X-L-ST: Restricted classical BRST domains, exact free recovery, and complete residual reports A conditional graded package on the unchanged I-B conical Enter spacetime

We construct a restricted classical graded continuation on the fixed I-B/X-A-ST host, separating nonlinear field nilpotence, action compatibility, free-shell descent, and finite reporting. The normalized zero-coupling field rules, gauge fermion, auxiliary–ghost Green matrix, and graded causal relations are re-established explicitly. Free causal recovery is an identity of the resulting free presentation, not a claim that a classical product has become an interacting quantum product.

The principal restricted construction is the centralizer of the classical master residual in the local variational BV algebra. Its Hamiltonian operator is nilpotent on that domain even when the master residual is nonzero. This gives a precise, nonempty restricted classical complex without asserting unrestricted symmetry restoration. A separate compactly tested free-solution witness proves that the prescribed nonlinear scalar rule does not descend to the unchanged unrestricted free-shell quotient. The witness works for every fixed scalar mass in this particular descent problem; it neither extends nor discharges the distinct prepared-insertion problem of XVI-N.

Conditional correction and quantum-presentation theorems state the additional identities required for descent, nilpotence, and exact free-fiber recovery. Finite rank and left-kernel tests are valid only after complete coverage of the declared correction space has been proved. The odd, ghost-number-one master residual and the even reciprocity pairing remain different targets.

Complete destructive residual-report instruments are recovered on identical installed finite interfaces, including arbitrary untouched references. Their adaptive composition, regular transport, resource fit, and error budget are proved locally. They encode supplied numerical evaluations, not Grassmann elements or acquired continuum fields.

The contribution is an explicit compatibility and obstruction package, not a new general BV formalism. Its potential use is to prevent a restricted classical differential, a successful report, or one cancelled diagnostic from being mistaken for interacting quantum BRST. The ramp width remains fixed and positive; it is not a thickness assigned to the conical worldsheet. No aligned-null contraction, renormalized interacting product, inherited characteristic quantum coefficient, metric reconstruction, or clock-free relational time is constructed.

status of the attempted continuation{ Constructed: the normalized free package, the nonlinear off-shell field differential, its restricted classical BV centralizer, the explicit free-shell non-descent witness, and complete finite report channels.

Conditional: correction-based quotient descent and a free-fiber-preserving quantum presentation.

Not constructed: a nontrivial interacting quantum BRST algebra realizing the prescribed nonlinear rules on the unrestricted free causal carrier, a renormalized interacting product, or the inherited characteristic quantum coefficient.}

The supplied development places X-L-ST after the free stage X-G-ST, the relative classical stage X-H-ST, and the later classical and finite-operational sheets X-I-ST, X-J-ST, and X-K-ST XGST,XHST,XIST,XJST,XKST. That ordering is documentary. It does not transfer every predecessor theorem or identify their carriers.

The present paper selects the restricted graded direction. Its nonlinear construction is classical. It also gives a conditional interface for a later quantum presentation, but does not assert that the missing quantum hypotheses have been satisfied.

All mathematical inputs used below are displayed as definitions, explicit operator normalizations, or conditional hypotheses. The cited sheets identify the intended comparisons. Their titles and catalogue placement are not substituted for proofs.

The supplied philosophy essay treats Enter as destructive computational-basis readout and insists that output type, accessible systems, and the information task remain explicit EntPhil,EnterI. In particular:

Consumption removes a measured quantum name from the declared continuation. It does not annihilate the host time coordinate or every physical degree of freedom in a possible apparatus.

A formal ghost is not a physical finite-register state. An odd classical residual cannot be inserted into a density matrix as if it were an ordinary real amplitude. Section sec:reports specifies the separate extraction of ordinary numerical coefficients before a report is installed.

The classical and quantum BV frameworks already distinguish equation ideals, products, and master equations FRclassical,FRquantum. The present paper makes no priority claim for those general structures.

Four points require explicit debugging.

1. The I-B conical defect is a codimension-two worldsheet. The positive width in the X track is a fixed analytical ramp or smoothing width, not a derived geometric core thickness.

2. Off-shell field nilpotence does not imply invariance of a prescribed action or descent to its free equations.

3. Ghost number is a grading. A filtration can be induced from it, but no convergence or spectral-sequence conclusion follows merely from saying “filtered by ghost number.”

4. Cancelling a single shell witness or one reciprocity number is not sufficient to define a quantum product. Complete relation, product, domain, and nilpotence identities must still be supplied.

Retain the prescribed time-oriented regular conical product

with completion worldsheet

The supported laboratory is disjoint from $\mathcal{W}$.

The independently declared analytical completion is

On the supported developing chart $\iota :\mathcal{L}\to {\mathcal{M}}^{\circ }$, retain

Retain the same switches, plateau $U⋐\mathcal{L}$, regulator prescription, and fixed positive width

Neither $\epsilon$, $\chi$, nor the host is varied.

