XVI-O1-ST-v5 Finite Enter Certificates with a Permanent Printed-Array Integrity Gate Complete readout, exact counterterm feasibility, an affine gauge propagator, and an audited failed regression

We develop an audit-hardened finite certificate architecture for a prescribed-host perturbative loop sector, preserving the distinction between mathematical proofs, printed data, supplied scientific records, and execution artifacts. The central v5 finding is a cross-artifact integrity witness: the v4 typeset D01 table has sixth coefficient $-4/3$, whereas its embedded and separately supplied regression has $+4/3$. With the unchanged gauge-table coefficient ${a^{(0)}}_6=4/3$, the regression relation fails by the exact residual $a^{(0)}+f^{\mathrm{reg}}=(8/3)e_6$. Python index 5 is displayed structure $B_6$. Both source values are retained; no physical coefficient is selected by editing a sign.

The supplied v4 transcript reports eight passing suites and one failed suite. Its phase failure occurs before that suite's variation-count and derivative checks, so those later predicates are not described as executed. Their printed-data proofs remain available. A proposed v5 regression checks the exact known discrepancy as a negative control and keeps data-integrity failure separate from successful regression behavior.

The retained mathematical results include complete destructive report instruments with untouched references, the sharp complex qubit-report half-diamond distance ${\max }_j|v_j-{\tilde{v}}_j|/(2B)$, its real binary specialization, original-minor and complete-kernel feasibility certificates, quotient descent versus non-feedback, primitive-enlargement criteria, and an exact affine general-$\xi$ inverse under the declared Q4S kinetic normalization. The old unchecked affine candidate still has residual $(0,0,1{)}^{\mathsf{T}}$; finite-field elimination and even/odd vertex-occurrence controls remain permanent audit components.

The supplied 31-entry second-readout comparison, mutation discrepancies $431/30,1/3,5/12$, 13-of-14 D01 variation, and restricted evanescent rank report $7\to 8$ are preserved with their precise evidence boundaries. The potential impact is a reusable release contract that prevents an expected software failure, a transcription inconsistency, or a missing scientific matrix from becoming a positive physics certificate. No novelty is claimed for elementary channel or rank mathematics. Q03 remains _CLOSED; full gauge rows and counterterm-derivative exactness remain ; the clean-machine rebuild remains . The new appendix is and unexecuted. The cone remains an input, bulk symmetry is not restored, XVI-N remains frozen, and O.1 is _CLOSED outside independently earned scopes.

boundary and principal correction{ The permanent discrepancy is not erased by changing the caption or overwriting a coefficient. The typeset vector, the regression vector, and the gauge vector are separately identified below.

This manuscript contains a self-contained proof of its finite mathematical statements and exact printed-data witnesses. It does not supply the absent loop programs, full gauge-image arrays, enlarged primitive response, or complete 23-row gauge-family matrix.

A supplied v4 run is an attributed execution record. It is not a v5 run. For the new listing, _status=PROPOSED, _executed=false, and _verified=false.}

Cheung's Enter calculus treats runtime quantum names as access rights to factors of a global state. Destructive computational-basis readout consumes the measured names from the returned quantum interface and supplies a classical record. Its measurement characterization is restricted to a declared strategy class, not an unrestricted derivation of continuum field measurements CheungEnter,CheungI.

The introductory essay The Philosophy of Enter distinguishes readout from recovery, a classical record from a quantum-return decoder, a channel from a thermodynamic machine, and an analogy from an identification CheungPhilosophy. These distinctions organize interpretation; they are not additional premises of the mathematical proofs here. Consumption of a name does not assert annihilation of every physical carrier used by an implementation.

I-B installs finite clocks, registers, calibration, controls, and resources on a prescribed conical spacetime CheungIB. En03-A separately declares geometry, locations, accessible algebras, and control/resource interfaces CheungEn03. We retain this shape but prove the needed finite instrument identities directly. Standard quantum instruments and completely bounded trace norms provide the general setting DaviesLewis,Watrous.

A numerical loop coefficient is not automatically a probability or a measurable field effect. The finite report channels below are explicitly constructed encodings of supplied numbers, not field-acquisition channels.

The evidence package supplied for v5 contains:

1. -O1-ST-v4.tex;

2. _xvi_o1_st_v4.py;

3. 4_verification_output.txt;

4. _SUMMARY_v4.txt;

5. the earlier supplied consolidated and second-readout manuscripts, together with version-qualified source excerpts.

