Landmark XVI-N Classical completion PreparedClassicalBVInterfaceThroughOrderKappa Science volume only (revision r12)

Phase I constructs the unprepared classical BV complex and its prepared and switch deformations through order $\kappa$, with reconciled odd-field antibracket couplings, the antifield rule $s{{\Phi }^{\star }}_A=(S,{{\Phi }^{\star }}_A)$, and the invariant baseline $S_{\mathrm{inv}}=S_D-V_{\sigma }-V_g$. Gate N-Ins uses the actual predecessor kernel $t_D$ of XVI-I, XVI-H, and XII-D as inherited by XVI-K. The plain divergence $D_{\mathrm{div}}$ is separated from the gauge-fixing divergence $D_{\mathrm{gf}}=D_{\mathrm{div}}\mathcal{P}$. The corrected root current is

not $E^A(D_{\mathrm{div}}\mathcal{P}\alpha )$. All values computed for the superseded vector are withdrawn.

For one massless special family, admissible smooth data give $H{|}_U=2m_0(e_0\otimes e_1+e_1\otimes e_0)$ and $C{|}_U=4m_0{p}^{\prime }(t-z)\mathrm{d}{x}^2$, with $m_0=\int S>0$, and hence $s_0{F}_t\ne 0$. A validated enclosure is $s_0{F}_t\in [3.3566,3.3637]\times 10^{-4}$. That nonvanishing is conditional on inherited reduction identities, re-checked pointwise and not re-proved, and on configuration-space admissibility. Lemma 8 is discharged. Route A is sufficient. Route B is not claimed discharged. Theorem 11 is a conditional obstruction of one regular, seed-preserving, antifield-linear completion of that one massless instance. It is not a universal completion. The certified small-mass interval is $0\le m<1.3715\times 10^{-4}$, with $\mathrm{Chi}\le 17843$. Arbitrary prescribed $m>0$ remains unresolved. N-Ins.4 remains unresolved as a universal antifield-linear completion. The prepared theory is not a BV complex. No ultraviolet completion, no loops, no graph closure, and no quantum coefficient are claimed. Numerical checks are corroboration, not certificates. No compilation is asserted. No symbolic execution is claimed.

Figure fig:scope-XVI-N, Figure fig:split-XVI-N, Figure fig:chain-XVI-N, Figure fig:pair-XVI-N, and Figure fig:gate-XVI-N name the stop. Not a proof, and not an admission of a quantum coefficient.

Scope of this revision. The massless obstruction is conditional on Route A. Route B is not claimed discharged. Arbitrary prescribed $m>0$ remains unresolved. The certified interval $0\le m<1.3715\times 10^{-4}$ is not that arbitrary-mass statement. N-Ins.4 stays unresolved as a universal completion. The prepared theory is not a BV complex. XV-B is not opened. The stops at H2, I3, and J2 remain unresolved. This sheet does not repair X-G or XI-G. No quantum coefficient is admitted.

A broken master identity is not a BV complex. The three objects kept separate are: the unprepared invariant action $S_{\mathrm{inv}}$ with nilpotent $Q_{\mathrm{inv}}$; the nilpotent diffeomorphism BRST operator $s=s_0+\kappa s_1$; and the prepared action $S_D=S_{\mathrm{inv}}+V_{\sigma }+V_g$, whose square is broken. The convention follows XIV-A: one may write $(S,S)=\kappa \int c^{\nu }{B}_{\nu }+\cdots$, and

Following that convention does not repair the square.

For the frozen generators,

Then $(S,{\Phi }^A)=-{\delta }^{R}S/\delta {{\Phi }^{\star }}_A$. The blanket identity $(S,\Phi )={\delta }^{L}S/\delta {\Phi }^{\star }$ is false for odd $\Phi$. The antifield rule is $s{{\Phi }^{\star }}_A=(S,{{\Phi }^{\star }}_A)$, not the reverse bracket.