All analytical equations, free fields, and Green operators below are defined on $\hat{M}$, not on the regular cone.

These are the prescribed-host conventions of I-B and X-A-ST IB,XAST. A formal tensor $q$ introduced later is not the conical host metric $G$.

The regular conical product is not globally hyperbolic. No global Green operator on that product is used here.

Proof. Fix $r_0>0$, a radial ray, and $z_0$. Choose $A>r_0/c_{\mathrm{h}}$. The causal diamond between the points with coordinates $(-A,r_0,{\phi }_0,z_0)$ and $(A,r_0,{\phi }_0,z_0)$ contains

This sequence approaches the excluded worldsheet and has no convergent subsequence in ${\mathcal{M}}^{\circ }$. The diamond is not compact.

The Green operators constructed below are instead the specified Minkowski operators on $\hat{M}$.

Nothing in Assumption ass:host gives the worldsheet a positive physical thickness. Such a thick-core model would require additional geometric data and would not follow from retaining $\epsilon >0$.

Use the field order

The fields $r,b,\varphi$ are even and have ghost number zero. The ghosts $c,\bar{c}$ are odd, with

All pairings use $\eta$, the coordinate density, and the bilinear formal transpose, without complex conjugation.

For a symmetric tensor $r$, define

In four dimensions,

Set

Then

Proof. Trace reversal is involutive in dimension four. Substitution into $\mathsf{C}Kv$ cancels the two divergence-gradient terms and leaves $\square v$. Compact integration by parts gives the transpose formula.

Consequently,

and hence

These formulas fix the X-E/X-F normalization retained in the supplied X-G-ST Green-matrix passage XGST. No normalization from the distinct XII–XVI inverse convention is substituted.

Define the odd derivation $s_0$, commuting with spacetime derivatives, by

It acts by the left graded Leibniz rule.

The derivation $s_0$ satisfies ${s_0}^2=0$ on the polynomial jet algebra and on finite products of compactly smeared local polynomial functionals.

Proof. Its square vanishes on every field generator. The square of an odd derivation is an even derivation, so vanishing on the generators proves the assertion.

The normalized free gauge fermion is

As usual, an action expression is understood variationally, or with compact variations and the corresponding relative support convention.

The gauge-fixing expression is

Eliminating $b$ gives

The reduced metric Hessian is exactly ${\mathsf{P}}_{\mathrm{gr}}$.

Proof. The odd Leibniz rule produces the minus sign in the ghost term. Equation (eq:operatoridentities) gives $s_0(\mathsf{C}r)=\square c$. The algebraic $b$-equation gives the stated value. The Hessian of $\frac{1}{4}\int (\mathsf{C}r{)}^2$ is $\frac{1}{2}{\mathsf{C}}^{\ast }\mathsf{C}$, which added to $A$ gives ${\mathsf{P}}_{\mathrm{gr}}$.

The free gauge-fixed action is therefore represented by

Its relative free BRST variation vanishes: the variation of $b\mathsf{C}r$ cancels that of $-\bar{c}\square c$, and $AK=0$.

Let ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}/\mathrm{A}}$ be the Minkowski Green operators for $P_m$, and let ${{\mathsf{E}}_0}^{\mathrm{R}/\mathrm{A}}$ be those for $\square$. Their compact-source inverse identities and causal support are the ordinary Minkowski wave-theory inputs BGP.

The bosonic $r,b$ Hessian is

For either support choice,

\mathcal G_{rb}^{\mathrm R/\mathrm A}
 &=
 \begin{pmatrix}
  \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
      &\frac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\mathsf C^*\\
  \frac12\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
      &\frac14\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
                                    \mathsf C^*-\frac12I
 \end{pmatrix}
 \nonumber\\
 &=
 \begin{pmatrix}
  -4\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}
                           &K\mathsf E_0^{\mathrm R/\mathrm A}\\
  -2\mathsf C\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}&0
 \end{pmatrix},
 \label{eq:Ghb}

where

Both inverse identities hold on compact tests.

Proof. The sourced equations are

Eliminate $b=(\mathsf{C}r-j)/2$. Then

Apply the metric Green inverse and substitute back.

The simplification follows from

These identities use ${{\mathsf{P}}_{\mathrm{tr}}}^2=I$, $\mathsf{C}K=\square$, and the compact-source Green identities. The same elimination on compact field pairs proves the left inverse. Differentiation preserves causal support.

For the ghost ordering $(c,\bar{c})$, use

The quadratic ghost expression $\frac{1}{2}(c,\bar{c}){\mathcal{L}}_{\mathrm{gh}}(c,\bar{c}{)}^T$ is $-\int \bar{c}\square c$, with the displayed ordering.

Thus the full free matrix is

Put

The free equations are

Define ${\mathfrak{A}}_0$ over $\mathbb{C}[[\hslash ]]$ by compactly smeared linear fields, equations (eq:freeeq), and the graded causal relations

Only finite polynomial products are used. No coincident nonlinear Wick field is included by this definition.

