X-H-ST: Relative classical BV structure and complete instruments for supplied residual reports A fixed-width plateau construction, a free-shell non-descent test, and the unchanged variational upgrade obstruction

We construct a relative classical Batalin–Vilkovisky structure and finite destructive-readout interfaces on the supported host retained by I-B and X-A-ST. All analytical fields and Green operators remain on the declared Minkowski completion, at fixed positive regulator width.

On the plateau, the X-G-ST field multiplet admits an explicit nonlinear differential $s=s_0+\kappa s_1+{\kappa }^2{s}_2$ that is nilpotent off shell. A parity-checked antifield lift and the natural Einstein–scalar reference action give a classical master action through order ${\kappa }^2$, as a local variational identity modulo divergences. The auxiliary field remains independent. The construction retains background tadpoles rather than deleting terms needed by the action identity.

Two corrections determine the scope. First, nonlinear nilpotence does not imply descent to the unchanged free equation quotient. A compactly tested free-solution witness shows that the natural first-order scalar transformation does not preserve that quotient. Consequently the relative classical complex is not an interacting differential on X-G-ST's free causal algebra. Second, the positive reciprocity pairing of X-G is a ghost-number-zero obstruction to an exact source-preserving variational upgrade, not the ghost-number-one classical master-equation residual. Both distinctions are proved rather than imposed terminologically.

We define finite dictionaries of supplied coefficient evaluations and construct separate complete destructive instruments for classical residual reports and for the reciprocity witness. Their output includes the classical code and the unmeasured finite residual, with arbitrary untouched references retained. The instruments report installed evaluations; they neither acquire continuum fields nor turn an odd BV residual into a physical field observable.

A first-order quantum correction is formulated only as a conditional consistency problem. Local fish ambiguities can be encoded once specified, but the missing nonlocal contraction and an interacting quantum BRST theorem are not supplied. The contribution is a controlled separation of local classical closure, free-carrier descent, variational integrability, and complete report implementation. This provides explicit tests against false upgrades while preserving the earlier tree identity. The release is local Grade S, not spacetime emergence.

corrections to the proposed plan{ A plateau classical differential can be constructed on off-shell local fields. It is not thereby a differential on the unchanged free causal algebra.

The classical master residual and the X-G reciprocity pairing have different parity, ghost number, and arguments. They are not identified.

A complete instrument for a supplied number realizes an encoded report observable. Without a field-to-apparatus coupling, it is not a measurement of an unknown continuum field or a physical BRST cohomology class.

The quantum continuation remains conditional. A classical master action and a finite report channel do not define a missing nonlocal quantum product.}

The supplied X-G-ST constructs a normalized free auxiliary–ghost multiplet and proves an exact source-preserving variational upgrade obstruction XGST. Its parent is Cheung's Landmark X-G XG. The present manuscript continues the question of a restricted classical interacting structure without assuming that restriction automatically produces an interacting differential on the free causal carrier.

The distinction is substantive. An off-shell vector field can square to zero while failing to preserve the equations used to define a quotient. Section sec:descent gives an explicit example for the proposed nonlinear scalar transformation. Accordingly:

1. a relative local classical BV complex is constructed;

2. its identification with the unchanged free equation quotient is not made;

3. the natural proposed descent to that quotient fails already at first order;

4. no interacting quantum differential is released.

X-G supplies the principal normalization and reciprocity mechanism. X-F and the supplied X-F-ST supply the separate equation-level tree construction and its background identity XF,XFST. They are not reinterpreted here as variational Møller maps.

X-H-ST denotes this proposed manuscript. No parent X-H theorem, repository installation, registration, submission, acceptance, or successful compilation is assumed or asserted.

Cheung's philosophy distinguishes destructive readout, quantum return, complete implementation, and geometric identification EnterPhilosophy,EnterI. We use that distinction in four precise ways:

1. a defined differential or residual is not acquisition of unknown field data;

2. a complete instrument is not a formal cohomology class;

3. reference-preserving transport is not entanglement generated by gravity;

4. the prescribed conical host is not derived from a BRST or BV complex.

