents the native nonlocal lapse automatically FR.

For notational clarity, $A_{\xi }W$ means that the differential operator $A_{\xi }$ acts in the first field slot. Formal adjoints refer to $(\cdot ,\cdot {)}_0$.

On ${\mathcal{P}}_{\mathrm{reg}}$,

where

In particular,

Proof. The operation ${\mathcal{V}}_{\xi }$ is a derivation of ordinary polynomial multiplication. Differentiating its action with respect to $z$ gives

In (eq:star), $W$ is fixed. The terms in which ${\mathcal{V}}_{\xi }$ differentiates ${F_z}^{(1)}$ or ${G_z}^{(1)}$ cancel between the three expressions. The two extra derivative terms are exactly (eq:Leibniz).

There exist real compact tests $a,g$, a compact spacetime vector field $\xi$, and a real spacelike-compact solution $U=Eg$ such that

Proof. Fix $t_0\in I$. Use the normally hyperbolic Cauchy problem for $\mathcal{L}$ to choose compact Cauchy data with scalar component $\phi (t_0)=0$ and $\dot{\phi }(t_0)=b$, where $b$ is a nonzero real compact spatial bump. The other initial components can be chosen zero. No physical subsidiary condition is required for this gauge-fixed comparison.

Finite propagation gives a real solution $U$ with uniformly compact spatial support on compact time intervals. Choose compact $\xi$ whose time component is nonzero where $b\ne 0$. Then the scalar component of $A_{\xi }U$ is nonzero at $t_0$, and hence on a spacetime neighborhood. A real compact test $a$ supported there detects it under the nondegenerate pairing.

Choose a smooth time function $\eta$ which is zero before an interior time and one after a later interior time, both chosen before $t_0$, with compact derivative support in $I$. Put $g=\mathcal{B}(\eta U)$. Since $\mathcal{B}U=0$, $g$ is supported in the transition region; finite propagation makes it spacetime compact.

Cauchy uniqueness gives

Their difference is $Eg=U$. Formal adjunction proves (eq:witness).

On the unfiltered centrally extended algebra above, no operation of the form

can be a ${\star }_W$-derivation for every compact $\xi$, while retaining the stated central second-jet relations and canonical kernel $E$.

Proof. The relation $[M_a,Z_g{]}_{{\star }_W}=0$ would imply

At leading order,

The resulting first-order commutator is $2i\hslash E({A_{\xi }}^{†}a,g)\mathrm{id}$.

Terms from $\hslash R_{\xi }$ have no first-order commutator contribution because ordinary multiplication at order zero is commutative. Lemma lem:witness makes the displayed coefficient nonzero, a contradiction.

Equivalently, a first-order correction has a symmetric ordinary Hochschild coboundary, whereas the antisymmetric part of (eq:Leibniz) contains $-2iE({A_{\xi }}^{†}a,g)$. This theorem excludes the specified unfiltered construction, not all quantum treatments of second jets.

Assign weight one to every $Z_g$ and weight two to every $M_a$. For a regular polynomial $F$, let ${\mathrm{deg}}_{\mathrm{w}}F$ be its weighted degree. Define

The coefficients may be formal in $\hslash$. This is a regular comparison algebra, not admission of a singular or noncompact coefficient.

The contraction in (eq:star) lowers weighted field degree by two. The operation ${\mathcal{V}}_{\xi }$ lowers weighted degree by at least one and does not lower the explicit power of $s$. Thus the product, operation, and quotient in (eq:Rees) are well defined.

On ${\mathcal{R}}_2$, ${\mathcal{V}}_{\xi }$ is a derivation of the retained first-order product. The obstruction in Theorem thm:unfiltered first occurs at total order $\hslash s^3$.

In particular, the unfiltered centrality obstruction cannot by itself exclude a quantum jet comparison claimed only through common total order $s^2$.

Proof. A nonzero term in (eq:Leibniz) requires a $\zeta$-derivative of one coefficient and a $z$-derivative of the other. The former coefficient must have weighted degree at least two and thus explicit order at least $s^2$. The latter must have weighted degree at least one and explicit order at least $s$. Their product is therefore divisible by $s^3$. Every Leibniz defect vanishes in ${\mathcal{R}}_2$.

