Landmark XVI-O A finite quantum BV sector for invariant Einstein–scalar gravity, and the obstruction-preserving audit of the inherited insertion Science volume (o70-sync: four D11 deliverables)

Revision o70-sync is a proof and build consistency correction so that the kernel counterterm evidence earns acceptance under a stated prescription. The intermediate wording is retained: the D11 Wick and determinant normalization is established; the tensor product, the scaling witness, and the off-diagonal Ward verification were tracked separately and are now constructed for the kernel; Gate 3 spanning stays Partial; bulk O.1 stays NOT_CLOSED.

Three verdicts are never collapsed. (1) Bulk construction is partial: the classical ${\kappa }^2$ master is Partial, four D11 prerequisites are Constructed for the kernel, the full ${\ell }_4$ polynomial is Unresolved, and O.1 is NOT_CLOSED. (2) Symmetry and counterterms: the subsector is restored under the prescription (kernel_two_point_Ward); the bulk matched set is empty; bulk symmetry_restored stays false. (3) The inherited insertion stays frozen by XVI-N, thm:massless-obstruction, and is distinct from $S_{\mathrm{inv}}$ and from the prepared defect $(S_D,S_D)$.

Coefficients $\alpha =A^2/60$ and $\beta =A^2/18$ are derived. All $256$ symbolic components are a primary check; invariant structures are documented and are not an independent claim. On the open set $U$, ${{\Pi }_{\mathrm{tr}}}^R{|}_U=(i{\kappa }^2/4)T$. The only active reference is W_w_jet_pullback_via_Pi_tr_o68. The kernel two-point Ward identity is not functional ST4b. $s_0{C}_L$ is not evaluated. The subsector verdict is Constructed_subsector_under_prescription for the kernel only. En01 and Em01 are not premises. Corpus $\mathcal{P}$ is not reused for $P_{\mathrm{tr}}$. No quantum coefficient is admitted. No compilation is asserted.

Landmark XVI-O connects XVI-M selected-graph renormalization to the XVI-N classical framework without treating the obstructed insertion as a completed quantum observable. En01 and Em01 are not premises. Predecessors XVI-A through XVI-N are read-only pins. The baseline is $S_{\mathrm{inv}}$, not $S_D$.

The three release verdicts are never collapsed.

1. Bulk quantum-sector construction. The finite target lives on $S_{\mathrm{inv}}$. The classical ${\kappa }^2$ master is Partial. Four D11 prerequisites are Constructed for the kernel. The full ${\ell }_4$ generating polynomial is Unresolved. O.1 is NOT_CLOSED.

2. Renormalized symmetry and counterterm compatibility. The subsector is restored under the prescription kernel_two_point_Ward. The bulk matched set is empty. Bulk symmetry_restored stays false. ST4a and functional ST4b are not claimed.

3. Inherited-insertion status. The classical obstruction of the declared class is preserved. A renormalized product is not closure.

The diagram inventory is manually specified and partially checked. It is not a completeness claim. Numerical checks are corroboration, not certificates. Sampling is not a universal identity.

The quantum sector is built on the invariant baseline

not on the prepared action $S_D$. A broken master identity is not a BV complex. The prepared square $(S_D,S_D)$ is a different object from $S_{\mathrm{inv}}$ and from the insertion obstruction.

XVI-N non-claim 12 leaves graph-to-action vertex identification unsettled. Sharing the cubic density ${\ell }_3$ does not identify XVI-M's selected graph with an action graph. Any future map needs its own certificate. Selected-graph coefficients are comparison data unless an explicit morphism into this sector is constructed.

Inherited here: the XVI-N classical interface through order $\kappa$; XVI-M Target I comparison data; the XVI-I seed $F_t$; the XII-C and XII-D kernels as named sources, not as new theorems; and the XIV-A bracket orientation. Written here: the provisional quantum target, the convention sheet, the provisional diagram inventory, the classical ${\kappa }^2$ scaffold, and the insertion freeze. Checked as package evidence, not as certificates of this note: pin hashes, and normalization tests that forbid silent ${\ell }_3$ identification and invalid ghost-number $-1$ source assignments. No numerical value in the archived scripts is a certificate.

Foundation controls retained from the o61 calculation: the inverse-metric series satisfies $q{q^{-1}}_{\mathrm{series}}=1+O({\epsilon }^5)$, and a negative diagonal $h_{11}=B$ gives $q^{11}=-1-\epsilon B-{\epsilon }^2{B}^2-\cdots$. The quartic density is ${\ell }_4=-2[{\epsilon }^4](\sqrt{-q}R)$, and $S_{\mathrm{EH}}=-2{\kappa }^{-2}\int \sqrt{-q}R$. Sparse ${\ell }_4$ probes are not the proof. No $s_2$ is invented.