The presentation (eq:freeeq)–(eq:CCR) is a nonzero associative graded polynomial carrier. The derivation $s_0$ preserves its defining relations and induces a nilpotent graded derivation of ${\mathfrak{A}}_0$.

With the displayed normalization, the generator map to the supplied X-G-ST free presentation is the identity, and its inverse is the same generator map.

Proof. The causal Green difference annihilates the free equations. It defines a graded antisymmetric central form on the compact-test quotient. The enveloping algebra of the corresponding central Lie superalgebra, with its central element identified with the unit and the form scaled by $i\hslash$, gives the stated polynomial carrier.

For equation stability, $s_0(Ar+{\mathsf{C}}^{\ast }b)=AKc=0$, and

The antighost equation maps to $\square b$. The metric Bianchi identity and

show that $\square b=0$ follows from the bosonic equations.

The nontrivial initially zero causal relation is the $r,\bar{c}$ relation:

The remaining checks use the zero cross blocks and the zero $cc$ and $bb$ blocks. Nilpotence descends from Lemma lem:s0nil.

The X-G-ST normalization in the supplied passages is exactly (eq:s0), (eq:Ghb), and the same free causal presentation XGST. Identical generators and relations induce mutually inverse maps. This comparison does not verify other results of that manuscript.

No positive physical Hilbert-space interpretation of the entire ghost carrier is asserted. The finite report systems of Section sec:reports are separate positive matrix algebras, not a CPTP realization of the ghost algebra.

Let $\mathcal{P}(U)$ be the graded algebra of smooth-coefficient polynomial jet expressions on unrestricted smooth germs in $U$. Let ${\mathcal{F}}_{\mathrm{ml}}(U)$ be the algebra generated by compactly smeared local polynomial functionals under finite pointwise products.

The latter is multilocal: a product of two single-integral functionals is generally not a single-integral local functional. Both algebras are completed coefficientwise in the even formal parameter $\kappa$.

Fix smooth classical backgrounds $h,k$ and write

The formal inverse-metric coefficients are permitted coefficientwise. Neither $q$ nor $\bar{q}$ is the host metric $G$.

Define

The differential commutes with spacetime derivatives and uses the left graded Leibniz rule. In particular,

The operator $s$ is a ghost-number-one odd derivation, preserves the stated local and multilocal classical domains, and satisfies $s^2=0$.

Its zero-coupling field operator is exactly $s_0$.

Proof. For the odd vector field $c$, set

The graded coordinate calculation gives ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. Terms with symmetric second derivatives cancel against antisymmetric ghost coefficients. Therefore

Since $\bar{q}$ is fixed and formal multiplication by $\kappa$ is injective before truncation, $s^{2}r=0$.

Similarly,

Expanding $s(c^{\nu }{\partial }_{\nu }{c}^{\mu })$ gives zero after substituting $sc=\kappa c\partial c$. The nonminimal pair is nilpotent by definition.

Every fixed coefficient uses finitely many field factors and derivatives. Compactly smeared local terms remain compactly smeared local terms under $s$, and the Leibniz rule preserves finite multilocal products. The leading coefficient is (eq:s0).

These are the relative field rules displayed in X-H-ST and X-I-ST XHST,XIST. The proof does not infer nilpotence from either source's title.

The unrestricted smooth-germ domain avoids a support mismatch in $s\bar{c}=b$. If compact support is imposed on $\bar{c}$, the same support requirement must be imposed on $b$. A compact-antighost condition with unrestricted $b$ does not define an $s$-stable configuration domain.

Every fixed coefficient has a finite ghost-number decomposition. One may set

on such finite expressions; $s{F}^p\subseteq F^{p+1}$. No completeness, convergence, or spectral-sequence degeneration claim is attached to this filtration.

The field rules do not determine an action. Let $S_{\mathrm{ret}}(\kappa )$ be the separately specified, even, ghost-number-zero classical action of the retained relative problem, including its actual switches, backgrounds, and declared classical deformation terms.

For the free-recovery statement its quadratic, zero-coupling part is required to be (eq:freeaction). An equality modulo a relative support convention is used only on the support class where it has been established. Switch derivatives are not deleted by calling a region a plateau.

Variational action notation is used in its usual local sense: compact variations determine the Euler derivatives. It is not an assertion that an integral of a noncompact configuration is a finite number FRclassical.

Introduce antifields ${{\Phi }_A}^{‡}$ of parity $|{\Phi }_A|+1$ and ghost number $-1-\mathrm{gh}({\Phi }_A)$. Use the odd canonical bracket convention

The bracket extends by the odd biderivation rules. Antifields are written on the right in

Consequently

Reordering (eq:kinham) with antifields on the left introduces the factor $(-1{)}^{|{\Phi }_A|+1}$. A different canonical bracket orientation requires its corresponding overall convention change.