A measured clock encoding is absent from the returned quantum interface. The complete output retains its classical code and the unmeasured finite system. This is an access statement, not physical annihilation of all implementing degrees of freedom IB,En09.

We call the finite object constructed below an encoded residual-report observable. We do not call the master residual itself a physical field observable merely because its supplied numerical coefficient has been placed in a table.

No Enter characterization theorem is reconstructed from an abbreviated list of axioms. The finite instruments use specified Born semantics, and their channel identities are proved directly.

Both $\kappa$ and $\hslash$ are formal parameters. Congruence modulo ${\kappa }^3$ means equality of coefficients through order two, not a numerical remainder estimate.

Odd fields have coefficients in an exterior algebra. Positivity and thresholds will be applied only to specified real coefficient extractions, never to an uninterpreted Grassmann-valued residual.

Local action identities are taken modulo integrated divergences. Arbitrarily multiplying a divergence representative by an observation function can change its integral; that operation is not silently used to evaluate a variational class.

Retain the instance already declared in X-A-ST and used by X-E-ST, X-F-ST, and X-G-ST:

1. the regular time-oriented conical product $({\mathcal{M}}^{\circ },G)$, its stipulated worldsheet $\mathcal{W}$, and its prescribed future relation;

2. an injective supported developing chart

   $$\iota :\mathcal{L}⟶{\mathcal{M}}^{\circ }$$

   with the retained compact reserve disjoint from $\mathcal{W}$;

3. the independent analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3,\Sigma =\{t=0\};$$

4. the signature and time ledger

   $${\iota }^{\ast }G=-\eta ,t=c_{\mathrm{host}}(T-T_{\Sigma });$$

5. a nonempty observation region

   $$\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3,J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

Every global Green operator and free field below belongs to $\hat{M}$. The regular conical product is not used as a globally hyperbolic Cauchy spacetime.

The defect is a worldsheet, not a worldline. No host-update map, second cone, or global conical Green inverse is introduced IB,XAST,XGST.

Fix the compact scalar switch $g$, an observation test $f$, a plateau $U$, and the positive onset width:

Here $\chi$ is smooth, zero on $(-\infty ,0]$, one on $[1,\infty )$, and flat at the endpoints. Thus ${\sigma }_{\epsilon }=g=1$ near $\overline{U}$. There is no shrinking-layer argument.

For compact supports write

Set

and use positive retarded/advanced inverses ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ of $\square$ and ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ of $P_m$. They are used only on the completion BGP.

For covariant symmetric tensors and vector fields define

With the tensor–density pairing,

In particular,

The fields, parities, and ghost numbers are

The independent auxiliary field is retained. The free differential and gauge fermion are

Thus

Eliminating $b$ gives $b=\mathsf{C}r/2$ and the Feynman weight $(\mathsf{C}r{)}^2/4$.

For reference, the retained Green blocks are

The scalar Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}/\mathrm{A}}$, not the positive equation inverse.

These matrices invert, respectively,

For the first block this follows by substituting $b=(\mathsf{C}r-j)/2$ and solving ${\mathsf{P}}_{\mathrm{gr}}r=J+{\mathsf{C}}^{\ast }j/2$. For the ghost block it follows by matrix multiplication. This records the normalization without constructing a new free theory XG,XGST.

The free causal algebra ${\mathfrak{A}}_0$ is the graded polynomial algebra of compactly smeared linear fields with these causal relations and the equations

Its equation ideal is denoted ${\mathcal{J}}_0$. X-G-ST's nilpotent causal derivation $s_0$ is retained. No interacting action on ${\mathfrak{A}}_0$ follows from that free result.

All constructions in this section are local on $U$. Fields are unrestricted smooth local sections before equations are imposed. Backgrounds $h,k$ are fixed smooth covariant symmetric tensors, not dynamical fields and not the host metric.

Declare the formal local reference split

The use of $\bar{q}$ through second order fixes a local reference expansion. It does not assert a new retarded prescription for X-F.

Let $\mathcal{P}(U)[[\kappa ]]$ be the graded differential algebra of local polynomial jet expressions, allowing the formal inverse metric coefficients determined by (eq:split). At each formal order the expressions have finite jet order and finite field degree.