The authoritative scalar action is the single source XI-A, quoted and not re-proved:

At $m=0$ the free Hessian is $-\square$, not the bare $\square$. The cubic and quartic vertices $V_3=(\kappa /2)a_1\partial \varphi \partial \varphi$ and $V_4=({\kappa }^2/2)a_2\partial \varphi \partial \varphi$ are derived from the same ${\mathcal{L}}_q$. The old catalogue density $-\sqrt{-q}{q}^{\mu \nu }\partial \varphi \partial \varphi$ is withdrawn. The antifield split keeps ${V^{(0)}}_{\mathrm{af},h}=h^{\ast }{R}_0(c)$ distinct from ${V^{(1)}}_{\mathrm{af},h}=\kappa a{h}^{\ast }{\mathcal{L}}_{c}h$, with ghost normalization $a=\frac{1}{2}$ as in XVI-N. That $a$ is not the D11 coefficient pair $(\alpha ,\beta )$.

Let $S=S_0+\kappa S_1+{\kappa }^2{S}_2$. Under compact variations and total-derivative equivalence, the retained classical master through order ${\kappa }^2$ is

Indexed evaluation gives $(S_0,S_0)$ Constructed, $(S_0,S_1)$ Constructed by inheritance of XVI-N through order $\kappa$, and $2(S_0,S_2)+(S_1,S_1)$ Partial: the scalar sector is Constructed by finite-$\kappa$ invariance of $\int {\mathcal{L}}_q$ plus Taylor expansion, while the gravitational component stays Partial. The full ten-component ${\ell }_4$ generating polynomial, the explicit gravitational expansion at order ${\kappa }^2$, and the complete ghost and antifield order ${\kappa }^2$ on $D[h]$ are Unresolved.

Gate 0 freezes the following conventions. The metric is $q_{\mu \nu }={\eta }_{\mu \nu }+\kappa h_{\mu \nu }$, with $\eta =\mathrm{diag}(+---)$. XI-A writes $\gamma$; the map is $h\leftrightarrow \gamma$. The massless scalar action is

and $B^{\mu \nu }=\sqrt{-q}{q}^{\mu \nu }=\eta +\kappa a_1+{\kappa }^2{a}_2$ with the XI-A coefficients $a_1,a_2$. The path integral is $e^{iS/\hslash }$, with Fourier convention ${\partial }_{\mu }\to i{k}_{\mu }$ and $\square \to -k^2$. The Green function and the Feynman propagator are separate. $G_F$ is a Hessian inverse. XIV-A uses $G^R=-P{E}^R$ with coefficient $-1$. XI-B is not a Hadamard statement of this kernel. After stripping $\hslash$, the Wick propagator is $D_F(k)=i/(k^2+i0)$, and $(-\square )D_F=i\delta$. The conversion ${\Delta }_F=-D_F$ is written once. The effective action is $\Gamma =S+\hslash {\Gamma }_1+\cdots$, with ${\Gamma }_{1,\mathrm{matter}}=-i\log Z$. The factor of $i$ inside $D_F$ is not counted again in $-i\log Z$. The scope is the massless scalar matter loop on a background metric. It is not complete bulk BV, not $F_t$, and not an insertion completion. En01 and Em01 are not premises. Track II stays frozen. O.1 stays NOT_CLOSED. Bulk symmetry_restored stays false. Every later factor follows this sheet. No convention is changed midway.

The subsector verdict is Constructed under the stated prescription, for the kernel only. Gate 3 spanning stays Partial because the covariance spanning set is still a candidate list. D11_kernel_counterterm is earned only after the four mathematical prerequisites pass. A development mark of Conditional is not a second verdict.

Gate 1 is the mixed scalar Noether identity, not an autonomous master equation $(S_{\mathrm{sc}},S_{\mathrm{sc}})=0$. With coupling-stripped densities $S_{\varphi ,i}=\frac{1}{2}\int {a_i}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi$ and $a_0=\eta$, ghost normalization $a=\frac{1}{2}$ distinct from $\alpha ,\beta$, and $sq=\kappa a{\mathcal{L}}_{c}q$, the densitized identities

integrate, under compact support, to

Three objects stay separate: the invariant baseline $S_{\mathrm{inv}}$; the prepared-action defect $(S_D,S_D)$; and the insertion-class obstruction of XVI-N, thm:massless-obstruction.