Conventions: $\eta =\mathrm{diag}(1,-1,-1,-1)$, $q=\eta +\kappa h$, $a=\frac{1}{2}$, and $\mathcal{P}$ is trace reversal. Parentheses symmetrization includes the factor $\frac{1}{2}$. The predecessor gauge map is written ${R^{\mathrm{pred}}}_0$, and $R_0(c)={R^{\mathrm{pred}}}_0(ac)$. The letter $K$ denotes only the kinetic operator $-\square \mathcal{P}$. It is never the gauge map.

Write ${S^{[1]}}_{\mathrm{inv}}={S^{[2]}}_{\mathrm{EH}}+{S^{[3]}}_{\mathrm{EH}}+{S^{\mathrm{sc}}}_{\mathrm{inv}}+S_{\mathrm{af}}$, $V_{\sigma }=\kappa \int ({\sigma }_{\epsilon }-1){\ell }_3[h]$, and $V_g$ for the switched-scalar deformation. Then ${S^{[1]}}_D={S^{[1]}}_{\mathrm{inv}}+V_{\sigma }+V_g$. Equivalently,

Setting ${\sigma }_{\epsilon }\equiv 1$ removes only $V_{\sigma }$. The switched theory $S_D-V_{\sigma }=S_{\mathrm{inv}}+V_g$ still carries the breaking $B_g$. It is not $S_{\mathrm{inv}}$.

The cubic density and Euler source are quoted from XII-C, not re-proved: ${\ell }_3=-2(r_3+b_1{r}_2+b_2{r}_1)$ and ${\mathrm{Euler}}_h({\ell }_3)=J_3$. The operators $A=2G_{\mathrm{lin}}$, $Q_{\nu }$, $N_{0,\nu }$, and the ramp defect are likewise quoted. Gate 4 records an exact-arithmetic sampling of unrestricted retained jets. That sampling is corroboration. It is not a universal-identity certificate.

In the linear-antifield truncation,

Here $B_{\nu }=N_{0,\nu }(Z)=-w{Q}_{\nu }(Ah)+{\mathfrak{R}}_{{\sigma }_{\epsilon },\nu }$ is the metric-ramp piece, and $B_{g,\nu }$ is the switched-scalar breaking retained in the prepared square. The scalar variations cancel between the metric and scalar sectors. The value ${\sigma }_{\epsilon }=1$ removes $V_{\sigma }$ but not $B_g$.

Degree zero in antifields is the constructed action variation. Degree one vanishes by the inherited nilpotence of $s_0$ on generators. Degree at least two is absent by the truncation ${S^{\prime }}_{\ge 2}=0$. The graph-to-action vertex identification stays unsettled. Free action, gauge fixing, $K=-\square \mathcal{P}$, and the propagator interface are normalization inputs under N-Normfree, which is constructed. They are not a graph closure.

${Q_D}^{2}F=(B_{\mathrm{tot}},F)$ with $B_{\mathrm{tot}}=\frac{1}{2}(S_D,S_D)$. At order $\kappa$, $B_{\mathrm{tot}}$ is determined by (eq:master-square). The prepared theory is not a BV complex.

Gate N-Ins is the classical insertion interface through order $\kappa$. The seed is $F_t=⟨t_D,h\otimes h\otimes \varphi ⟩$. The option that sets $\varphi =0$ is rejected, because that specialization kills $F_t$. The completion space is the minimal sector $(h,\varphi ,c;h^{\star },{\varphi }^{\star },c^{\star })$, with

seed $I_t{|}_{{\Phi }^{\star }=0}=F_t$, and no new inverse operators.