The manuscripts and artifacts are cited under their supplied identifiers SuppliedV3,SuppliedBot,SuppliedV4,V4Artifacts.

The summary is explanatory metadata. It is not an independent execution record or a substitute for examining statement order in the script. No authorship of this anonymous manuscript is inferred from the authorship of cited sources.

We distinguish:

1. Specifications: objects, conventions, scopes, and acceptance predicates fixed before a calculation.

2. Mathematical proofs: written arguments from explicit hypotheses.

3. Exact synthetic witnesses: specified finite data with exactly established properties. This category includes the printed-array comparison, without identifying it with loop data.

4. Supplied scientific records: attributed calculations or scientific conclusions in supplied manuscripts.

5. Execution artifacts: supplied program text, environment descriptions, and transcripts, with provenance qualifications.

6. Listed implementations: new program text without an executed run.

These categories are not interchangeable. A supplied transcript is not an independently reproduced run; a displayed exact residual is not merely a software assertion.

The predecessor's successful mathematical results are retained. Two audit corrections are added:

1. the failed phase predicate is localized to a difference between the typeset table and the regression transcription;

2. predicates after the failing instruction are distinguished from predicates actually reached in that run.

The new regression contract records the discrepancy exactly. It does not redefine the false equality to be true.

No elementary rank theorem, instrument theorem, encryption mechanism, or general propagator technology is claimed as new. The contribution is their joint use in a scope-explicit counterterm and data-integrity release architecture.

Fix $0<\mu <2\pi$, $a=1-\mu /(2\pi )$, and $c_{\mathrm{host}}>0$. Set

with prescribed metric

Increasing $T$ is future directed. The completion defect is

The supported laboratory avoids $\mathcal{W}$; the inherited width $ϵ>0$ remains fixed CheungIB,CheungXAST.

The metric has signature $(-,+,+,+)$, and $\mathcal{W}$ is a two-dimensional worldsheet. On an injective angular chart,

gives $G=-\mathrm{d}{t}^2+\mathrm{d}{x}^2+\mathrm{d}{y}^2+\mathrm{d}{z}^2$. The regular product is not globally hyperbolic.

Proof. The signature and dimension are immediate, and differentiation gives the local development. For $r_0>0$, take in geometrized time

The points $x_n=(0,r_0/n,{\vartheta }_0,z_0)$, $n\ge 2$, belong to $J^{+}(p)\cap J^{-}(q)$ by radial causal curves. They have no convergent subsequence in the regular product, since their completion limit has $r=0$. The diamond is not compact.

Declare separately

on the supported chart. Free fields, Green operators, loop integrals, and local counterterms use this analytical carrier and its prescription, not an assumed global Cauchy theorem on $M^{\circ }$ CheungXAST,CheungXGST. A signature reversal does not authorize independent changes of action-Hessian or Green-function signs.

A conserved traceless tensor need not satisfy Einstein's equation for the prescribed host.

Proof. For a nonzero constant ${\rho }_0$, let

The forms are parallel, so $\tau$ is conserved; its $G$-trace vanishes. The regular host is locally flat and has zero Einstein tensor. Thus $\mathrm{Ein}(G)={\kappa }_{\mathrm{grav}}\tau$ fails for nonzero ${\kappa }_{\mathrm{grav}}$ at zero cosmological constant. No distributional source equation at the defect is constructed.

The conical product is not globally identified with Minkowski spacetime. The analytical completion is not an inferred geometry.

A scope is

specifying host, action or observable, loop order, power of $\kappa$, external sector, mass, gauge fermion, analytical prescription, counterterm presentation, complete residual space, and separately imposed consistency tests. Equality includes versions, conventions, and identification maps.

The supplied XVI-Q Q11 record, read with XVI-P p23, attributes closure only to the one-loop, order-${\kappa }^2$, massless bulk two-point retained sector of $S_{\mathrm{inv}}$, with frozen gauge fermion, QdS loop/external conventions, Q4S vertex indices, $\overline{\mathrm{MS}}$, and the p22 evanescent contract. All 23 named identity rows are reported cancelled by local counterterms. The consistency statement covers the bulk 11 four-dimensional directions. The used presentation is 11 p14r directions over $\mathbb{Q}(\mathrm{i})$ plus 10 p22 evanescent operators, 21 in total CheungXVIP,CheungXVIQ.