Gate 2 normalizes the D11 prefactor. Route A, the Wick route, independently yields

with ${\Gamma }_{1,\mathrm{matter}}=-i\log Z$ and ${\Pi }_{\mathrm{bubble}}=(i{\kappa }^2/4)VV(CC+CC)$. Route B expands $\log Z=-\frac{1}{2}\mathrm{Tr}\log H$ to $(-{\kappa }^2/4)\mathrm{Tr}(D_F{H}_1{D}_F{H}_1)$. Symmetry of $a_1$ gives ${\mathrm{comb}}_B=\frac{1}{2}$, which is Constructed and not unresolved, and

The tensor witness is $T_{0000}=24A^2/r^8$ at $z=(0,r,0,0)$. That witness is corroboration of the stated point, not a universal identity by sampling.

One analytic finite-part scheme is frozen:

Frozen data: $\eta =\mathrm{diag}(+---)$; the boundary value $Q-i0$; the principal logarithm; $\mu >0$ as an inverse length; the finite part as the Laurent constant term; and no additional finite contact $c\delta$. The proof identifier is lem:analytic_FP_u_mu_R_o70. It is the same finite-part prescription as the o68 and o69 identifiers, with the constructive holomorphy argument added. It is not momentum-space minimal subtraction, and it is not XVI-J's value-jet extension of $d(z{)}^2$.

The family $u_{\zeta }$ admits a distributional meromorphic continuation near $\zeta =0$ against compact smooth tests, with a simple pole. On tests supported in the open set $U$, the pairing at $\zeta =0$ equals $⟨(Q-i0{)}^{-2},f⟩$. The pole part is local,

and the residue, obtained only after the holomorphy premise below, is

The extension ${u_{\mu }}^R$ is Lorentz covariant and even under $z\mapsto -z$. Its scaling degree satisfies $\mathrm{sd}({u_{\mu }}^R)\le 4$. Residual contact freedom is frozen by excluding any added $c\delta$. XVI-J, prop:extension, is freedom only and is not this contact term.

Proof. Do not argue from the zero of $4\zeta (\zeta -1)$ alone, and do not argue from the Wick Jacobian alone. Let $F(\zeta )=(Q-i0{)}^{\zeta }$. This is a holomorphic family of distributions near $\zeta =0$ by instantiating Dütsch–Fredenhagen–Keller–Rejzner, arXiv:1311.5424, Definition 4.2 and Sections 4.1–4.2, with hypotheses $Q-i0$, the principal logarithm, tests in $\mathcal{D}({\mathbb{R}}^4)$, and the Gate 0 normalization. Classical support is the Gelfand–Shilov and Hörmander fact that complex powers $(x\pm i0{)}^{\lambda }$ are entire in $\lambda$. Constant-coefficient differentiation under the pairing gives $\square {\varphi }_{\lambda }=4\lambda (\lambda +1){\varphi }_{\lambda -1}$, hence $\square F(\zeta )=4\zeta (\zeta +1){\varphi }_{-1+\zeta }$. Since $F(0)=1$, one has $\square F(0)=0$, so holomorphy of $F$ yields $\square F(\zeta )=\zeta G(\zeta )$ with $G$ holomorphic in ${\mathcal{D}}^{\prime }$. Therefore

is holomorphic near $\zeta =0$. Only after that premise, Gate 0 gives $D_F=-A(Q-i0{)}^{-1}$ with $A=1/(4{\pi }^2)$ and $(-\square )D_F=i\delta$, hence $\square (Q-i0{)}^{-1}=4{\pi }^{2}i\delta$. The second recurrence $\square {\varphi }_{-1+\zeta }=4\zeta (-1+\zeta ){\varphi }_{-2+\zeta }$ then produces the residue $r=-i{\pi }^2$. The scale ${\mu }^{2\zeta }$ affects only the finite part. The operational Laurent identity is

Near a nonzero null point, ordinary absolute integrability of $(Q-i0{)}^{-2+\zeta }$ requires $\mathrm{Re}\zeta >1$. The pairing at $\zeta =0$ on $U$ is analytic continuation, not an ordinary integral. A Euclidean radial integral with $|S^3|=2{\pi }^2$ is an independent sign check, not the justification.