Let $\mathcal{B}$ be the coefficientwise classical local variational BV algebra, together with finite products of its compactly supported functionals, on which these brackets are defined. No distributional quantum contraction is used.

The Hamiltonian (eq:kinham) satisfies

Proof. Its Hamiltonian vector field restricts to $s$ on the fields. The bracket of two antifield-linear Hamiltonians is antifield-linear, and its coefficient at ${{\Phi }_A}^{‡}$ is the coefficient of $s^2{\Phi }^A$, with the fixed canonical ordering. Theorem thm:kinematic makes every coefficient zero.

Define

On the declared variational domain,

The residual has parity one and ghost number one. It is field-dependent and antifield-independent.

It need not vanish merely because $s^2=0$.

Proof. Field-only functionals have zero mutual antibracket. Lemma lem:kinmaster removes the kinematic self-bracket. The two mixed brackets coincide because both Hamiltonians are even, and give $(S_{\mathrm{kin}},S_{\mathrm{ret}})=s{S}_{\mathrm{ret}}$. The other two identities are the odd Jacobi identities. The parity and ghost-number statements follow from the degree of the odd bracket.

Define

All identities are coefficientwise in $\kappa$, on the fixed relative variational domain.

The domain ${\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}$:

1. is a graded subalgebra for the classical product and the odd bracket;

2. is preserved by $\mathcal{Q}$;

3. carries the nilpotent differential $\mathcal{Q}{|}_{{\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}}$;

4. contains the complete field-only classical algebra, on which $\mathcal{Q}=s$;

5. is the largest subdomain of $\mathcal{B}$ on which the same Hamiltonian operator has square zero.

This construction does not set ${\mathcal{R}}_{\mathrm{cl}}=0$ in the ambient algebra.

Proof. The map $({\mathcal{R}}_{\mathrm{cl}},\cdot )$ is an even derivation of both the product and bracket. Its kernel is therefore a graded subalgebra for both operations.

By (eq:curvature) and Jacobi,

Thus $({\mathcal{R}}_{\mathrm{cl}},F)=0$ implies $({\mathcal{R}}_{\mathrm{cl}},\mathcal{Q}F)=0$. Equation (eq:curvature) then proves nilpotence.

Both ${\mathcal{R}}_{\mathrm{cl}}$ and a field-only functional depend on no antifields, so their antibracket is zero. On such functionals $\mathcal{Q}=s$.

Finally, if the same $\mathcal{Q}$ has square zero on an element $F$, then (eq:curvature) forces $({\mathcal{R}}_{\mathrm{cl}},F)=0$. Every square-zero domain is contained in (eq:centralizer).

The theorem supplies a restricted classical differential algebra. It does not prove that its cohomology is the physical observable algebra, that a Koszul–Tate resolution is acyclic, or that unrestricted bulk symmetry has been restored.

The centralizer construction is an application of the classical Jacobi identity, not a new general BV theorem.

At $\kappa =0$, the field rules are precisely (eq:s0), and the normalized quadratic action and Green matrix are (eq:freeaction) and (eq:fullGreen). The free causal presentation therefore recovers ${\mathfrak{A}}_0$ by Theorem thm:freeBRST.

This recovery has three explicit steps:

It is not an algebra homomorphism from a commutative classical field algebra to the noncommutative causal algebra.

The constructed field-only special fiber is exactly the free off-shell field algebra with differential $s_0$. The free causal quantization of its displayed linear equations is exactly ${\mathfrak{A}}_0$.

No flat family of interacting quantum algebras follows. In particular, the special fiber of the full antifield centralizer need not equal the centralizer of the specialized residual.

Proof. The field-only algebra lies entirely in ${\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}$, so specializing its formal coefficients gives every free field expression, with the kernel consisting of multiples of $\kappa$. Theorem thm:freeBRST gives its separate free causal presentation.

Kernels need not commute with specialization. For example, the kernel of multiplication by $\kappa$ on $\mathbb{C}[[\kappa ]]$ is zero, whereas the specialized zero map has kernel $\mathbb{C}$. Hence no specialization theorem for the entire centralizer can be inferred without additional hypotheses.

Let ${\mathcal{I}}_{\mathrm{eq}}\subset {\mathcal{F}}_{\mathrm{ml}}$ be the graded ideal generated by compact smearings of (eq:freeeq). Let ${\mathcal{I}}_{\mathrm{sol}}$ be the ideal of functionals vanishing on all smooth free solutions, interpreted with finite Grassmann coefficient algebras. Then

No equality between these ideals is needed.

For every fixed $m\ge 0$, the nonlinear rule

does not preserve ${\mathcal{I}}_{\mathrm{sol}}[[\kappa ]]$. It therefore does not descend to the unchanged unrestricted classical free-shell quotient.

A compactly smeared scalar equation detects the failure exactly at order $\kappa$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in U$ and a sufficiently small product neighborhood inside $U$. Define

Thus $u_m^{\prime \prime }+m^2{u}_m=0$, and $u_m^{\prime }>0$ on a sufficiently small time interval about $t_{\ast }$.