The v5 status remains _CLOSED. This record does not update XVI-O's different prescription, restore bulk symmetry, discharge XVI-N, or certify higher arity or loop order.

QdS denotes the quasi-dimensional scheme, not de Sitter spacetime. The symbolic dimension and evanescent algebra must be handled before the prescribed Laurent extraction. Consistent DRED does not license unrestricted use of four-dimensional tensor identities Stockinger.

A package supplies:

1. the action, gauge fermion, field/antifield normalizations, Fourier and Grassmann conventions, and subtraction prescription;

2. the complete inventory and the derivation of every zero;

3. complete row assembly and all computational dependencies;

4. counterterm generators, relations, and reconstruction maps;

5. exact arrays or explicit missing entries;

6. original minors, complete kernels, residuals, negative controls, error conventions, and budgets;

7. a manifest binding exact versions and bytes.

An unfinished task or resource refusal is not zero.

A digest identifies bytes relative to a manifest. It does not prove that the manifest is trustworthy or that the bytes implement the declared physics.

Following the En03-A shape CheungEn03, retain:

1. external geometry (eq:host) and the separate carrier (eq:completion);

2. an actual location map for physical laboratory registers;

3. accessible finite matrix algebras and classical record algebras;

4. the complete controls, arithmetic, storage, routing, calibration, and failure interface.

The 12 inventory graphs and 23 named rows are logical tasks, not spacetime ports. The 21-direction analytical presentation is not a quantum observable algebra.

Work over $\mathbb{K}=\mathbb{Q}(\mathrm{i})$, unless an enlarged exact field is declared. Let

For each named row choose a complete finite coefficient space $W_{\alpha }$, and put $W={{⨁}_{\alpha =1}}^{23}{W}_{\alpha }$. Twenty-three named rows need not mean 23 scalar equations.

Let $A:V_{\mathrm{pres}}\to W$, $d\in W$, and let $L_0:V_{\mathrm{pres}}\to J_0$ encode only the consistency tests actually claimed. Define

The response descends to reconstructed counterterms iff

A zero tested coefficient residual implies the asserted functional identity when the coefficient/test map is injective on a known subspace containing all possible tested residuals. Without separation, this implication can fail.

Proof. Equal reconstructions differ by a reconstruction-kernel vector; their responses agree exactly when $M$ annihilates it. The coverage assertion is injectivity. A nonzero element in the test map's kernel supplies the counterexample without it.

No full O.1 rank or original minor is deduced from the numbers 23 and 21.

Fix a finite dictionary within declared representation and resource bounds. Let $C\cong {\mathbb{C}}^D$, $D\ge 2$, have calibrated orthonormal basis $\{|{\tau }_j⟩{\}}_{j=1}^D$. Let $A_{\mathrm{lab}}$ contain all surviving laboratory registers, and install finite words $q_j$ carrying scope, package, evidence, status, and coefficient codes.

Put

The record retains $j$ even when two words coincide. Unused labels and failures have explicit words.

The map (eq:instrument) is CPTP, and for every finite untouched reference $E$,

For a state $\rho$,

The positive-probability conditional state is ${\sigma }_j/p_j$. No normalized state is assigned to a zero branch. The returned quantum interface excludes $C$.

Proof. The Kraus maps $K_j=|j,q_j⟩\otimes D_j$ satisfy

Writing $X={\sum }_{j,k}|{\tau }_j⟩⟨{\tau }_k|\otimes X_{jk}$, the output blocks are $X_{jj}$, whose sum is ${\mathrm{Tr}}_{C}X$. The same calculation with $A_{\mathrm{lab}}\otimes E$ proves the reference identity. Positivity supplies the branch rule.

This is an I-B/En09-A supplied-report interface CheungIB,CheungEn09. It does not evaluate unknown loop integrals by measurement or acquire an unknown continuum field.

The report instrument has no universal left inverse. Its classical marginal does not determine its surviving quantum state.

Proof. The distinct coherent clock states $(|{\tau }_1⟩\pm |{\tau }_2⟩)/\sqrt{2}$, tensored with the same laboratory state, have identical outputs. Also, $|{\tau }_1⟩⟨{\tau }_1|\otimes {\rho }_A$ and $|{\tau }_1⟩⟨{\tau }_1|\otimes {{\rho }^{\prime }}_A$, with ${\rho }_A\ne {{\rho }^{\prime }}_A$, have the same deterministic record and different residuals.