Define $s$ by fixing $\bar{q},h,k$, commuting with total spacetime derivatives, and setting

The Leibniz rule is left graded: $s(FG)=(sF)G+(-1{)}^{|F|}F(sG)$. In particular,

The coefficient operators are explicit:

The leading field differential is exactly (eq:freeBRST).

The derivation (eq:nonlinear) satisfies $s^2=0$ on $\mathcal{P}(U)[[\kappa ]]$. Consequently its truncation through second order is nilpotent modulo ${\kappa }^3$, without imposing free or interacting equations.

Proof. For the odd vector field $c$, put $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. Expansion of the tensor Lie derivative with the odd Leibniz rule gives

Terms containing symmetric second derivatives cancel against antisymmetric ghost coefficients. It follows from $sc=\kappa w$ and (eq:sq) that

Since $q=\bar{q}+\kappa r$ and $s\bar{q}=0$, this gives $s^{2}r=0$ coefficientwise.

For the scalar,

by the same cancellation. Direct expansion of $s(c^{\nu }{\partial }_{\nu }{c}^{\mu })$ gives zero after substitution of $sc=\kappa c\partial c$. The nonminimal pair is nilpotent by definition. The square is an even derivation, so the generator checks prove the assertion on the algebra.

This is the natural diffeomorphism kinematics already distinguished from interacting quantum BRST in X-G-ST XGST. The theorem constructs a classical local differential. It does not yet claim an action symmetry or quotient descent.

For any compact $A⋐U$, simultaneous compact support of all dynamic fields in $A$ is stable under (eq:nonlinear). In particular, restricting $\bar{c}$ requires the same support restriction on $b$, because $s\bar{c}=b$.

One may instead work with arbitrary smooth germs and compact smearing tests. That is the main local algebraic interpretation here. Compact configurations will be used only as an evaluation sector for finite reports.

A space with compact antighost but unrestricted auxiliary field is not used: $s$ need not be tangent to such a space.

Let a local Lagrangian mean a horizontal four-form modulo a horizontal divergence:

Euler differentiation annihilates divergences. Generalized actions act on compactly supported local functionals by their local Euler derivatives.

We use the algebra generated by compact local functionals under ordinary multiplication. It is a multilocal algebra, denoted ${\mathcal{F}}_{\mathrm{ml}}(U)$; products of two single integrals are not falsely declared to be single local integrals. The classical variational bracket is defined on this algebra with its usual integration-by-parts identifications FRBV.

No time-ordered product, Wick extension, or BV Laplacian is required for this classical operation.

Fix the curvature convention by

Write $\mathcal{I}[q]=J_{q}R[q]$, $J_q=\sqrt{-\det q}$, and

The local reference invariant action is the class

These integrals denote variational classes, not an assumption of globally finite action.

The coefficient of ${\kappa }^{-1}$ is

a divergence. It is removed before the action is viewed as a nonnegative formal series. Thus no manipulation of ${\kappa }^{-1}$ in a nilpotent coupling ring is involved.

The background de Donder vector and fermion are

In particular,

The independent $b$ has not been eliminated.

On $U$, $sI=0$ as a local variational class.

Proof. The curved densities are natural under $sq=\kappa {\mathcal{L}}_{c}q$ and $s\varphi =\kappa c\partial \varphi$:

The background-only subtraction is $s$-inert. The resulting variation is a divergence. Theorem thm:kinematic gives $s(s{\Psi }_{\bar{q}})=0$.

On the plateau the original switched scalar density equals ${\mathcal{L}}_q$, so this is its relative local action identity. It is not a new claim of unrestricted symmetry of the globally switched action XG,XGST.

For $j=0,1,2$, define

Here $[{\kappa }^n]$ is Taylor coefficient extraction, equivalently the $n$-th derivative at zero divided by $n!$. The curvature, split, and fermion have all been specified. Thus (eq:Ij) is a finite definition of every vertex required at this order, not an unspecified choice of a cubic or quartic action.

In particular, $I_2$ retains the Einstein–Hilbert coefficient through fourth geometric order. Omitting that coefficient would generally invalidate the second-order master identity.