Take the Grassmann algebra with one odd generator $\vartheta$, and the smooth free configuration

All equations (eq:freeeq) hold.

Choose a nonnegative $f\in {C_c}^{\infty }(U)$, supported in that neighborhood, normalized by

The functional

belongs to ${\mathcal{I}}_{\mathrm{eq}}$. On the displayed free configuration,

and therefore

It follows that

Thus $s{F}_f\notin {\mathcal{I}}_{\mathrm{sol}}[[\kappa ]]$.

The detecting smearing is compact. The free solutions used in the proof are unrestricted smooth solutions and need not have compact spacetime support. No claim about XVI-N's separately specified Route B is made.

Suppose a corrected field derivation has

commutes with spacetime derivatives, and satisfies $(C_1\varphi ){|}_{\mathrm{free}}=0$ as a smooth function on every complete free configuration. Then it still fails unchanged free-shell descent on the witness of Theorem thm:nondescent.

Proof. On a free configuration, the identically zero smooth function $C_1\varphi$ and all of its spacetime derivatives vanish. Hence $P_m(C_1\varphi )=0$ there. The order-$\kappa$ value remains $2\kappa \vartheta$.

This is an obstruction to a named correction class, not to every possible changed equation system or interacting carrier.

Suppose a proposed differential on an unchanged free quantum carrier is required to have the displayed nonlinear scalar rule as its classical normalization, including the compact equation functional $F_f$. If its classical reduction preserves the unchanged free equations, such a proposal is incompatible with (eq:witness).

Proof. A well-defined differential sends the zero equation class to zero. Its assumed classical reduction would therefore make $s{F}_f$ vanish on every free solution. Equation (eq:witness) contradicts that requirement.

No nonlinear quantum composite has been formed in this argument. The corollary is conditional on the proposed quantum map having the stated classical normalization.

A smooth homogeneous Minkowski wave with compact spacetime support is zero. Consequently, restricting ghosts to compact spacetime support does not preserve the full unrestricted free ghost solution space.

Proof. A compactly supported wave has zero Cauchy data on a sufficiently early time slice. Cauchy uniqueness makes it identically zero.

The witness does not contradict relative off-shell nilpotence in X-H-ST. It tests a different operation: descent to the unrestricted free shell.

Let $\mathcal{D}$ be a declared graded subalgebra of ${\mathcal{F}}_{\mathrm{ml}}[[\kappa ]]$, and let $J\subset \mathcal{D}$ be a graded ideal. Let $C$ be an odd ghost-number-one derivation of $\mathcal{D}$, beginning at positive order in $\kappa$, and define $d=s+C$.

Then $d$ induces a square-zero derivation of $\mathcal{D}/J$ if and only if

Both conditions refer to the entire declared ideal and domain, not to a selected witness.

Proof. A derivation descends through a quotient exactly when it preserves the defining ideal, proving (eq:idealtest). Since $s^2=0$,

Its induced square vanishes exactly under (eq:niltest).

The same statement applies to a subalgebra of ${\mathcal{B}}_{{\mathcal{R}}_{\mathrm{cl}}}$, with $\mathcal{Q}$ in place of $s$, provided all operations are defined there.

If a ghost-number-zero correction $a$ is added to a classical action, it changes the master residual by $sa$. The exact action-compatibility equation is

on the specified relative domain. Such a correction is not permitted merely because the equation can be written: its admissible support, normalization, and action-preservation requirements must be fixed first.

Write

Square-zero requires

For example,

These are additional equations. A first-order linear solve does not discharge the second equation.

Let $V$ be the free graded compact-test generator space. Let $\mathfrak{T}$ be its formal tensor algebra over $\mathbb{C}[[\hslash ,\kappa ]]$, completed coefficientwise. Let $J_0$ denote the free equation and causal-relation ideal giving ${\mathfrak{A}}_0$.

A proposed interacting presentation supplies:

1. an explicitly defined graded two-sided ideal $\hat{J}\subset \mathfrak{T}$;

2. an odd ghost-number-one derivation $\hat{d}$ on $\mathfrak{T}$, with actual values in that tensor algebra;

3. the exact free-fiber identities

   $$(\hat{J}+\kappa \mathfrak{T})/\kappa \mathfrak{T}=J_0,\hat{d}\mathrm{mod}\kappa =s_0;$$

4. all analytical definitions needed to represent the proposed nonlinear field expressions on its intended domain.

The symbol for a local nonlinear quantum field is not accepted as an element merely by analogy with a classical jet.

For Definition def:quantuminput, the quotient

has a well-defined nilpotent induced derivation precisely when

Under the stated free-fiber identities,

with the identity on free generators. If additionally

the quotient has no $\kappa$-torsion.

These conditions do not prove causal factorization, renormalized time ordering, positivity, or existence of the intended analytical representation.