If distinct readings $t_j$ must equal a fixed event time $T_e$ on every positive-probability branch of a normalized input, there is one deterministic matching outcome. A prepared basis clock separately reads its installed word deterministically. Finite matrices cannot satisfy $[T,H]=\mathrm{i}\hslash I$ for $\hslash \ne 0$.

Proof. At most one distinct reading matches $T_e$, and probabilities sum to one. Basis readout follows from (eq:instrument). The trace of the proposed commutator would give $0=\mathrm{i}\hslash D$.

A gauge-table comparison at $\xi =1$ is not this event-time calibration theorem. Repreparing a known basis state is not a universal inverse for an unknown coherent clock.

For real $r_j$ with $|r_j|\le B$, define the binary report

The coefficient is a bounded expectation, not an exact rational value recovered from one shot.

For complex $v_j$, $|v_j|\le B$, define

The Bloch bound makes ${\tau }_B(v_j)$ a state. This interface returns a classical task label and a quantum report qubit, not a wholly classical coefficient word. Both maps are channels: they measure the clock and append valid classical or quantum states while retaining the branch block.

With $d_{\diamond }(\Phi ,\Psi )=\frac{1}{2}{\Vert \Phi -\Psi \Vert }_{\diamond }$ and one common admissible $B>0$,

These equalities include arbitrary finite untouched references.

Proof. The difference of the two qubit states has eigenvalues $\pm |v_j-{\tilde{v}}_j|/(2B)$, and hence trace norm $|v_j-{\tilde{v}}_j|/B$. For any operator $X$, including a reference extension, the task blocks are orthogonal. Thus the output difference has trace norm

Clock pinching is an average of unitary conjugations, so ${\sum }_j{\Vert X_{jj}\Vert }_1\le {\Vert X\Vert }_1$. This proves the diamond upper bound. A basis clock input at a maximizing index attains it. For binary reports, the two scalar differences are $\pm (r_j-{\tilde{r}}_j)/(2B)$; the same calculation applies.

Distinct exact symbolic words are orthogonal outputs and have half-diamond distance one on a distinguishing task. The formulas above do not describe that symbolic metric. Splitting a complex coefficient into separate real and imaginary binary tasks gives the maximum component discrepancy on the expanded dictionary, not automatically the complex modulus.

An error confined to one task can attain the full maximum in (eq:complexerror) or (eq:realerror), regardless of its rarity under a chosen sampling law.

Proof. Prepare the affected basis task.

For compatible ideal and implemented complete channels,

Proof. Telescope the difference, use contractivity under channel composition and the triangle inequality, and use the maximum one for half-diamond channel distance Watrous.

Records, failures, residual systems, and references remain in the compared interfaces. Oai283-A is a documentary comparison for this resource/error discipline, not a thermodynamic or loop premise CheungOai283.

The presentation follows II/II-B's original-minor discipline without importing their numerical certificate CheungII,CheungIIB.

Let $M\in {\mathbb{K}}^{n\times p}$, $y\in {\mathbb{K}}^n$, and $0\le r\le \min (n,p)$. Suppose original row and column index sets $I,J$, both of size $r$, and $L\in {\mathbb{K}}^{(n-r)\times n}$ satisfy

Then

A solution $c_{\star }$ gives the complete affine family $c_{\star }+\ker M$. A nonzero component of $Ly$ gives an outside-span separator.

Proof. The minor gives $\mathrm{rank}M\ge r$. Since $\mathrm{im}M\subseteq \ker L$ and $\dim \ker L=r$, the opposite inequality holds and $\mathrm{im}M=\ker L$. This proves membership and augmented-rank equality. The affine assertion follows by subtracting two solutions. A nonzero row pairing separates the target from the image.

The empty minor has determinant one. An annihilating matrix is not necessarily a complete left kernel.

If $\mathrm{rank}M=r$, $MZ=0$, and $Z\in {\mathbb{K}}^{p\times (p-r)}$ has rank $p-r$, then $\mathrm{im}Z=\ker M$. With a verified particular solution, all solutions are $c_{\star }+Zt$.

