Landmark XVI-P One-loop power counting and a renormalized two-point sector, without bulk symmetry and without closing O.1 Science volume (p23: O.1 requirement recorded; P.6 inventory not evaluated)

Revision p23 records the O.1 requirement copied from the XVI-O ledger and checks one-loop power counting on the frozen inventory. The enlarge rule is not triggered: there is no one-loop three-point sector at ${\kappa }^2$. The P.6 entry is an inventory of fifteen one-loop ${\kappa }^3$ three-point graphs. That inventory is not evaluated. The P.5 calculation is renormalized at one loop and at ${\kappa }^2$ for the two-point functions with QdS external legs. QdS is an external-leg scheme, not a de Sitter state. Bulk symmetry is not claimed. O.1 stays NOT_CLOSED, because O.2 and the bulk identity are outside XVI-P. The inherited insertion stays frozen by XVI-N. Route B is not discharged. XVI-P does not update XVI-O: the classical ${\kappa }^2$ master there stays Partial, and symmetry_restored stays false. The appendix, the certificates, and the archived scripts are package evidence, not a series certificate. No quantum coefficient of the inherited characteristic graph is admitted. No compilation is asserted.

The notes below are the science volume, in release order from p23 back through the inherited D11 context. This section is a reading map for those notes. It does not add a theorem.

The XVI-O ledger O_quantum_target.json (o70-sync, read only) states the O.1 target: the one-loop ${\kappa }^2$ bulk two-point sector on $S_{\mathrm{inv}}$ closes under a verified retained quantum identity. Closing it requires O.2, O.4, O.5, O.6, and O.7, and an enlargement of the target if the identity constrains more legs at ${\kappa }^2$. XVI-P records that requirement and treats O.4, O.5, the projected part of O.6 and O.7, and the enlarge rule as in scope. O.2, the classical ${\kappa }^2$ master including the gravitational ${\ell }_4$, and the bulk ST4b identity stay with XVI-O and XVI-N and are not touched. O.1 is therefore not closable inside XVI-P, and it stays NOT_CLOSED.

Every frozen interaction vertex carries ${\kappa }^{\mathrm{legs}-2}$, with antifield sources counted as legs. A one-loop graph is then ${\kappa }^E$ with $E$ external legs. The one-loop identity at ${\kappa }^2$ uses only $E=2$ and $E=1$. A mechanical enumeration reproduces the frozen ${\kappa }^2$ inventory exactly, nine graphs out of nine (D01–D06, D11, D12, D13), and the three graphs of the one-point row. The enlarge rule is not triggered: there is no one-loop three-point sector at ${\kappa }^2$.

Three-point functions first appear at ${\kappa }^3$. The one-loop ${\kappa }^3$ three-point inventory has fifteen graphs: $hhh$ five, $h\varphi \varphi$ four, $c\bar{c}h$ two, $Kch$ two, $c^{\ast }cc$ one, and ${\varphi }^{\ast }c\varphi$ one. This is a topology-level inventory. The amplitude rules, the counterterm domain, the evaluation, the three-point defect, and ${\sigma }_0$ closure are not done. No defect outcome is claimed for P.6.

P.5 calculation status is renormalized at one loop and at ${\kappa }^2$ for the two-point functions $hh$, $K^{\prime }\cdot c$, and $\bar{c}c$, in the $\overline{\mathrm{MS}}$ scheme in DRED, with the p22 evanescent counterterms, and with external legs in QdS. The status does not cover higher-point functions, two loops, bulk symmetry, or O.1. p20 retracts the p19 claim that full ${\sigma }_0$ closure of the $\overline{\mathrm{MS}}$ counterterm fails. The corrected complete one-loop ${\kappa }^2$ counterterm, including the ghost partner, is ${\sigma }_0$-closed in the ghost $+1$ sectors that the note names. That closure is the projected two-point counterterm. It is not bulk symmetry and not O.1.