At order zero,

Indeed, the quadratic Einstein–Hilbert coefficient is $\frac{1}{2}⟨l,\mathsf{A}l⟩$, and the background-only part has been subtracted.

The Hessian of $I_0$ is the retained free Hessian, but for general off-shell $h$ its metric Euler equation has the additional source $\mathsf{A}h$. At $h=0$, $I_0$ is exactly the free action (eq:freeaction).

Equation (eq:tadpole) is not silently replaced by the fluctuation-independent-source subtraction used in X-F's response prescription. Such a replacement would change the action being used in this section. Therefore this reference BV action is not asserted to be an exact variational realization of that prescription.

Adjoin antifields ${\Phi }_A^{\ast }$ with

They are dual tensor densities of the appropriate bundle type.

Use the canonical odd bracket with

extended as an odd biderivation and by integration by parts on local functionals. For an even Hamiltonian $\mathcal{S}$, its left action is $\mathcal{Q}F=(\mathcal{S},F)$.

Let $R^A=s{\Phi }^A$. Define

In components this is

One has $(H_s,{\Phi }^A)=R^A$ for every field. Moreover,

as a local variational class.

Proof. Put ${ϵ}_A=|{\Phi }^A|$. The biderivation rule and the canonical coordinate bracket give

The prefactor in (eq:Hs) therefore gives $(H_s,{\Phi }^A)=R^A$, for even and odd species. In particular the minus signs in the ghost and antighost terms of (eq:Hsexplicit) are required.

The bracket square of an antifield-linear Hamiltonian is antifield linear. Its action on every field is the square of the induced evolutionary vector field, by the graded Jacobi identity. That square is zero by Theorem thm:kinematic. Consequently the bracket square is zero modulo a divergence: a variationally trivial antifield-linear functional has no nonzero antifield-independent constant.

Let

be its nonnegative formal expansion through second order. Then on $U$,

as a local variational class.

The Hamiltonian operator

is an odd differential on the classical multilocal BV algebra modulo ${\kappa }^3$. Its restriction to fields is (eq:nonlinear) through that order.

Proof. The antifield-independent functional $I$ satisfies $(I,I)=0$. By Lemma lem:sign, $H_s$ acts on antifield-independent functionals as $s$. Since the bracket of two even arguments is symmetric,

The last equality is Lemma lem:action. The action has no negative formal coefficients after removal of the specified divergence. Thus truncation preserves this equality through order two. Explicitly,

The graded Jacobi identity yields

The generator action follows from Lemma lem:sign.

The construction uses the standard classical cotangent-lift mechanism FRBV. It provides a differential, not a theorem of Koszul–Tate acyclicity, physical cohomology, or a physical state. For off-shell $h$, its order-zero antifield equations include (eq:tadpole); they are not identified with the unchanged free equations.

An odd derivation $D$ of an off-shell algebra induces a derivation on its quotient by an ideal $\mathcal{I}$ only if $D\mathcal{I}\subseteq \mathcal{I}$. The equation $D^2=0$, even off shell, does not imply this condition.

Proof. If $F$ and $F+J$ represent the same quotient element, their proposed images differ by $DJ$. Well-definedness requires $DJ\in \mathcal{I}$. Nilpotence tests a different composition and imposes no such requirement.

The differential (eq:nonlinear) does not preserve the unchanged free equation ideal, even locally on the plateau. In particular its natural first-order action cannot induce an interacting derivation on ${\mathfrak{A}}_0$ merely by using the unchanged free equations.

This failure occurs at order $\kappa$, so a second-order correction leaving $s_1\varphi$ unchanged cannot remove it.

Proof. Work in coordinates centred at a point $p\in U$, with $\tau =t-t_p$ and $y^1=x^1-{x_p}^1$. Take a nonzero odd generator $\theta$, a number $\lambda >0$, and

Set $r=b=\bar{c}=0$. These are smooth local free configurations: $\square c=0$, $P_m\varphi =0$, and (eq:freeequations) holds.