Proof. The inclusion and dimensions agree.

For (eq:system), this proves exactly the declared cancellation and consistency equations. Functional interpretation additionally requires descent and coverage. Coefficient uniqueness is presentation-relative, not physical or cohomological uniqueness.

Let

The old fallback $c_{\mathrm{old}}=(1,1,0{)}^{\mathsf{T}}$ has residual

The complete correct family is

Proof. Direct multiplication gives the residual and the corrected solution. The original minor on rows $1,3$, columns $1,2$, is $-1$. The complete left kernel is generated by $(-2,1,0)$. The displayed nonzero right-kernel vector spans the one-dimensional kernel. Apply Theorem thm:rank and Proposition prop:right.

Adding a kernel vector to the wrong particular solution preserves its nonzero residual. A solver exception does not validate a fallback vector.

The second-readout predecessor attributes its rules to XVI-Q's 6q05/gravity.py, derived from the declared action rather than importing XVI-P's vertex tables, and pole extraction to 04/uvpole.py SuppliedBot,CheungXVIQ. Its reported conventions include

The corresponding ghost and antifield couplings, auxiliary elimination, R09 vertex checks, and fresh same-machine run are supplied scientific records, not newly executed derivations here.

The record discloses a common author, machine, and extraction route. A second folder is not a clean-machine or heterogeneous rebuild.

Use

For $E_i=a_i\otimes a_i$, the structures are

\begin{array}{c|l@{\qquad}c|l}
1&xyQ^2&8&u^2v^2\\
2&xv^2Q&9&u^2y_tQ\\
3&xy_tQ^2&10&uvz_tQ\\
4&z^2Q^2&11&yx_tQ^2\\
5&zuvQ&12&v^2x_tQ\\
6&zz_tQ^2&13&x_ty_tQ^2\\
7&u^2yQ&14&z_t^2Q^2
\end{array}

and are denoted $B_1,\dots ,B_{14}$. Hats and tildes are scheme contractions, not host geometries.

The v4 typeset table uses $v_{01,j}={\tilde{v}}_{01,j}=\mathrm{i}{f_j}^{\mathrm{tex}}$ and $v_{03,j}={\tilde{v}}_{03,j}=\mathrm{i}{g}_j$. Its mutant values are $\mathrm{i}{f_j}^{\mu },\mathrm{i}{g_j}^{\mu }$.

For D06, use

with

These retain all $14+14+3=31$ displayed comparison entries.

For the printed frozen/own arrays, the advertised comparison discrepancy is zero. The component-mutant comparisons give

At common bound $B=32$, the complex-report distances are

The D01 changed-entry count and maximum discrepancy are the same if the regression vector $f^{\mathrm{reg}}$ defined below is used instead of $f^{\mathrm{tex}}$.

Proof. The displayed own/frozen entries coincide. For D01, entry 5 changes by $431/15$, the maximum. For D03, the maximum is $2/3$, at entries 5 and 10. For D06, it is $5/6$, at entry 2. All magnitudes are below 32, so Theorem thm:error applies. Changing only D01 entry 6 from $-4/3$ to $+4/3$ changes that mutant discrepancy from $7$ to $13/3$; neither is the maximum, and both are nonzero.

This is arithmetic of printed arrays. The original loop provenance is supplied, not re-established. Moreover, the v4 regression does not contain independently constructed own and frozen arrays for all 31 entries; its passing mutation checks are not an independent replay of the 31-entry scientific comparison.

The recorded ghost mutation changes selected transport terms in ${\mathcal{L}}_{c}h$, and the graviton mutation changes one Christoffel contribution used in $V_3$. They are not common reversals of entire vertex rules.

For amplitudes multilinear in vertex occurrences,

A $V_3\times V_3$ bubble is blind to a common cubic-rule reversal. A two-copy ghost loop is similarly blind. For $B_{ga}=n_{ga}\mathrm{mod}2$, sign vectors in ${\ker }_{{\mathbb{F}}_2}B$ are invisible to these graphwise comparisons.

Proof. Multiply one factor $-1$ for each occurrence. The simultaneous parity condition is $Bs=0$.

Zero amplitudes and cancellations create further blind spots. Odd-occurrence tests, direct differentiated vertices, or independently normalized tree identities supply additional controls. They are not claimed newly executed loop tests.