For the witness pair, use $s^2{M}_a$ and $s{Z}_g$. The nonzero coefficient of Theorem thm:unfiltered is then multiplied by $s^3$, proving the exact order statement.

XX-E's centrality test is a valid test of the unfiltered algebra. It is not an obstruction at every truncation. Theorem thm:filtered corrects that possible overreading. It does not construct native Q3 or an interacting Ward map.

The gauge-fixed free quotient imposes $Z_{\mathcal{B}g}=0$. The weighted version includes $s{Z}_{\mathcal{B}g}=0$.

For compact $\xi$ with $K\xi \ne 0$, the ordinary coupled operation does not preserve the weighted free-equation ideal, even through total order $s^2$. Indeed,

and this is nonzero for some compact $g$.

Proof. Formal self-adjointness gives (eq:quotient). If $\mathcal{B}K\xi =0$, then the compactly supported section $K\xi$ solves the normally hyperbolic equation $\mathcal{L}K\xi =0$. Its zero Cauchy data before its support imply $K\xi =0$, contrary to the hypothesis. Nondegeneracy of compact testing supplies $g$. The scalar coefficient is of order $s$, not $s^3$.

Similarly, compact solutions of $Q\xi =0$ are zero by Cauchy uniqueness. Testing only that residual subgroup gives a vacuous identity. The full compact geometric gauge group is the relevant group in a future native comparison.

The leading two-jet operation alone is not the whole infinitesimal jet algebra. For a compact descriptor $\theta$, define

With commutators of operators on functionals,

These are the retained pullback, or anti-action, sign conventions.

Proof. On $z$, the first commutator is zero. On $\zeta$, it is

The operations are ordinary polynomial derivations, so their values on generators determine them. The second commutator is zero on both generators.

A future consistency or anomaly equation must use a constructed quantum action with its actual bracket conventions. Classical jet brackets do not supply that action.

Write $⟨\mathbf{x}⟩=1+|\mathbf{x}|$. For $q>1$, define decreasing-source seminorms

Sources are smooth and possess the finite seminorms needed for each conclusion. These conditions are weaker than requiring every Schwartz seminorm to be finite.

Let

For the rods, $b^{ia}=(\text{calCz}{)}^{ia}$.

Suppose $q>1$ and all required seminorms (eq:weighted) are finite. Then $\mathsf{N}b$ is absolutely defined, smooth, and tends to zero at spatial infinity. It solves

and is the unique smooth decaying solution.

Uniformly for $t\in J$, its undifferentiated output, and each fixed time derivative with the same source bound, obey

For a bounded cutoff profile ${\chi }_R=\chi (\cdot /R)$ equal to one on the radius-$R$ ball, put ${\mathsf{N}}_{R}b=\mathsf{N}({\chi }_{R}b)$. For every compact output region $K$ and fixed derivative order $m$,

for all sufficiently large $R$.

Proof. Near $\mathbf{y}=\mathbf{x}$, the dipole kernel has locally integrable size $C|\mathbf{x}-\mathbf{y}{|}^{-2}$. At infinity, for compact $\mathbf{x}$, its product with the source has radial majorant $C{r}^{-q}\mathrm{d}r$, integrable for $q>1$.

Local smoothness follows by writing $\mathbf{r}=\mathbf{x}-\mathbf{y}$, splitting with a smooth local cutoff in $\mathbf{r}$, and transferring output derivatives to the smooth source in the singular part. The nonsingular part may be differentiated directly. Weighted source derivatives justify all these operations.

For the decay estimate let $r=|\mathbf{x}|$ be large. In $|\mathbf{y}|\le r/2$, the kernel is bounded by $C{r}^{-2}$. The source integral is bounded by $C{r}^{3-q}$ for $q<3$, $C\log r$ for $q=3$, and $C$ for $q>3$.

In $r/2<|\mathbf{y}|<2r$, bound the source by $C{r}^{-q}$ and the dipole integral over a ball of radius $3r$ by $Cr$. In $|\mathbf{y}|\ge 2r$, use the radial integral $C{{\int }_{2r}}^{\infty }{\rho }^{-q}\mathrm{d}\rho$. These give (eq:weighteddecay). Time derivatives are treated by differentiating only the source.