Proof. Ideal preservation is exactly representative independence. The second condition is exactly square-zero after quotienting. The free-fiber statement is the third isomorphism theorem applied to $\mathfrak{T}/(\hat{J}+\kappa \mathfrak{T})$. The saturation condition is precisely the assertion that $\kappa [F]=0$ implies $[F]=0$. The additional analytical and physical properties do not occur among these algebraic hypotheses.

interacting quantum gate is not passed{ No nontrivial $\hat{J},\hat{d}$ satisfying the prescribed nonlinear classical normalization and all required analytical conditions is supplied here.

The unchanged free-shell target is obstructed by Theorem thm:nondescent. A changed carrier would require its own construction and comparison.

Transporting a known free algebra by an arbitrary invertible linear change would only give a transported free presentation until its relation to the retained interaction had been proved.}

The classical master residual satisfies

By contrast, a reciprocity defect for an even classical source or response map $J$, on even variations, has form

and is even with ghost number zero.

If $J$ is the variational derivative of a twice differentiable classical functional on the declared even variation domain, then

for every admitted pair $u,v$. Neither this identity nor its failure is a ghost-number-one master equation.

Proof. The two terms are the mixed second variations of the same functional. Their equality is symmetry of the second derivative on even variations.

The ghost-number and parity assignments follow from the arguments in (eq:reciprocity); they differ from those of ${\mathcal{R}}_{\mathrm{cl}}$.

The supplied X-G-ST record contains a positive free-solution reciprocity witness and X-H-ST contains finite reports of its installed evaluation XGST,XHST. For use here its exact source normalization is kept as a declared data package:

No numerical value or unsupplied component formula is invented. Results using this positivity are conditional on that explicitly identified witness package. The newly proved scalar witness (eq:witness) is independent of it.

Suppose permitted source corrections have reciprocity changes in a declared vector space $\mathcal{B}$. If

then no permitted correction restores reciprocity on a domain containing the witness.

Proof. Every corrected pairing has evaluation ${\beta }_{\mathrm{G}}+0>0$. It cannot be the zero bilinear defect.

Cancelling ${\beta }_{\mathrm{G}}$ at one pair would remove that detector only. It would not prove vanishing of the entire pairing.

A homogeneous ghost-number-one residual cannot cancel a homogeneous ghost-number-zero reciprocity defect. Their sum vanishes in a graded direct sum only if both components vanish. Neither residual is a loop anomaly merely because it is an obstruction.

Proof. Distinct homogeneous summands of a graded direct sum are linearly independent. Both displayed residuals are classical constructions with no loop expansion. A quantum anomaly statement would require its own defined quantum product and master-Ward framework FRquantum.

For one specified linear correction problem:

1. all candidate corrections are parameterized by $z\in {\mathbb{C}}^n$, without omitted admissible directions;

2. the complete residual belongs to a proved finite-dimensional vector space $W$;

3. a specified linear evaluation map $E:W\to {\mathbb{C}}^m$ is injective;

4. the residual depends affinely on $z$.

Odd and even residual sectors are kept as separate graded summands.

Under Assumption ass:coverage, the complete linear cancellation problem is equivalent to

It is solvable if and only if

equivalently if and only if

A nonzero $\mathrm{rank}M$-minor and a spanning left kernel give an exact algebraic certificate. If $z_{\ast }$ is a solution, all solutions are $z_{\ast }+\ker M$.

Proof. Injective coverage makes zero evaluated residual equivalent to zero residual in $W$. The matrix statement follows by choosing bases. The left-kernel condition is the finite-dimensional criterion $b\in \mathrm{im}M$. The affine solution description follows by subtraction.

This is the rank–minor–left-kernel shape compared with II-B IIB. No source-specific rank, minor, coefficient, or numerical count is imported.

Without proved coverage, a successful finite fit is only a fit. A normal-jet order bound does not by itself make a tangential distribution space finite dimensional. Even a complete linear fit does not supply the nonlinear equations (eq:MCorders), the quantum relations, or an undefined graph product.

A report package supplies:

1. a named compact evaluation configuration and all classical source data;

2. a finite Grassmann coefficient convention;

3. an ordinary real extraction from that evaluation;

4. a finite encoding, calibration basis, and program;

5. every returned quantum residual, failure, and reference.

For example, if

in a specified finite Grassmann algebra, a report may use

with $I_{\ast }$ fixed in advance. A reciprocity report may instead use the ordinary real number ${\beta }_{\mathrm{G}}$ in (eq:recinput).

These numbers are different reports. No map from an odd Grassmann element to a positive density matrix is asserted.

Let

have a calibrated orthonormal basis $\{|{\tau }_k⟩\}$. Let $S$ contain every unmeasured finite output, and set

The channel

is CPTP and equals

It satisfies

for every untouched reference $E$.

The quantum clock names are consumed. A positive-probability block is normalized only by dividing by its own trace; no normalized state is assigned to a zero-probability branch.