The v4 typeset table and v4 embedded/extracted regression do not encode the same D01 baseline vector SuppliedV4,V4Artifacts.

Using displayed positions $1,\dots ,14$, they are

The unchanged gauge-table constant vector is

Python index 5 is displayed row 6, namely $B_6=\widehat{a_1\cdot a_2}\widetilde{a_1\cdot a_2}{Q}^2$.

For the arrays in Record rec:phase,

Thus the v4 regression assertion $a^{(0)}=-f^{\mathrm{reg}}$ is false at precisely Python index 5. There is no single scalar $\lambda$ with $a^{(0)}=\lambda f^{\mathrm{reg}}$.

If these arrays are encoded as the corresponding complex reports, the cross-artifact coefficient discrepancy is $4/3$. At common bound $B=32$, its half-diamond distance is $1/24$.

Proof. All entries cancel except the sixth, where $4/3+4/3=8/3$. The first entry forces $\lambda =-1$, which fails at the sixth. Under the convention $v=u/\mathrm{i}$, compare $\mathrm{i}{f}^{\mathrm{reg}}$ with $-\mathrm{i}{a}^{(0)}$; their difference is $(8\mathrm{i}/3)e_6$. Apply Theorem thm:error.

The predecessor's proposed caption change from $+\mathrm{i}{u}^{(0)}$ to $-\mathrm{i}{u}^{(0)}$ follows algebraically from $v=u/\mathrm{i}$, since $1/\mathrm{i}=-\mathrm{i}$. It remains a documentary normalization observation. It cannot make $a^{(0)}=-f^{\mathrm{reg}}$ true.

The typeset and regression values are both retained. Their physical origin cannot be decided from this comparison: it does not identify which underlying loop artifact, if any, contains the correct coefficient. In particular, it is inaccurate to describe the witness as a failure of every entry in the typeset gauge table. The failure is specifically cross-artifact.

Let

For the retained arrays, $P_{\mathrm{consistent}}$ is false and $P_{\mathrm{witness}}$ is true. Passing a regression that enforces $P_{\mathrm{witness}}$ does not establish $P_{\mathrm{consistent}}$.

Proof. The right sides are different because $8/3\ne 0$. Equation (eq:phaseresidual) proves the assertions.

The integrity gate therefore records _RECORDED and _consistent=false. A later reconciliation requires a new versioned source record, the actual original arrays, a justified convention map, and a fresh exact comparison. Merely treating the old failure as expected does not discharge that obligation.

The supplied output identifies Python 3.10.21, SymPy 1.14.0, and a Linux platform, and reports eight passing suites and one failure V4Artifacts. The complete short transcript is retained in Appendix app:v4log.

Its reported source digest is

9f661c8f3eb1dc75b580eb756174f4ff5711f172230ae369b17dbf5a2fbb106.

This is a digest printed by the supplied record, not a digest independently recomputed here from authenticated original bytes. No execution date, signed provenance chain, or separately observed process exit code is supplied.

In the supplied v4 script, the mutation-count, maximum-discrepancy, and common-bound checks precede the phase assertion. The 13-varying-structure and derivative checks follow it. Consequently, in the reported failing invocation:

1. the earlier mutation checks were reached;

2. the phase assertion rejected the data;

3. that invocation did not reach the subsequent variation-count and derivative predicates in the same test function.

The runner continued to the remaining suites.

Proof. The function executes sequentially and raises at (all(x == -y for x,y in zip(a0,f)), ...). There is no local handler resuming inside that function. The outer runner catches the exception and proceeds to the next listed test.

Thus the supplied summary's claim that all other table checks “held” is too broad if read as execution coverage. The 13-of-14 count and derivative formulas are nevertheless proved directly from the printed coefficients in Section sec:gaugetables; they are not lost.

The reported 1397 evaluated predicates include expected rejections and internal checks. That count does not establish that every advertised predicate was reached.

The earlier affine example had a false advertised equation despite attributed PASS. The v4 phase suite instead rejected a false equation and reported aggregate failure. The first is an acceptance-semantics defect; the second is evidence that the relevant fail-closed check worked.

Neither statement establishes every predicate of the earlier software or the correctness of the loop data.

The supplied R07–R09 descriptions report six even cubic classes, four odd classes before a Schouten quotient, and two afterwards SuppliedBot,CheungXVIR. Counts do not provide representatives or response maps.