Since $s$ commutes with spacetime differentiation,

On free inputs the right side is

At $p$ it equals

Choose a compact real scalar test supported in a small neighbourhood of $p$ on which the coefficient has one nonzero sign. The corresponding free equation functional vanishes on all free configurations, but its $s$-image does not.

Every element of the free dynamical equation ideal has zero normal symbol on smooth free configurations: equation factors vanish there, and their contractions vanish by the free bisolution identities. Hence the displayed image cannot be an equation-ideal element. No term beginning at order ${\kappa }^2$ changes its nonzero order-$\kappa$ coefficient.

Thus Stage A is constructed on off-shell fields, and Stage B is constructed on a relative classical BV algebra. The stronger target of an interacting differential on the unchanged free causal algebra has not been supplied.

A smooth ghost coefficient compactly supported in spacetime and satisfying $\square c=0$ on the completion is zero. Consequently restricting ghosts to ${C_c}^{\infty }(U)$ does not produce a nontrivial interacting action on the unrestricted free ghost carrier.

Proof. A Cauchy surface earlier than the compact support has zero position and velocity data. Wave uniqueness gives the zero solution.

The theorem in Section sec:stageA and this support observation are compatible. The former uses off-shell local ghosts; the latter concerns homogeneous free solutions. Compact gauge parameters are not nonzero compact free ghost configurations.

A future retarded transport to an interacting equation space would require a separate map, a compatible interaction prescription, and its domain theorem. The X-G obstruction prevents assuming that the unchanged post-variation source map already supplies that variational transport.

On the completion, retain

Use the same kinematical transformation and the natural gauge fermion, but no post-variation source switching.

For compact ghost coefficients, direct variation gives

Here

To verify the formula, use

and ${\partial }_{\mu }{T_0}^{\mu }{}_{\nu }=(P_m\varphi ){\partial }_{\nu }\varphi$. All switch derivatives remain XG,XGST.

Let ${\mathcal{S}}_{\mathrm{sw}}$ be this natural reference action plus its signed antifield lift. The argument of Theorem thm:BV now gives

The result has ghost number one and odd parity.

For example, with

one has $P_m{\varphi }_m=0$ and ${T_0}^{00}=1/2$. For a nonzero compact switch with a plateau, choose $\beta \ge 0$, nonzero and compactly supported where ${\partial }_{t}g>0$, and put $c=\theta \beta {\partial }_t$. The leading coefficient of (eq:swCME) is

Such a transition exists on the completion along a vertical line reaching the plateau. Calling this detector a retained laboratory test additionally requires its support to lie in $\mathcal{L}$, as qualified in X-G-ST.

The full off-shell breaking is not confined to $\mathrm{supp}\mathrm{d}g$, because of the term containing $1-g$. On $U$, all terms in (eq:breaking) vanish.

Let $Q_w$ be the first background coefficient of the natural metric Hessian including background de Donder gauge:

Its tensor types give the identity

It follows by differentiating naturality under simultaneous background and fluctuation pullback XF,XG.

X-F's mixed equation-level source map is

For compact $w$, $Q_w$ is formally symmetric, but ${N_w}^{\mathrm{T}}=Q_w{\sigma }_{\epsilon }$. A symmetric action Hessian $L_w$ reproducing $k_w[u]$ on every smooth free metric input would necessarily satisfy

Indeed, equality of retarded fields and application of ${\mathsf{P}}_{\mathrm{gr}}$ imply $L_{w}u=N_{w}u$; symmetry then gives (eq:reciprocity). Compact background coefficients make these pairings well defined.

Choose nonnegative $\zeta \in {C_c}^{\infty }((0,\epsilon ))$, nonzero where ${\sigma }_{\epsilon }^{\prime \prime }\ne 0$, and

Put

The data (eq:recipdata) give

The background can be supported in the retained regular reserve, and the detecting free inputs can be replaced by free inputs with compact Cauchy data.