For compact output and sufficiently large $R$, the cutoff difference is supported away from the kernel singularity. Each output spatial derivative of $P_i$ improves its tail; time derivatives act on $b$. Bounding all such terms by the undifferentiated radial estimate gives (eq:cutoffestimate).

Compact cutoff sources satisfy the distributional Poisson equation. The local smooth convergence, also for two output derivatives, passes it to the limit. A difference of smooth decaying solutions is entire harmonic and vanishes by the maximum principle.

The $q>1$ bound guarantees existence and decay, not necessarily an inverse-square rate. The stronger $q>3$ condition gives that rate for the undifferentiated rod.

At $q=1$, the pointwise bound $|b(\mathbf{y})|\le C⟨\mathbf{y}{⟩}^{-1}$ does not universally ensure absolute Newton-divergence convergence.

Proof. Take a smooth source which outside a ball is

At $\mathbf{x}=0$,

Its spatial integral diverges logarithmically. The source has the displayed rate and can be smoothly filled in inside the ball.

This is a sharp threshold for the stated uniform sufficient absolute-convergence class. Conditional cancellations or different domains can give other constructions.

Let $D$ be the fixed symmetric subtraction remainder in the XX-D construction. Its internal equal-time rod integrand is

Suppose $d$ is smooth near equal-time coincidence and, for compact output $K$, its required output derivatives satisfy

outside a fixed ball. Then the internal rod correction

exists, is cutoff independent, and has cutoff error $O(R^{1-q})$ in the stated output differentiability norms.

Proof. Near coincidence, change to relative spatial coordinates and use smoothness of $d$ and local integrability of $P_i$. At infinity, output differentiation is legitimate and the same radial estimate as (eq:cutoffestimate) applies.

For an external compact probe $g$, a sufficient alternative to XX-D's Schwartz condition is that $\mathcal{C}W(\cdot ,g)$ have the decreasing-source bounds of Theorem thm:weighted, uniformly on bounded probe families. That theorem then defines its original rod. Any Ward-transformed kernel needs the same estimates for its actual transformed entries.

No selected native representative satisfying these conditions is constructed here. The weaker criteria are new usable sufficient conditions, not verification of Q3.

For finite $m$, consider

with smooth coefficients and compact tangential support. Let the source reconstruction map be $c\mapsto b$. It is not assumed injective.

Define

The reconstructed source has the exact unique form

It is zero if and only if every ${j_{\gamma }}^{ia}$ is zero.

The original Newton-divergence image is

Thus it depends on the reconstructed source, not on an arbitrary choice of coefficient presentation.

Proof. Smooth multiplication of delta derivatives gives

Collect coefficients to obtain (eq:contactnormal).

To extract a given coefficient, test in the relative variable $\mathbf{y}-\mathbf{z}$ with

where $\theta =1$ near zero, and multiply by an arbitrary compact tangential test in $(t,\mathbf{z})$. Every relative derivative at zero except $\gamma$ vanishes. This extracts $j_{\gamma }$, up to its fixed sign, proving uniqueness and the null test.

For (eq:Newtoncontact), the relative variables $\mathbf{x}-\mathbf{y}$ and $\mathbf{y}-\mathbf{z}$ are independent coordinates. Their tensor-product distribution is defined before pushforward. The contact restricts the integration to $\mathbf{y}=\mathbf{z}$, making the operation proper on compact output tests. Its derivative signs give

This proves the image formula.

The normal-form shape agrees with the reconstruction discipline in Cheung's XV-B and XIV-J, but the present contact variables and Newton map are independently displayed and proved CheungXVB,CheungXIVJ.

The kernel in (eq:Newtoncontact) is a distribution across $\mathbf{x}=\mathbf{z}$. An additional coincident-point restriction is not supplied. The zeroth derivative already has a noncompact dipole tail. Higher derivatives can include singular contact pieces as well as their off-diagonal tails.