QdS names the external-leg scheme of the earned two-point sector. A finite term that appears if external traces are taken with $g_4$ (Q4S) is disclosed as a scheme choice. Neither scheme is Em02-E's relational lapse, a de Sitter state, a cosmological observer algebra, or horizon entropy.

bulk_symmetry_restored stays false. The kernel two-point Ward identity of XVI-O is not this calculation, and this calculation is not functional ST4b. Bulk BV restoration is not claimed. XVI-P does not update XVI-O. On XVI-O the classical ${\kappa }^2$ master stays Partial, Gate 3 spanning stays Partial, and O.1 stays NOT_CLOSED.

The inherited insertion stays frozen by XVI-N, thm:massless-obstruction. Route A of that obstruction is not reopened. Route B is not discharged. Existence of a renormalized product of the bare seed $F_t$ is not BV or BRST closure of $F_t$. Success on the two-point sector does not complete $F_t$. XV-B is not opened. XIV-J stays closed. H2, I3, and J2 stay unresolved. No quantum coefficient of the inherited characteristic graph is admitted.

Certificates, listings, and the hash manifest are package evidence in the sense of XVI-O conv:package. An archive is not a mathematical certificate. The same-box rebuild record is not an independent mathematical certificate and is not a compilation of this source. No compilation is asserted.

.1, exactly. The XVI-O ledger O_quantum_target.json (o70-sync; verbatim copy in provenance/XVI-O, sha pinned, read only) states the O.1 target: the one-loop ${\kappa }^2$ bulk two-point sector on $S_{\mathrm{inv}}$ (graviton, scalar, ghost, including the gh$=0$ BV source seeds; frozen gauge fermion; $m=0$). O.1 closes iff this sector closes under a verified retained quantum identity. Every checklist item must be earned: O.2 (classical ${\kappa }^2$ master, including grav ${\ell }_4$), O.4 (inventory), O.5 (graphs evaluated), O.6 (identity verified) and O.7 (remainder cancellation / counterterm solution). The target must first be enlarged if the identity constrains more legs or orders at ${\kappa }^2$. In XVI-P scope: O.4, O.5, the projected part of O.6/O.7, and the enlarge rule. Out of scope: O.2 and the bulk ST4b identity, which belong to XVI-O/XVI-N and stay untouched. Hence O.1 is not closable inside XVI-P, and it stays NOT_CLOSED.

one-loop power counting. Every interaction vertex of the frozen inventory carries ${\kappa }^{\text{legs}-2}$, with antifield sources counted as legs. This is checked vertex by vertex. A one-loop graph has $I=V$, so it is ${\kappa }^E$ with $E$ external legs. The one-loop identity at ${\kappa }^2$ is $(S^{(0)},{{\Gamma }_1}^{(2)})+(S^{(1)},{{\Gamma }_1}^{(1)})$; the term $(S^{(2)},{{\Gamma }_1}^{(0)})$ vanishes because ${{\Gamma }_1}^{(0)}$ is field independent. Only $E=2$ and $E=1$ enter. A mechanical enumeration of all one-loop 1PI graphs (declared propagators $hh$, $\varphi \varphi$, $c\bar{c}$) gives the following:

• for $E=2$ it reproduces the frozen ${\kappa }^2$ inventory exactly (D01–D06, D11, D12, D13: 9/9);

• for $E=1$ it gives the 3 graphs of the one-point row;

• it detects four mutations (a wrong $\kappa$ power, two dropped vertices, a spurious propagator).

The enlarge rule is not triggered: there is no one-loop three-point sector at ${\kappa }^2$.