Proof. On a free input, (eq:typed) gives $Q_{w}u=-{\mathsf{P}}_{\mathrm{gr}}{\mathcal{L}}_{{\xi }_{\mathrm{on}}}^{\mathrm{cov}}u$. The relevant Lie components are

There are also $0i$ components but no spatial–spatial components. For tensors of this form, $({\mathsf{P}}_{\mathrm{gr}}a{)}^{00}=-\square a_{00}/8$. Spatial Laplacians integrate to zero, and

Thus $W=(5M_{\chi }/8)\int {\sigma }_{\epsilon }{\psi }^{\prime \prime }\mathrm{d}t$. Two integrations by parts yield (eq:W).

Choose an observation event $(t_o,{\mathbf{x}}_o)\in \mathcal{O}$. Since $t_o>2\epsilon$, a small spatial ball $B$ satisfies

Choose ${\chi }_0$ supported in $B$. For compact-Cauchy replacements, choose initial data equal to those of $u,v$ on the initial causal shadow of a neighbourhood of $\mathrm{supp}w$, and compactly cut off the data outside a larger ball. Finite propagation preserves $u,v$ near $\mathrm{supp}w$, so the local pairing is unchanged BGP.

This is Cheung's X-G witness, with its normalization retained XG,XGST. It excludes the exact source-preserving variational upgrade class, not every possible changed classical action. Algebraic auxiliary elimination leaves a symmetric reduced Hessian and does not remove the contradiction. Corrections beginning at positive powers of $\hslash$ cannot cancel a nonzero classical formal coefficient.

The pairing $W$ is not the classical master-equation residual in (eq:swCME). In particular, the statement that the unrestricted master equation fails “exactly by $W$” is not valid for the frozen data.

Proof. The master residual is odd with ghost number one. The pairing $W$ is an even real number of ghost number zero, bilinear in two free metric probes. No coefficient-extraction or comparison map identifying those arguments has been specified.

There is also a direct separation. In the natural reference action, set the scalar and ghosts to zero and all antifields to zero. The master residual is then zero, including in the compact onset background of (eq:recipdata). The independent reciprocity experiment in that same background still has $W>0$. The pairing detects the absence of a symmetric Hessian reproducing a different, source-switched map; it is not evaluation of a master action that has not been constructed.

Nonconstant onset backgrounds do not all carry the same evaluated defect. The theorem gives a specified witness excluding an all-background upgrade. It does not say that every nonconstant background has nonzero $W$.

The instrument in Stage E will therefore report reciprocity incompatibility, not a quantum anomaly or an identified master residual.

The X-F-ST direct identity acts on the classical background coefficients while holding the free fields fixed. For $\xi ,{\zeta }_p\in {\mathfrak{g}}_p$,

It is a different operation from $s$, which acts on the field variables.

For the mixed response,

The proof uses (eq:typed), ${\sigma }_{\epsilon }=1$ on $\mathrm{supp}\xi$, and the retarded left inverse on the compact tensor ${{\mathcal{L}}^{\mathrm{cov}}}_{\xi }r$. After compact smearing, the last term is a graviton-slot equation image.

This remains the cancellation supplied by the mixed tree. The matter and self trees are inert under that background transformation XF,XFST. No additional copy of the scalar second-order pair, matter tree, or mixed cancellation is added.

The relative BV construction does not establish that the relational observable is $\mathcal{Q}$-closed. Action symmetry, observable closure, and the fixed-free-input background identity are different claims. The supplied tree identity is not reinterpreted as a variational Møller theorem or an interacting quantum Ward identity.

Fix a finite exterior algebra ${\Lambda }_N$ with a specified ordered basis. A coefficient extraction means a specified real linear map

for example the real or imaginary part of one basis coefficient. It is not a state, positivity functional, or ghost measurement.

There are two classes of classical report tasks:

1. a coefficient of the integrated classical master residual

   $${{\mathcal{R}}_{\mathrm{ME}}}^{[2]}=\frac{1}{2}({\mathcal{S}}^{[2]},{\mathcal{S}}^{[2]})$$

   evaluated on supplied compact field and antifield profiles supported in a common $A⋐U$;

2. a compactly smeared coefficient of

   $$s^2{\Phi }^A\text{or}({\mathcal{Q}}^{[2]}{)}^{2}F$$

   through order two, evaluated on supplied local profiles, optionally satisfying the free equations near the smearing support.