The primary constructive definition of $t_D$ is quoted from the predecessors and is not redefined. In the companion transcription the relevant XVI-I labels are eq:t-test and prop:Ft; the science volume's name prop:test is that predecessor statement, not a new label. XVI-H thm:trep is the equivalent polarized form. XII-D thm:root and lem:root-kernel supply the compact root. XVI-K retains the construction data and does not supply a new closed-form kernel. Component tensors of the cubic source remain implicit. They are not a single numerically expanded kernel. Sha256 strings printed beside those files in the science volume are provenance pointers. They are not a byte-identity claim for the copies registered here, and they are not a new premise.

On the invariant free shell $Ah=0$ and $(\square +m^2)\varphi =0$, with compact ghost supported in the plateau where ${\sigma }_{\epsilon }=1$, the science volume records the reduction

The cubic-variation step cites XI-E thm:cubicvar and is re-checked pointwise. It is not re-proved. The nilpotence step cites XI-B lem:nilpotence as a written derivation. Neither citation is a declared input of a new theorem.

Two divergence operators are kept distinct. $D_{\mathrm{div}}T={\partial }_{\mu }{T}^{\mu \nu }$ and $D_{\mathrm{gf}}r={\partial }_{\mu }(\mathcal{P}r{)}^{\mu \nu }=D_{\mathrm{div}}\mathcal{P}r$, with formal adjoints ${D^{\star }}_{\mathrm{div}}v=-{\partial }_{(\mu }{v}_{\nu )}$ and ${D^{\star }}_{\mathrm{gf}}=\mathcal{P}{D^{\star }}_{\mathrm{div}}$. From the displayed linearized Einstein operator,

For $a_{\mathrm{root}}=-E^A\mathcal{P}\alpha$,

The superseded vector $E^A(D_{\mathrm{div}}\mathcal{P}\alpha )$ is wrong for traceful $\alpha$. Every r7 value computed for that vector is withdrawn. Since $D_{\mathrm{div}}A=0$, one has $D_{\mathrm{div}}H=0$.

For symmetric $H$, the root current of the reduction is

and $s_0{F}_t=⟨C,\xi ⟩$ for $\xi$ supported in the plateau. On $U$, equivalently, $s_0{F}_t=-2⟨w,D_{\mathrm{gf}}({\mathcal{L}}_{\xi }h)⟩$.

Let $T=4$, $\epsilon =\frac{1}{2}$, and $U=(0.9,1.1)\times (-0.1,0.1{)}^3$. Let $S\in {C^{\infty }}_c(\mathbb{R};[0,1])$ with $\mathrm{supp}S=[2,3]$, not claimed inside $(2,3)$, and let the cutoff profile of the science volume produce $g=\sigma =1$ on a neighbourhood of $U$, with $U\subset \mathrm{Int}\{g=1\}\cap \{\epsilon <t<T\}$. For the massless special family with $\varphi =x^1$ and $v_f=\psi$,

globally on the slab, and ${\mathrm{supp}}_t(D_{\mathrm{div}}\alpha )\subset [2,3]$. On the open set where the cutoffs equal one,

Every advanced cone from a point of $U$ meets the source slab only inside that plateau set. The future cones from $U$ therefore see exactly that model source. The collar of the compact root equals one on the cones that are used, and the independent switch of XVI-J and XVI-K does not enter $t_D$. The local formulas on $U$,

follow from the actual compact construction by four-dimensional Huygens support of the advanced kernel, not from a polynomial surrogate. A numeric comparison of the actual source with the model source is corroboration only.

The free data of the witness are $\varphi =x^1$ and the transverse-traceless wave $h_{12}=h_{21}=p(t-z)$, admitted by Route A. Route B is not claimed discharged.

On $U$, for the massless special family of Lemma lem:support,

Hence, for every $\xi \in {C^{\infty }}_c(U;T{\mathbb{R}}^4)$,

The integral is strictly positive whenever $p^{\prime }>0$ on $\mathrm{supp}\xi$ and $0\le {\xi }^2\not\equiv 0$. In particular $C{|}_U\ne 0$. The moment $m_0$ is positive because $S\ge 0$ and $S\not\equiv 0$.