.6 entry (honest partial). Three-point functions first appear at ${\kappa }^3$. The one-loop ${\kappa }^3$ three-point inventory has 15 graphs: $hhh$ 5, $h\varphi \varphi$ 4, $c\bar{c}h$ 2, $Kch$ 2, $c^{\ast }cc$ 1, ${\varphi }^{\ast }c\varphi$ 1. This is a topology-level inventory only. The amplitude rules, the counterterm domain over $\mathbb{Q}(i)$ (including evanescent operators), the evaluation, the three-point defect and ${\sigma }_0$ closure are not done in p23, so no defect outcome is claimed.

from the p22 review. (i) Polynomial-domain refusal (p23 contract revision, one block appended to dred.py). From outside sympy, polynomial algebra that can evaluate $d$ raises: rem, prem, div, resultant, compose, Poly.shift, Poly.rem, and similar. Internal renames of $d$ are admitted only to a Dummy, so Matrix.subs(d$\to$y) raises. A live probe runs 9 forms on 4 expressions, and rational-function algebra is unchanged. (ii) The gate now checks the values $\rho =1/2$, $E_2$ coefficient $-1/16$, SE coefficient $+1/8$, and the summed h.h coefficients against both pinned values and the archived p22 certificate. (iii) The README title line carries “two-point functions with QdS external legs”.

revision (disclosed). p22 carries one disclosed contract revision, listed in p22_conventions.CONTRACT_REVISION: (i) evanescent counterterms are admitted outside the p14r 4D domain, and (ii) code/p5_entry/dred.py gains one delimited additive block (the runtime $d$ guard). Outside the block the file is byte-identical to the p18 release; contract_registry.dred_revision checks this, and the p22 gate pins the exact sha256 of the revised file.

-M2: the matching statement. XVI-O has no evanescent counterpart, so the eight evanescent ($\tilde{g}$) structures of D11 cannot be matched to XVI-O. We tested three candidate statements exactly, with $d$ opaque: (a) {match a $d$-dimensional continuation of the XVI-O kernel} is refuted for both natural continuations ($g_4\to \hat{g}$ and $g_4$ kept), because the exact D11 tensor differs from either continuation of its own 4D part on $\stackrel{~}{}$ external legs. XVI-O defines no other continuation, so (a) is not a usable statement. (b) Pure local evanescent operators: proven. Every evanescent pole (D01, D03, D11) is a polynomial in $p$ and independent of $d$ after pole extraction. (c) Vanishing in the renormalised result: proven in a stronger form. After subtraction, every evanescent component is identically zero on 4D (hatted) external legs at every $d$, not merely at $O(ϵ)$. The scope is D11_matching_verified_scope = “4D components: M2 discharged (P18_D11_M2, unchanged); evanescent components: EV-M2 proven; NOT matched to XVI-O”.

-CT: the counterterms. The coefficient is $-(\text{UV pole})$ in the $N_{ϵ}{\kappa }^2/ϵ$ normalisation of P14. There are three pieces:

• h.h operators for the eight $\tilde{g}$ structures, summed over D01+D03+D11: $g_4\tilde{g}(p^2{)}^2$: $-7i/120$ (twice); $g_{4,mr}{\tilde{g}}_{ns}(p^2{)}^2$: $-i/160$; $\tilde{g}\tilde{g}(p^2{)}^2$: $i/160$ (twice); $\tilde{g}pp{p}^2$: $7i/120$ (twice), $i/160$. Per-graph values are in certificates/P22_evanescent.json.

• the K${}^{\prime }$.c operator for D13 $E_2$, with coefficient $-1/16$;

• its ghost-equation partner $\bar{c}{\Sigma }_{\mathrm{SE}}c$, with coefficient $+1/8$.