Proof. The recorded Kraus maps are $|k{⟩}_R\otimes D_k$, and their squared moduli sum to ${\sum }_k{P}_k\otimes I_S=I_{CS}$. Expansion of $X$ in the calibration basis shows that both formulas return its $kk$-block. Summing gives the partial trace. Tensoring the Kraus maps with $I_E$ proves the reference statement. The codomain has no quantum $C$-factor.

This is the explicit I-B/En09-A interface IB,En09, not a field-measurement theorem.

If fixed complete continuations ${\Lambda }_k$ are specified, then

is CPTP. Actual loss and rejection are retained as outcomes rather than postselected away.

Identify an X-H-ST residual-report or reciprocity-report interface with the same:

Then the recovered channel is exactly (eq:continuation) on all input operators and every reference extension.

If an X-K-ST finite continuation is appended on matching interfaces, its complete history channel is the same channel obtained by composing these maps.

Proof. Theorem thm:instrument identifies each subnormalized block on all operators. Applying the identical ${\Lambda }_k$ gives equality of every recorded output block. Finite induction gives equality after identical complete continuations.

The theorem recovers the supplied X-H-ST report channels under identical installed data XHST. It does not reconstruct an unavailable source evaluation or infer that its numerical calibration has been physically acquired.

For an ordinary supplied real number $a$, fix an encoding scale ${\sigma }_{\mathrm{e}}>0$, distinct from $\epsilon$, and define

This is the finite amplitude shape used in X-Enter-ST XEnterST. It is a preparation from supplied data, not a continuum acquisition map.

Computational destructive readout of $\tau (a)$ has probability $p(a)$ for code $1$. For every $a,b\in \mathbb{R}$,

A fixed subsequent complete channel cannot enlarge this bound, including with an arbitrary independent input–reference state.

Proof. The probability is the squared second amplitude. The Bloch vector is

Its derivative has Euclidean norm $\mathrm{sech}(a/{\sigma }_{\mathrm{e}})/{\sigma }_{\mathrm{e}}\le 1/{\sigma }_{\mathrm{e}}$. Qubit trace distance is half the Euclidean distance between Bloch vectors, proving (eq:encodingerror). Tensoring with a state preserves trace norm, and channels contract trace distance.

One attempted readout gives one random code, not the exact value $a$ or its exact probability. No finite-sample inference theorem is included.

There are complete one-outcome instruments with identical record laws and residual trace distance $1/2$ on one qubit input.

Proof. Compare the identity residual channel with computational dephasing, each carrying the same single classical symbol. On $|+⟩⟨+|$, their outputs have residuals

Their difference has eigenvalues $1/2,-1/2$, so its trace distance is $1/2$.

This is the complete-residual distinction also documented in En03-A and related source records En03.

The destructive channel has no left inverse on the full clock input space.

Proof. With the same surviving input, the distinct clock states

have identical outputs. No fixed left inverse can recover both.

If two continuum configurations induce the same installed finite input state, every fixed report program has identical complete output on them. It cannot distinguish differing target values on that pair.

This does not exclude every acquisition model. It states the missing input requirement of the present report package.

Proof. A fixed linear channel has the same output on the same input. A claim to distinguish the target values would contradict that equality. An additional continuum-to-register acquisition map could change the premise, but no such map is constructed here.

At each node $v$ of a finite public-outcome tree, supply Kraus maps

The index $a$ is public; $\mu$ is not accessible unless separately reported. All actual memory, residuals, and failure flags belong to the node interfaces.

Each clock input is a live, unconsumed factor or a new factor returned by an explicitly specified preparation or restoration map. Reusing the spelling of a consumed name does not supply a quantum input.

The complete leaf-history channel is

where $K_{a,\mu }$ is the compatible ordered product along that refined history. All failures and reference extensions are included.

Proof. At each node, pull the completeness identity of its child continuations back through its Kraus maps. Backward induction over the finite tree gives the identity in (eq:history). Tensor every map with $I_E$ for the reference statement.

This re-establishes the needed X-K-ST finite composition law XKST. A single Kraus map per public history is not assumed.

Choose a straight developed segment of length $\ell >0$ inside the retained regular tube. Install $n+1$ equal $p$-qubit payload registers $A_0,\dots ,A_n$, with independent blanks at $A_1,\dots ,A_n$. Use successive permitted SWAPs, each over developed duration $\ell /(n\upsilon )$, where $0<\upsilon \le 1$.

After hop $j$, every payload–reference input has complete route state

Arrival occurs after developed duration $\ell /\upsilon$, or host duration $\ell /(c_{\mathrm{h}}\upsilon )$. The route stays in the supplied regular tube.

Proof. Each SWAP moves the entire payload and all its correlations into the next independent blank, leaving the previous slot blank. Induction proves the formula. The specified spacings and durations certify future-causal cross-port dependencies. Equation (eq:signature) gives the host duration.