The identity is conditional on the inherited formulas XVI-I records for $s_0{F}_t$ and on XI-E thm:cubicvar, the latter re-checked pointwise and not re-proved. Scope: one special family, and only the instance $m=0$.

For the explicit compact ghost of the science volume, with width $d=0.09$ and $p(u)=u$, monotone interval Riemann sums give

and $0$ is not in that interval. Support and regularity are exact by construction. The quadrature error is enclosed. The enclosure is corroboration of the analytic sign. It is not a substitute for Proposition prop:nonvanishing, and it is not a certificate that every kernel is obstructed.

A one-plus-one planar auxiliary model has an exact nonzero variation after the divergence correction. That model establishes no frozen four-dimensional completion obstruction. Planar data are not compact in four dimensions, and the one-plus-one Green function is not the four-dimensional advanced kernel. The toy witness and the inadmissible value of $g-1$ on ghost support are withdrawn.

Route A quotes the frozen contract: fields are ambient smooth configurations, no spatial decay is required, and metric Cauchy data are unrestricted. The displayed free solutions $\varphi =x^1$ and $h_{12}=p(t-z)$ are therefore admitted. Route A is discharged and is sufficient for Theorem thm:massless-obstruction.

Route B, spatially compact Cauchy data, is not claimed discharged. A previous draft used domain-of-dependence cutoffs relative to $\mathrm{supp}(g-\sigma )$. For $t>\epsilon$ one has $\sigma =1$, while $g=0$ outside a compact spatial support, so $g-\sigma =-1$ on an unbounded spatial region. Cauchy shadows of that set are not spatially compact, and the draft is invalid as written. A repair that replaces $\mathrm{supp}(g-\sigma )$ by an explicitly compact region actually entering the seed is not carried out. The algebraic transverse-traceless gadget remains a constraint-compatible generator. It does not by itself discharge Route B.

Let $A_0=\int (h^{\star \mu \nu }{U}_{\mu \nu }+{\varphi }^{\star }V+{c^{\star }}_{\mu }{W}^{\mu })$ be antifield-linear of ghost numbers $(1,1,2)$, with regular multilocal coefficients and no new inverse operators. Let $P_0$ project to antifield number zero and let $Q_0=(S_0,\cdot )$ be the free differential of the minimal free extended action. Then

and therefore

A graded lattice check is corroboration of the algebra. It is not a substitute for the continuum statement.

Assume the inherited seed, the reduction identities as re-checked and not re-proved, Route A, the plateau and root-cutoff requirements, Lemma lem:support, and Lemma lem:field-only. For the specified massless special-family instance of Proposition prop:nonvanishing, no regular, seed-preserving, antifield-linear completion $I_t=F_t+A_0+\kappa A_1$ satisfies $Q_{\mathrm{inv}}{I}_t=O({\kappa }^2)$.

Proof. On the witness, with $p^{\prime }>0$ on $\mathrm{supp}\xi$ and $0\le {\xi }^2\not\equiv 0$, equation (eq:s0Ft) gives $s_0{F}_t>0$. Closure at field-only degree and order ${\kappa }^0$ requires $s_0{F}_t+P_0{Q}_0{A}_0=0$. Lemma lem:field-only gives $P_0{Q}_0{A}_0=0$ on the free shell, so $s_0{F}_t=0$, a contradiction. The higher term $A_1$ cannot repair a contradiction already present at order ${\kappa }^0$.

The theorem covers one constructed instance. It does not obstruct every admissible kernel, and it does not obstruct a fixed $m>0$ without further work. It does rule out a universal completion prescription that would have to cover this instance. N-Ins.4 therefore remains unresolved as a universal statement. The obstruction is not an ultraviolet subtraction, not a loop identity, and not a graph closure. It does not open XV-B. It is not the physical characteristic limit through the aligned-null locus, and it is not H2, I3, or J2.