On the open set $U$, the generated tensor is $T=\mathcal{K}u$ in ${\mathcal{D}}^{\prime }(U)$, with

The coefficients are derived from the generated $T$. All $256$ symbolic components are the primary check. Invariant structures are documented and are not an independent basis claim. The operators $P_{\mathrm{tr}}$ are not the corpus trace-reversal operator $\mathcal{P}$. Then ${T^R}_{\mathrm{tr}}=\mathcal{K}{u_{\mu }}^R$ and

with $\mathrm{sd}(\mathcal{K}{u_{\mu }}^R)\le 8$. A missing scaling degree stays Unresolved and is never defaulted to $8$. Wavefront existence is not equality of distributions.

The only active reference for the weighted kernel is the jet pullback

prescription W_w_jet_pullback_via_Pi_tr_o68. The old cutoff construction is archived. Equivalence of that cutoff with this pullback is not claimed. The false pointwise bound $|T|\le C|z{|}_E^{-8}$ is removed. Annular seminorms replace it. The covariance spanning set remains a candidate list, so Gate 3 stays Partial even though Deliverables 1–3 are Constructed under this reference.

Contact moments and the local correction are

Odd moments vanish by evenness. Then ${\Pi }_w+L={\Pi }_{\mathrm{tr}}$, and $\partial ({\Pi }_w+L)=0$ on both pairs. That identity is kernel_two_point_Ward. It is not functional ST4b. The distribution $L$ has order at most $4$. The kernel-divergence contact bound on $\partial L$ is five. The pairing is a formal transpose, not an $L^2$ adjoint. Lorentz covariance of $L$ is not claimed. The counterterm density $C_L=\frac{1}{2}\int hLh$ has ghost number $0$. The variation $s_0{C}_L$ is not evaluated. A kernel counterterm is not bulk BV restoration. A missing one-point moment D12 stays Missing and is never defaulted to $0$. Under the stated massless dimensional prescription the recorded value is zero; that record is not a default.

The inventory status is manually specified and partially checked. It accepts the o60 repairs: mixed D04 and D06, $V_{\mathrm{af},\varphi }$ at order ${\kappa }^1$, $m=0$, and leg balance. D04 freezes the kinetic structure $h\partial \varphi \partial \varphi$, with each vertex carrying one external and one internal scalar, so the count is four legs, two internal, two external, and mass zero. That placement is not all-internal. Global leg balance is not connectedness and is not one-particle irreducibility. Classical tree seeds D07, D08, and D09 are excluded from one-loop counterterm free slots. The Einstein–scalar $hh$ sector retains the D11 scalar loop and the D12 scalar tadpole. D11 is evaluated as the matter-loop subsector under the prescription above. D12 is the massless dimensional-regularization prescription, not a default zero inserted when the ledger is absent.

Renormalized amplitudes, at this milestone, are the kernel statements already recorded. Route A and Route B give the same prefactor $\Pi =(i{\kappa }^2/4)T$. Extension of the D11 kernel under the active $W_w$ reference is Constructed. The contact Ward identity is Constructed as kernel_two_point_Ward. Gate 3 overall remains Partial only because the spanning set is a candidate list. Bulk amplitudes are not Constructed. O.1 stays NOT_CLOSED.

The retained identity is a statement, not a verification. For the finite one-loop, order-${\kappa }^2$, bulk two-point sector on $S_{\mathrm{inv}}$, with antifields and sources retained only through vertex-derived ghost-number $0$ seeds, the organization is

at one loop and order ${\kappa }^2$ in the retained sector, followed by a projection and a counterterm equation. Here $(\cdot ,\cdot )$ is the classical antibracket of XIV-A and XVI-N, and $A_1$ is the one-loop defect. Vanishing of the projected defect, up to admitted finite counterterms and BV-exact changes, is the goal of the symmetry restoration problem. It is not a claim that restoration has occurred.

This identity is not the classical statement $(S_{\mathrm{inv}},S_{\mathrm{inv}})=O({\kappa }^2)$. Classical consistency is infrastructure, not a substitute. It is not an automatic identification of an effective-action identity with a time-ordered-product identity. It is not a prepared $S_D$ identity. It is not XVI-M selected-image matching. Free equations of motion are not used: the free-EOM ledger is Unresolved. Verification of (eq:identity) is not Constructed. No claim is made that $A_1=0$ after counterterms. Bulk symmetry_restored stays false.

The symmetry-defect ledger keeps ST4a, local ${\Gamma }_1$ matching, separate from ST4b. They are tasks to be tested, not a forced reduction. The defect has ghost number $1$. An admitted counterterm has ghost number $0$. The organization D01+D02+D03 is a candidate only. The matched counterterm set is empty.