For (R1), one integrates the complete variational bracket, not an arbitrarily weighted representative of its divergence class. On compact evaluation profiles its divergence current is compact, so its integral is zero. Equivalently, one may insert a cutoff equal to one near the support of that current; no derivative of the cutoff then contributes.

For (R2), the phrase “free-shell projection” means evaluation of the already defined off-shell square on supplied free configurations. It does not mean that $s$ descends to the free quotient. Theorem thm:nondescent excludes that inference.

Every task (R1) or (R2), at the released orders, has real extracted value zero. The evaluations are continuous in their smooth data on common compact support stages. A nonempty finite dictionary of these tasks exists.

Proof. Theorems thm:kinematic and thm:BV give the zero identities before evaluation. The compact evaluation rule respects the divergence identification.

At each formal order the expressions are finite local differential polynomials. Compact integration and a fixed coefficient extraction are continuous in the smooth seminorms of their finitely many input jets.

For nonemptiness choose a nonzero real bump $\rho \in {C_c}^{\infty }(U)$, any compact tensor profile for $r$, and, for example,

with compact antifield profiles and a finite exterior algebra containing the indicated generators. These are off-shell test profiles. At least one such task can be installed without requiring a nonzero compact free ghost.

The theorem proving zero is analytical. A later table readout does not independently prove that universal identity. It reports a task whose value has been supplied and justified.

A dictionary entry contains:

1. a task identifier, quantity type, formal order, and coefficient-extraction rule;

2. the support and input-description contract;

3. a supplied real value $a_j$ together with a rational enclosure

   $$I_j=[L_j,U_j],L_j\le a_j\le U_j;$$

4. a declared rational threshold ${\tau }_j>0$.

For the proved zero tasks one can use $I_j=[0,0]$.

Define the status

If the installed enclosure is valid, PASS implies $|a_j|\le {\tau }_j$, and FAIL implies $|a_j|>{\tau }_j$. No conclusion is drawn from UNDETERMINED.

Proof. The value belongs to its enclosure. The three cases follow from the stated set inclusions.

No algorithm for obtaining enclosures of arbitrary smooth field descriptions is assumed. The finite dictionary is supplied data. Distinct deterministic numerical words need not define nearby channels when the numbers are close. No false continuity claim for thresholding or orthogonal report words is made.

Let the finite dictionary have $J$ entries. Choose $d_C=2^{n_C}\ge J$, $n_C\ge 1$, and a calibrated orthonormal clock basis $\{|{\tau }_j⟩{\}}_{j=1}^{d_C}$. Unused labels receive an explicit $\mathrm{UNUSED}$ report.

Let $A$ contain all unmeasured finite quantum registers. Let $q_j$ be a finite classical word containing the entry identifier, enclosure, status, and order tags. Put

Distinct output labels retain $j$, even when several entries have the same zero residual.

Define

The map (eq:instrument) is CPTP and

The identities hold on all input operators and after adjoining any untouched reference $E$.

For clock preparation $|{\tau }_j⟩⟨{\tau }_j{|}_C\otimes {\rho }_{AE}$, the output is

For an arbitrary correlated input, the returned quantum block is the subnormalized conditional block $D_{j}X{D}_j^{†}$, not an assumed unchanged conditional state.

The clock factor $C$ is consumed from the returned quantum interface.

Proof. A Kraus family is

This proves complete positivity and trace preservation.

Expand $X={\sum }_{j,l}|{\tau }_j⟩⟨{\tau }_l|\otimes X_{jl}$. The $j$-block in (eq:instrument) is $X_{jj}$; their sum is ${\mathrm{Tr}}_{C}X$. The same expansion with $A\otimes E$ proves the reference identities and the basis-input formula.

The quantum codomain is $A$, not $C\otimes A$. The measured clock names are therefore unavailable as quantum inputs to a continuation through this interface.

This is the I-B/En09-A complete instrument pattern, with the present report words explicitly attached IB,En09. Conditional reference steering is compatible with the unchanged unconditional reference marginal. Only positive-probability branches are normalized.

For a chosen rational displayed value ${\hat{a}}_j$ of each report, the bounded register observable is

The Heisenberg pullback is

It is an observable of the finite task encoding.