Keep the metric, ghost, cutoff, and test region frozen, and vary only the scalar mass through $f_m=(\square +m^2)\psi$ and ${\varphi }_m=x^1\cos (m(t-5/2))$, with advanced solution $v_{f_m}=\psi$. Do not replace every advanced operator by its massive counterpart. The tensor root and the coordinate completion remain massless. Only the scalar leg and the explicit $m^2$ term in the scalar source carry the mass.

Write $W(m)=-⟨a_m,A({\mathcal{L}}_{\xi }h)⟩$. The difference $W(m)-W(0)$ splits into a local channel and a nonlocal double-advanced channel. On the cone window of Lemma lem:support the rational factors are $K_1=1113/500$ and $K_2=11197151/10000000$. They are not free parameters. Inclusion-preserving interval arithmetic produces the release constants

with $\mathrm{Chi}\le 17843$. Exact rational arithmetic yields $C_{\mathrm{release}}{m_{\mathrm{release}}}^2<W_{\mathrm{release}}$, and the release window sits under the certified massless lower endpoint. Hence

The bound is conditional on the same inherited reduction identities as $W(0)$. It is an obstruction on a proved mass interval. It is not a theorem for every prescribed $m>0$.

The frozen contract fixes $f$, $g$, ${\sigma }_{\epsilon }$, and $U$ for each instance. It does not grant a free length $L$ that may be varied after the instance is chosen. Rescaling would change the supports and the relative geometry of the dip and the plateau. The localized scaling route is unavailable as an automatic extension. Arbitrary prescribed $m>0$ remains outside (eq:small-mass). Route B remains unresolved.

N-Ins.5 is structural and contingent on invariant completion:

The second source order is excluded. The identity is not itself the missing invariant completion.

N-A is constructed, including odd signs, the antifield prose, $S_{\mathrm{inv}}$, and the separation of $D_{\mathrm{div}}$ from $D_{\mathrm{gf}}$. N-Ins.1, N-Ins.2, and N-Ins.5 are constructed, the last only contingently. N-Ins.3 is analytic four-dimensional nonvanishing for the massless special family, conditional on Lemma lem:support, Route A, Proposition prop:nonvanishing, and Theorem thm:massless-obstruction. N-Ins.4 is unresolved as a universal antifield-linear completion. N-Normfree is constructed. N-B through N-E are not claimed. Unresolved normalization items of XVI-M beyond N-Normfree are untouched.

No ultraviolet completion is claimed. No loops, no graph closure, and no singular extensions are claimed. The toy kernel and the inadmissible value of $g-1$ on ghost support are withdrawn. On shell, the analytic nonvanishing is for one massless special family and is conditional on inherited reduction identities. Bump membership is $\mathrm{supp}S=[2,3]$ with $S\in {C^{\infty }}_c(\mathbb{R})$. The one-plus-one calculation is a model variation, not an obstruction. Values for the superseded vector are withdrawn. Numerical evidence is corroboration, not a certificate. Sampling is not a universal-identity certificate. The graph-to-action vertex identification is unsettled. Theorem thm:massless-obstruction covers one constructed instance, not every kernel and not a fixed $m>0$. The prepared theory is not a BV complex. XV-B is not opened. No quantum coefficient is admitted. No compilation is asserted. No symbolic execution is claimed.

The science volume is the registrable manuscript. The companion appendix of listings, note-source archives, and provenance dumps is not this text and is not searched as a premise. The public package records a regression taxonomy that separates mathematical checks from packaging checks. This companion does not run that regression, does not execute a pipeline script, and does not treat an exit code, a log, or a hash as a theorem. A numerical enclosure already stated in the science volume is cited as that stated enclosure. It is not recomputed here, and it is not promoted to a quantum coefficient, an ultraviolet completion, a physical characteristic limit, an opened XV-B, or a derivation of a metric or of Einstein dynamics.