Closure. ${\sigma }_0$ of the counterterms is zero in every sector, so full closure holds with the evanescent counterterms included. In h.c, the per-graph $p\cdot T$ of D01 and D03 cancel in the sum, D11 is transverse, and $E\cdot E_2=0$. In c.b, the ghost equation with $\rho =1/2$ (calibrated on the 4D pair) holds exactly in $d$ for $E_2$ and ${\Sigma }_{\mathrm{SE}}$. c.cbs is unreachable. Identity. The h.h slot values of the counterterm sum are zero in all six external modes, and the K.c slot values are zero, so the projected identity is unchanged. Remnants. The finite evanescent parts are nonzero, but they vanish identically on hatted external legs. With QdS external legs there is no $O(ϵ)\times$pole effect. If external traces are taken with $g_4$ (Q4S), a finite term $7i{p}^2(\hat{a}\cdot \hat{b}{p}^2-\hat{a}\cdot p\hat{b}\cdot p)/60$ arises. It is disclosed as a scheme choice. Mutations. Each of the following is detected: dropping the D01 or D03 counterterm; doubling D03; dropping the SE partner; flipping the sign of the $E_2$ counterterm; and putting a trace operator in place of $s_{0}h$. Status. P.5 calculation status is renormalized (one loop, ${\kappa }^2$, the two-point functions hh / K${}^{\prime }$.c / $\bar{c}$c, MS-bar in DRED with the p22 evanescent counterterms, external legs in QdS). It does not cover higher points, two loops, bulk symmetry or O.1.

{Runtime $d$ guard.} The guard intercepts the following calls whenever the caller is outside sympy/mpmath/dred: subs, xreplace (including on a bare $d$), replace, evalf/n, the Poly value and coefficient accessors, coeff, as_independent, as_coeff_exponent, series, collect, limit, Lambda, and $d==d$. Any of them that would give $d$ a value raises PrematureD4Substitution. Authorised paths (substitute_d, authorised_d) still work. A live probe in the gate runs 30 forms on 4 expressions: every reviewer form from p21, the power-base hole, alias and free_symbols lookups, Symbol.__new__ and Function calls. All of them raise, and ordinary algebra is unchanged. Some bypasses cannot be intercepted at runtime: identity (is) tests, string round-trips and Poly.rep. These are disclosed and covered only by the secondary static allow-list scan.

. The tools/ and notes prefix exemptions are removed. Every changed, added or removed file relative to the p21 zip must be listed in DISCLOSED_REVISIONS_P22. The only exceptions are the generated reference values and the pinned archival/p21 records. Earlier-layer gates accept the registered p22 revisions only through contract_registry.

.{p21 is a small release from p20 with no scope expansion. It follows the findings of the p20 independent rebuild (PASS; report sha256 e8f358f9, bound to tree 2c0aa153{} and zip ca32ce01; same box, not a second machine). No contract revision; the p15–p19 layers are as in p20; the disclosed p21 revisions of the p20 layer are d11_pin.py and release_p20.py. bulk_symmetry_restored is false regardless. O.1 stays NOT_CLOSED.}

D11 pin (p20: OPEN). The sign and magnitude of the D11 UV pole are derived from $\frac{1}{2}\mathrm{Tr}\ln$ of the minimal scalar loop under the package's declared conventions, each checked verbatim in the code:

• $G={\Gamma }^{(2)}$: the amplitude rules declare $i{\Gamma }_1=\int \mathrm{pref}N/\prod (q^2+i0)$ and G6 declares $G={\kappa }^2{N}_{ϵ}\hat{G}$, $\hat{G}=-i\mathrm{pref}/{\kappa }^2\int N/(D_1{D}_2)$; hence $G=-i\cdot i\Gamma =\Gamma$.

• Wick direction: $N_{ϵ}=i{S}_{ϵ}/(16{\pi }^2)$ and the certified bubble pole $\hat{I}(1,1)=+1/ϵ$ give $d^d{l}_M=+i{d}^d{l}_E$; with $\eta =(+,-,-,-)$ and $S_{\varphi }=\frac{1}{2}\int \sqrt{-g}{g}^{\mu \nu }\partial \varphi \partial \varphi$ this gives $i{S}_M=-S_E$, so ${L^{\mathrm{eff}}}_M=-{L^{\mathrm{eff}}}_E$.