The instrument above does not measure an unknown continuum master residual, select a physical BRST state, or turn a ghost-number-one expression into a self-adjoint field operator.

Proof. Its input has only the declared finite factors. Its Kraus maps contain the installed table but no map from continuum configurations to those factors. Two hypothetical continuum inputs attached to the same finite preparation therefore give the same output.

The only self-adjoint observable constructed for the report is the finite diagonal operator in (eq:reportobservable). Coefficient extraction from an exterior algebra is not a positive field representation. No physical cohomology or such representation has been specified.

Thus the instrument licenses the phrase “observed encoded residual report” on its declared interface. It does not supply the stronger physical identification suggested by calling the unsmeared BV residual an observable of an interacting quantum theory.

The finite residual $A$ returned by the instrument and the mathematical master residual ${\mathcal{R}}_{\mathrm{ME}}$ are different objects despite the shared word “residual”.

Choose a separate finite dictionary of the reciprocity tasks (eq:W), with its own clock $C_{\mathrm{rec}}$. A task can contain the fixed analytic expression for $W$, a supplied rational enclosure $[L,U]\subset (0,\infty )$, and the tag $RECIPROCITY\_INCOMPATIBLE$.

Such an enclosure exists for each fixed witness: the proof gives $W>0$, so rational numbers $0<L\le W\le U$ can be chosen and supplied. This existence is not a general evaluation algorithm. For example, if $|{\sigma }_{\epsilon }^{\prime \prime }|\ge a>0$ on $\mathrm{supp}\zeta$, then

is an explicit lower-bound interface.

Define ${\mathcal{M}}_{\mathrm{rec}}$ by the same Kraus formula as (eq:instrument), using these separate report words and clock names.

The map ${\mathcal{M}}_{\mathrm{rec}}$ is CPTP, consumes its clock encoding, retains the unmeasured finite residual, and satisfies (eq:drop) including every reference. On a task-basis input it returns its certified positive reciprocity report with probability one.

It does not cancel $W$, evaluate a master action that does not exist, or identify the report with a quantum anomaly.

Proof. Theorem thm:instrument applies to any finite installed report table. The last assertions follow from Proposition prop:distinct and from the fact that readout changes no source equation.

An exact arbitrary real value is not asserted to have a finite numerical code. The returned word is its installed identifier and certified enclosure, or an exact number when one has actually been supplied.

Nontrivial dictionary labels may be calibrated as history or task labels and read at one later event. They are not all declared to equal the time of that same event. If distinct calibrated labels must equal the coordinate of one fixed reading event on every positive branch, only one label can occur IB.

Neither clock is returned. The geometric coordinate $t$, the host time $T$, and the prescribed future relation remain. No relational unitary evolution follows from clock extraction alone; En09-A's additional stationarity and clock-covariance hypotheses are not assumed here.

Under the specified flat Feynman conventions, the local scalar–metric fish components are

They are defined off zero and have scaling degree at most six. Finite-scaling-degree extensions exist, and their internal ambiguity has the form

This is the local extension interface retained from X-G and the standard extension theorem XG,XGST,BF.

For a separately chosen extension and finite coefficient list, the resulting local counterterm functional and its classical BV variation can be specified and evaluated on compact smooth inputs. All external differentiations and derivatives of test coefficients must be retained. The internal derivative bound is not automatically a bound of two on the total derivative order of the resulting local density.

None of these local data defines the missing nonlocal coordinate contraction or the entire one-loop master residual.

The following statement fixes the logic without introducing an undefined BV Laplacian.

Assume a later construction supplies, on the same relative domain and through the same formal order:

1. all quantum products and insertion maps needed to define the first-order residual;

2. an admitted odd ghost-number-one functional ${\mathfrak{A}}_1$, including every normalization and transport contribution;

3. a convention in which adding an even ghost-number-zero counterterm $C_1$ changes that residual by ${\mathcal{Q}}^{[2]}{C}_1$;

4. the consistency identity ${\mathcal{Q}}^{[2]}{\mathfrak{A}}_1=0$ modulo ${\kappa }^3$.