Let ${\mathcal{P}}_q$ be local polynomial functionals homogeneous of field/antifield degree $q$, with any permitted relations required to preserve that grading. Let $P_2$ be homogeneous projection and $e:{\mathcal{P}}_2\to {\mathbb{K}}^{21}$ a declared quadratic coefficient map.

For ten representatives $u_a\in {\mathcal{P}}_3$, the matrix of $e{P}_2$ is

The original minor is empty with determinant one. The zero map descends through every relation subspace.

In a nonnegative formal $\kappa$-series, an order-${\kappa }^3$ addition cannot change the order-${\kappa }^2$ coefficient under operations that do not lower the filtration.

Proof. Homogeneity gives $P_2{u}_a=0$. The full left kernel and empty minor certify rank zero. A zero map annihilates all relations. For the formal-order assertion, products of nonnegative powers cannot create a term below the order of the addition.

The reported cubic slots $Kch,\bar{c}ch,c^{\ast }cc,hhh$ have three field slots; spectator vectors representing an epsilon tensor are not fields. The predecessor attributes a slot-count check on 821 even coordinates and the odd terms to 1_1c.py; that archive is not newly inspected.

This theorem concerns direct graded re-expansion, not Wick contraction, expansion around a nonzero background, or an unspecified changed identity response. Such operations need their own maps. A formal zero projection does not establish the validity of a strictly four-dimensional Schouten quotient in DRED.

Let $R_U\subseteq U$, let $T:V_{\mathrm{allowed}}\to W_{\mathrm{full}}$ be the allowed counterterm response, and let $F:U\to W_{\mathrm{full}}$ be the complete review response. For a complete left kernel $L_T$ of $T$,

where $\pi :W_{\mathrm{full}}\to W_{\mathrm{full}}/\mathrm{im}T$. The descended obstruction vanishes on every supplied class iff

Proof. Descent requires annihilation of relations after quotienting. Theorem thm:rank gives $\mathrm{im}T=\ker L_T$. Vanishing on all classes requires containment of $F(U)$, not only $F(R_U)$, in that image.

Consistency restrictions belong in $V_{\mathrm{allowed}}$ or in augmented response rows. No projection may discard outside-span components first. Parity mutations test selected software behavior; they do not replace these hypotheses.

The gauge family is a new scope. Take real $\xi \ne 0$, scale the eliminated gauge fixing by ${\xi }^{-1}$, and hold the declared vertices and prescription fixed. An auxiliary/gauge-fermion realization must reproduce this normalization, not merely share its name.

On symmetric Q4S tensors define

For momentum $k$, put

Stars denote algebraic adjoints for the stated pairings, not positive Hilbert-space adjoints. Then

For $k^2\ne 0$,

Also

where

Proof. Since $J^2=4J$, ${\mathsf{A}}^2=I$. For a covector $v$,

This gives $\mathsf{FR}=k^{2}I/2$. Furthermore, ${\mathsf{R}}^{\ast }{\mathsf{F}}^{\ast }={\mathsf{R}}^{\ast }\mathsf{AR}=\mathsf{FR}$, which proves the projector identity. Expanding $\mathsf{R}{\mathsf{R}}^{\ast }$ gives the four terms.

For

the rational inverse at $k^2\ne 0,\xi \ne 0$ is

Proof. Write ${\mathsf{K}}_1=k^2\mathsf{A}$, ${\mathsf{P}}_1=\mathsf{A}/k^2$, and $a_{\xi }={\xi }^{-1}-1$. Then

Multiplying by ${\mathsf{P}}_1$ gives

This is a general algebraic proof in the declared trace-four algebra, not an interpolation from finite momenta. The DRED contraction prescription and overall Feynman factor remain scientific inputs. On $k^2=0$, a prescribed boundary value is required; this is not pointwise inversion there. At $\xi =0$, the eliminated kinetic expression is undefined. The predecessor's action-Hessian check is attributed to 1_kincheck.py, not newly executed.

Put $\zeta =\xi -1$. With unchanged vertices, a graph with $n_h$ graviton lines has degree at most $n_h$ in $\zeta$ under linear pole extraction.