The counterterm ledgers are parameterized bookkeeping slots, not a certified independent basis. independence_proven stays false. The matched set stays empty. Three layers stay separate: frozen classical source vertices, which are excluded from one-loop free slots; calculated one-loop source amplitudes, which are Unresolved; and admitted counterterms with explicit $\hslash$ and $\kappa$ orders. The flag for an ST4a local-matching slot is true only when the grading contains ghost number $0$ and the role is not a tree seed. The unconditional escape or True is withdrawn.

The seed is $F_t=⟨t_D,h\otimes h\otimes \varphi ⟩$ as in XVI-I, with no extra factorial. The regular seed-preserving antifield-linear class is $I_t=F_t+A_0+\kappa A_1$, with $A_0$ the integral of $h^{\ast }U+{\varphi }^{\ast }V+c^{\ast }W$. Under the XVI-N massless-obstruction hypotheses, on the constructed massless special-family witness, no such $I_t$ satisfies $Q_{\mathrm{inv}}{I}_t=O({\kappa }^2)$. Route A is sufficient and is discharged. Route B is not discharged. The certified small-mass interval $0\le m<1.3715\times 10^{-4}$ is XVI-N's interval, not a new theorem, and it is not arbitrary prescribed $m>0$.

No quantum completion is claimed whose classical leading term contradicts that obstruction. Extra antifields or nonminimal fields do not automatically remove the order-${\kappa }^0$ field-only contradiction $P_0{Q}_0{A}_0=0$ on the free shell. Existence of a renormalized product of the bare $F_t$ is not BV or BRST closure of $F_t$. Success on the bulk kernel, verdict 1, does not complete $F_t$, verdict 3.

Ultraviolet subtraction and finite symmetry repair are kept separate. Acceptance of the kernel counterterm is exactly the conjunction of the fixed extension, the identity on $U$, the active $W_w$ reference, and the moment formulae for $L$. That acceptance is not bulk restoration and not a completion of $F_t$. H2, I3, and J2 remain unresolved. XV-B is not opened. This is not an ultraviolet completion, not a derivation of a metric or of Einstein dynamics, not the aligned-null limit through $N$, and not a cosmological curvature singularity. Em01-G's isolated failure, Em02-A's localization powers, and Em02-B's simulated numbers are not this kernel.

The scripts, ledgers, certificates, notes, and withdrawn builders from https://github.com/cheungngo/XVI-O at revision o70-sync are archived as package evidence. That archive is not a theorem of this note, and it is not a certificate. It is not executed here. No compilation is asserted.

The science-facing modules are code/matter_loop/d11_scalar_extension.py, d11_global_representation.py, d11_Ww_reference.py, d11_contact_moments.py, and d11_subsector_summary.py, together with the Gate 0–2 modules convention_sheet.py, mixed_scalar_noether.py, d11_normalized.py, and d11_tensor.py. Classical scaffold modules live under code/classical/. The inventory builder is code/inventory/build_diagram_inventory.py. Tests are tests/test_matter_loop_gates.py, tests/test_quantum_normalization.py, and tests/test_inverse_metric_series.py. Machine-readable status is RELEASE_STATUS.json and the files in ledgers/ and certificates/. The builders in withdrawn/o62-matter/ and withdrawn/o66-tr-archived/ are not active.

Read from that status file, and not re-certified here: the subsector string is Constructed_subsector_under_prescription; D11_kernel_counterterm is earned only after the four prerequisites; bulk BV restoration is not_established; O.1 closure is NOT_CLOSED; the insertion status is unchanged; symmetry_restored is false; the matched counterterm list is empty; and independence_proven is false. A pass flag in a JSON file is not a proof. Missing scaling is Unresolved and is never defaulted to $8$. Missing D12 is Missing and is never defaulted to $0$. The appendix volume of listings is not part of these claims.

XI-A

N. Cheung, XI-A. Quoted scalar action. Not a new theorem of this note.

XVI-I

N. Cheung, XVI-I. Seed $F_t$ with no extra factorial.

XVI-J

N. Cheung, XVI-J. Extension freedom only. Not the contact $L$.

XVI-M

N. Cheung, XVI-M. Selected-graph comparison data. Not this kernel.

XVI-N

N. Cheung, XVI-N. Massless obstruction, read-only pin. Route B not discharged.

DFKR

Dütsch, Fredenhagen, Keller, and Rejzner, arXiv:1311.5424, Definition 4.2 and Sections 4.1–4.2. Analytic regularization instantiated for $F(\zeta )=(Q-i0{)}^{\zeta }$. Not a premise beyond that instantiation.