A record used at a distant controller must itself be delivered on a declared causal classical wire, or included in the transported payload. Treating a record as a mathematical reference preserves its correlations but does not deliver it.

The SWAP permission is an operational input. No microscopic gravitational Hamiltonian is derived.

For a complete block schedule, define the backwards region of an output by repeatedly taking the union of blocks meeting its current backwards region. Controllers and classical wires are included.

An output observable pulls back only to its scheduled ancestors. Equality of joint ancestor–reference reductions gives equality of the joint output reductions.

Metric containment follows if every actual dependency edge has a separately certified future-causal path in the retained host.

Proof. On a disjoint block the pulled-back observable is the identity, and a channel adjoint is unital. Only a meeting block can enlarge support. Backward induction proves the support claim. Concatenating the certified edge paths proves metric containment.

At every branch cut $c$, charge all live laboratory qubits, including reserved blanks and memory:

with shared allocations counted only once. Returned storage, supplier storage, classical tables, and untouched references have separate ledgers.

A host realization requires

and the corresponding classical-storage, location, permission, and duration inequalities. Finiteness alone does not prove these inequalities.

A certificate that passes these fit checks uses only its declared finite registers inside the retained reserve. Its operations change neither

nor the fixed switches, width, and regulator.

Proof. None of the command types has a host-update or regulator-update output. Induction preserves that metadata. The fit checks give the location and capacity bounds.

Use

These comparisons include arbitrary untouched references Watrous.

Suppose ideal and implemented stages are CPTP maps on matching complete interfaces, with

Then

For a matched finite adaptive tree with node allowances $e_v$, the bound is

Preparation and final-output allowances are added once on their actual comparison interfaces.

Proof. For a sequence, replace one factor at a time. Pre- and postcomposition by channels contract diamond norm; the triangle inequality gives the sum.

For a tree, a controlled continuation on classical direct-sum blocks has error at most the largest child error. Thus the recursive bound is

Backward induction gives the maximum root-to-leaf sum.

Hidden correlated memory cannot be priced by a carrier-only one-step marginal. It must be included in the actual step interface. A trace-distance bound for one tested input is not a diamond-norm bound.

The same budget bounds retained event probabilities and record TV distance. It does not validate a false supplied enclosure, certify an unretained arrival flag, or provide a finite-sample confidence theorem.

The X-F three-tree plateau identity remains counted once XF. Neither an additional report nor a second clock adds a duplicate tree coefficient.

The X-A-conv/X-Enter-ST convolution obstruction is unrepaired XConvST,XEnterST. The X-J-ST compact scalar coefficients remain classical supplied evaluations, not a whole-algebra convolution repair XJST. The X-D-ST fixed-input sharp-limit result is not metric admission XDST.

No Einstein equation follows from trace preservation, a classical record, or reference-preserving transport. The host and action remain inputs.

The internal amplitude constructed at XIV-I's I2 does not define its joint cross-line composition. The supported-sector reference in XIV-J treats $K_6+K_8+K_9$ on its stated off-locus domain, not the complete six-branch target XIVI,XIVJ.

For that target, the still-required chain is

The present classical differential is not any arrow of this chain. No differential is applied to the absent inherited coefficient.

XIV-H's H2 ending is historical and version-specific; XIV-I's narrower I2 advance is not withdrawn XIVH,XIVI. The supplied XV-B version 2 has its own B1 full-map boundary, while some older series summaries use F0 for another technical stopping convention. These labels are not silently identified XVB.

Technical lifting theorems in XV-B do not constitute opening its downstream quantum-admission exit. A reconstruction-null criterion does not construct the missing graph.

The standing value

\texttt{\detokenize{O.1 = NOT_CLOSED}}

is retained outside each independently earned scope. The supplied XVI-Q Q11 description reports a limited massless one-loop ${\kappa }^2$ two-point closure under its enlarged counterterm domain and XVI-P prescription; that record does not update XVI-O's different prescription or establish bulk symmetry restoration XVIrecords.

Accordingly,

\texttt{\detokenize{bulk_symmetry_restored = false}}.

XVI-N and its Route B are not discharged. The mass parameter in Theorem thm:nondescent belongs to a different scalar equation-descent witness, not to the prepared insertion of XVI-N XVIN.

The external time axis, event locations, preparations, and durations remain after every destructive readout. A finite clock-conditioned report is not quantized relational time without an external clock.

Reference-preserving transport preserves supplied correlations. It is not a product-to-entangled-state map generated by the metric. No Page curve, de Sitter state, cosmological bounce, vacuum-energy computation, ultraviolet completion, or holographic dictionary is constructed.

Oai282-A and Oai283-A are documentary comparisons for coverage, interface, and resource-accounting practices OaiRecords. OpenAI Families 282, 362, and 376 are not premises. No family proof, catalogue scope note, or reported formalization is independently verified here.

The labels below retain the numbering supplied with the plan. A status is a statement about the scope of X-L-ST, not a universal impossibility theorem.