The finite bound $|\gamma |\le m$ does not make the coefficient space finite dimensional: the smooth tangential functions $j_{\gamma }$ remain freely variable. Nor is the Newton map necessarily injective on sources; divergence-free source directions can be invisible.

In one relative variable $q$,

Indeed, its value on a test $\psi$ is $-{\partial }_q(q\psi )(0)+\psi (0)=0$. Thus a degree-one presentation and a degree-zero presentation can reconstruct opposite nonzero terms.

No leading counterterm, anomaly, or normalization space may be counted from presentation degree alone. The full reconstructed null relations must be respected. This local control does not construct the separate XIV characteristic graph CheungXIVI,CheungXIVJ,CheungXVB.

Suppose a future native package gives an admitted insertion and an operation on one specified domain:

The normalization data held fixed by ${\mathcal{V}}_{\xi }$ must be declared. Any additional variations of those data belong to the specified ${\mathcal{A}}_{\xi }$ or to another explicitly accounted operation.

If (eq:expansion) is constructed on the same native domain and its classical symbol is (eq:O), then the retained breaking is

If, in fixed normalization coordinates,

where ${\Gamma }_f$ is a defined field-independent scalar Wick correction, then

Neither $R_{1,f}$ nor ${\mathcal{A}}_{\xi }{O}_f$ may be set to zero merely because the free scalar correction has zero ordinary field variation.

Proof. Multiply the formal expansions. The order-zero term is ${\mathcal{V}}_{\xi }{O}_f=0$ by Lemma lem:classical. The order-one term is exactly (eq:breaking). Field independence gives ${\mathcal{V}}_{\xi }{\Gamma }_f=0$ in the stated fixed coordinates. It says nothing about the other two terms.

This theorem calculates a formal coefficient from constructed application data. It does not calculate the native $B_{\xi ,f}$ from XX-A–XX-E, because neither the required operation nor its complete quantum insertion coefficient has been supplied.

Quantum BV and anomalous Master Ward constructions distinguish the quantum operation from ordinary classical differentiation FRBV,BDFrR. An anomaly-consistency framework is not a value for the present nonlocal $B_{\xi ,f}$.

For a separately constructed Lie-algebra action $d_x=d_{0,x}+\hslash d_{1,x}$ satisfying $[d_x,d_y]=d_{[x,y]}$, and an insertion with $d_{0,x}O=0$, let $d_{x}I=\hslash B_x$. Then

Proof. Apply the representation identity to $I$ and take the first quantum coefficient.

For the pullback anti-action, the corresponding sign changes must be used. Equation (eq:consistency) is not imposed on an undefined defect. Nor does satisfying it prove that a defect is an admissible counterterm image. These distinctions are central to anomalous Ward analyses BDFrR.

Let $\mathcal{C}$ be a specified vector space of formal correction data after the native package has been constructed. Suppose its calculated insertion and operation changes are

Here $J_f$ and $M_{\xi }$ are actual linear maps on this correction space.

The retained breaking changes by

If the Ward operation is fixed, $M=0$. If the insertion correction is only a field-independent scalar in fixed normalization coordinates, its ${\mathcal{V}}_{\xi }{J}_f$ image is zero.

Proof. Expand the transformed operation on the transformed insertion through first order in $\hslash$. The cross term $M(c)J(c)$ has order ${\hslash }^2$ and is discarded. The remaining new terms are exactly (eq:image).

Thus scalar reordering changes alone cannot cancel a nonzero breaking in the class with a fixed ordinary leading operation and no other image. This is a statement about that declared class, not a verdict about an unevaluated native anomaly.

A zeroth-order covariance replacement $W\mapsto W+S$ is a new choice of free construction. It must be recomputed or transported on a verified domain. It is not automatically a correction of the form $\hslash J(c)$ to a fixed package.

Retain the free mixed defect

For real symmetric smooth bisolution changes,

Its smoothness does not imply either vanishing or compatibility with the native infrared domain.

Suppose the calculated combined correction map is

Its targets are the actual complete defects, with all tests and retained coefficients included.