The predecessor attributes the following to its gauge-family scripts SuppliedBot:

1. D03 and D11 have no graviton line and are $\xi$-independent;

2. D04 has zero UV-pole polynomial in the stated family;

3. D06 and D13 are affine, and D01 is quadratic;

4. $D06=-2D13[K\to T(\bar{c})]$ holds coefficientwise in $\zeta$, in the incoming-antighost convention.

These remain supplied scientific records. The D01 base-point identification across all v4 artifacts is subject to Section sec:integrity.

Here

Momentum orientation and the separately disclosed D13 slot orientation must be transported consistently CheungXVIQ.

In own-rule normalization $u=\mathrm{i}v$, D06 is

For $K=a_K\otimes a_K$, set

On $(sQt,s^2{u}_p,rsQ,A_K{u}_{p}Q)$, D13 has coefficients

The fourth structure first appears at order $\zeta$.

For D01 write

All printed coefficients are retained:

Exactly 13 of these 14 structures vary with $\xi$; $B_{14}$ is constant. The printed derivative is

These facts do not require correcting either version of $f$.

Proof. Only row 14 has both variable coefficients zero. Differentiate the finite polynomial.

The cross-artifact witness concerns the displayed constant coefficient. It does not alter the formal differentiation of the unchanged $a^{(1)},a^{(2)}$ arrays. This observation is not a new verification of their loop origin.

Massless one-point and tadpole zeros are retained under the supplied prescription. Scaleless dimensional integrals can vanish by UV/IR cancellation; the integral with integrand $1/(l^2{)}^2$ illustrates the logarithmic issue. No assertion that every scaleless tadpole has no logarithmic UV or IR contribution is used. Insertion problems and infrared rearrangements need their own separation SmirnovChetyrkin.

D04's supplied zero is not a theorem about every gauge-invariant scalar correlation function. Such observables require additional contributions and gauge-dependent cancellations Frob2018.

Before differentiating, identify the gauge descriptions and their basis transports. Declare a finite complex

The primitive space has ghost number $-1$; it is not the 21-direction ghost-number-zero counterterm space.

For a differentiable solution $A_{\xi }c(\xi )+d_{\xi }=0$ in the fixed identified presentation,

At fixed $\xi$, put $w_{\xi }=-d_{\xi }^{\prime }-A_{\xi }^{\prime }c(\xi )$. An exact tangent $z=Su$ satisfying differentiated cancellation exists iff

Every such tangent is $B_0$-closed.

Proof. Apply Theorem thm:rank to $S$. Differentiate cancellation and substitute $c^{\prime }=Su$. Finally $B_{0}Su=0$.

If the identified differential varies, use $S_{\xi }$ and state its transport. Parameter-dependent consistency equations must also be differentiated. A pointwise tangent is not an interval theorem without smooth choices and exceptional-parameter control.

Free BRST, relative classical BV, free-equation descent, and interacting quantum BRST remain different objects CheungXGST,CheungXHST. No X-G-ST or X-H-ST normalization is substituted for the P/Q action normalization.

The predecessor reports testing the $hh$ pole derivative against the image of

in its 13-generator dimension-two $X$-presentation. For the relaxed map allowing independent $X,X^{\prime }$ in

it reports on a 15-structure $hh$ space

Fixed-normalization failures, including the listed $\xi =1,2,\frac{1}{2}$ tests, are also reported. The four-dimensional projection is reported to have rank five on six structures and to accept both the derivative and the undifferentiated $\xi =1$ pole SuppliedBot.

The full generator list, ambient basis, matrices, original minors, and left-kernel rows are absent from the supplied artifacts. The relaxed rank report is not extended to additional parameters and is not newly re-certified.

Define the named source-relative obstruction

If the reported ranks hold for those actual arrays, this class is nonzero at $\xi =1,2$, because adjoining the target raises rank. That excludes only the tested class. The four-dimensional test is nondiscriminating in the reported comparison, not invalid for every purpose.

A complete rank-seven image certificate on 15 rows needs an original $7\times 7$ minor and an $8\times 15$ complete left-kernel matrix, with a nonzero target pairing.

Let $H:Y_{\mathrm{old}}\to W$ have a complete left kernel $L$, let $E:Y_{\mathrm{new}}\to W$ be a specified additional primitive response, and let $b\in W$. Then

Let $Q_E$ be a complete left kernel of $LE$, and set

Then

If $D_{E}b\ne 0$, a row $\ell$ with $\ell b\ne 0$ satisfies