Let ${\mathcal{C}}_{\mathrm{allow}}\subseteq \mathcal{C}$ be the set of corrections satisfying all declared constraints: domains, normalization, reality, subsidiary conditions, infrared bounds, and any required physical positivity or quotient conditions. It need not be a vector space or an affine space.

For constructed $B,D_{\mathrm{sub}},L,T$, a permitted simultaneous correction exists exactly when

where the notation on the left is understood as the ordered pair $(-B,D_{\mathrm{sub}})$.

More explicitly, if $c_{\ast }\in \mathcal{C}$ solves the unconstrained equation $\mathcal{M}{c}_{\ast }=(-B,D_{\mathrm{sub}})$, then the complete permitted solution set is

If the allowed set is the whole vector space, this reduces to the affine actual-image criterion.

Proof. A simultaneous permitted repair means $B+Lc=0$, $D_{\mathrm{sub}}-Tc=0$, and $c\in {\mathcal{C}}_{\mathrm{allow}}$. This is exactly membership in the displayed image. The difference of two unconstrained solutions belongs to $\ker L\cap \ker T$, and every element of that common-kernel translate is an unconstrained solution. Intersect with the allowed set to obtain (eq:solutions).

Equation (eq:constrained) denotes

The ordered pair, not a quotient or image closure, is the exact target.

Constraints on a covariance can make the allowed correction set nonaffine. Therefore solving a linear Ward equation in an enlarged vector space need not solve the constrained problem.

Proof. Already for real symmetric matrices, start from ${\mu }_0=\mathrm{diag}(0,1)$. The allowed changes preserving ${\mu }_0+S\ge 0$ do not form a vector space: $S=\mathrm{diag}(1,0)$ is allowed, but $-S$ is not. A linear equation demanding the latter direction is solvable in the unrestricted symmetric-matrix space and not in the positivity-compatible set. The example proves the logical distinction without identifying this matrix problem with the native physical quotient.

The actual-image discipline is consistent with Cheung's XVIII-A and the supplied XV-B interfaces CheungXVIIIA,CheungXVB. It is an application criterion, not a native cancellation proof here.

If a finite probe map $P$ satisfies $P(B+Lc)=0$, the full identity follows only with independently proved coverage or injectivity on the allowed residual space. Otherwise a nonzero element of $\ker P$ is invisible.

Pure-trace probes in particular miss the smooth compact metric-slot kernel

This blind sector does not make a completely specified causal Green operator nonunique. It prevents selected contractions from certifying an unrestricted kernel CheungXVIIIA.

A nonzero ${\hslash }^0$ residual cannot be canceled by a correction divisible by $\hslash$. This follows immediately from coefficientwise formal equality and applies to the classical cutoff residual of XX-E.

Reduction modulo $\hslash$, whenever the quantum insertion is admitted with the chosen symbol, recovers (eq:O). The original one-rod term survives, and $Y_2$ is absent for ${\partial }_i{N}_0=0$.

This is a formal classical-symbol statement, not a pseudodifferential principal-symbol theorem, an expectation-value limit, or a numerical $O({\hslash }^2)$ estimate.

Finite self-contraction of a quadratic regular polynomial terminates algebraically. That fact does not bound the missing interacting coefficient, its complete quantum remainder, or a regulator-removal family.

With one common amplitude parameter,

Consequently the first quantum contraction of two insertions through total $s^2$ is only

A single noncompact quadratic derivative can enter at total order $s^3$; two can enter at total order $s^4$. Coefficientwise products of the quadratic coefficients require their own domains.

The corrected total-order statement is XX-E's, not a result supplied by the earlier XX-D two-endpoint discussion.

Whenever two rods actually enter, the canonical constraint from XX-C is retained:

A domain preserving this nonzero canonical inverse-distance term cannot require both ordered two-rod covariances to be $O(r^{-2})$.

No two-endpoint covariance extension is proved by Theorem thm:weighted; its theorem concerns one Newton output with a controlled source. No two-rod Ward contribution is evaluated here.

For a nonzero nonnegative $\rho \in {C_c}^{\infty }((1,2))$, put $\eta (\mathbf{Y})=Y_1\rho (|\mathbf{Y}|)$. Let $\tau \in {C_c}^{\infty }(I)$, and take