• Heat kernel (Euclidean, $d=4-2ϵ$): $L_{E,\mathrm{div}}=-a_2/(32{\pi }^2ϵ)$, so $L_{M,\mathrm{div}}=+a_2/(32{\pi }^2ϵ)$ with $a_2=C^2/120+R^2/72$ ($\xi =0$).

• With $g=\eta +\kappa h$ the Minkowski quadratic forms are $C^2\to \frac{1}{2}{p}^{4}h{P}_{2}h$ and $R^2\to 3p^{4}h{P}_{0}h$.

Result: ${\Gamma }^{(2)}{|}_{\mathrm{pole}}={\kappa }^2/(16{\pi }^2ϵ)[p^4{P}_2/240+p^4{P}_0/24]$. This equals the package pole ${\kappa }^2{N}_{ϵ}\hat{G}/ϵ$ with $c_2=-i/240$, $c_0=-i/24$ exactly (ratio 1 for both spins). No convention choice is external; the external inputs are mathematics only (the $a_2$ coefficients and the heat-kernel pole formula). A $\times 2$ normalisation error and a sign flip keep the 10:1 ratio but are detected by the absolute pin.

. The top-level absolute_pin_status and ratio_pin_status (the keys typeset in this PDF) must equal the nested statuses. Run standalone, the p20 gate re-checks the p18 layer (quantum inputs earned) and the graph inventory instead of trusting the P19 layer record; the p21 gate re-checks all lower layers.

. The enumerated pattern list is complemented by an allow-list scan (tests/test_p21.py): expression-tree decomposition (.args, make_args, as_ordered_terms, ), sympy callable aliases, computed symbol names and value calls mentioning $d$ are refused everywhere except at reviewed sites. All eight forms reported by the reviewer are caught. A runtime check confirms that the stored exact-$d$ D11 amplitudes re-parse with the opaque $d$ and reproduce their poles only through substitute_d. It also found a limitation: expr.subs(d, v) is not intercepted when $d$ occurs only as a power base. The contract is not changed in p21; no package value path uses it, and the static rule refuses it. This is a check, not a proof.

. The p20 edit of manuscript/notes/P19_defect.tex is now listed in DISCLOSED_MANUSCRIPT_REVISIONS. The reference key /p19/change_log_vs_p18, which was recomputed on the current tree, is renamed p19_gate_change_log_vs_p18_recomputed_on_current_tree. The log as released stays in the pinned archival/p19/reference_values_p19.json.

.{p20 is a correction release of p19 with no scope expansion. It follows the findings of the p19 independent rebuild (PASS with findings; report sha256 1b01f1d8, bound to tree fcf90354{} and zip da924275; same box, not a second machine). There is no contract revision, and the p15–p18 layers are byte-identical. The disclosed p20 revisions of the p19 layer are defect.py and release_p19.py. bulk_symmetry_restored is false regardless. O.1 stays NOT_CLOSED; the insertion stays frozen; XV-B and XIV-J are unopened; Route B is not discharged; no $c\delta$ is added to ${u^R}_{\mu }$.}

K1 (the p19 text is retracted). p19 stated that full ${\sigma }_0$ closure of the MS-bar counterterm fails with the derived ghost-equation factor $+1$, that it needs $-1$, and that this was unresolved. The statement is wrong; it came from a Grassmann-orientation bookkeeping error.

• The route-G densities are generated in the order (c,cb). The P15 ghost self-energy coordinates $\Sigma$ use the (cb,c) orientation, and $V[c.cb\#k]=-V[(\mathrm{cb},\mathrm{c})\#k]$. p20 measures this ratio (via pw_vertex) as $-1$ for $k=0,1$.

• In route-G (c,cb) coordinates, $\Sigma$ has pole $(+3/8,-1/4)$ and local part $(2/3,-5/12)$.

• The derived factor is $+1$ (the ghost equation $\Sigma =2iD[G_{Kc}]$ reproduces the P15 pole). With it, the complete MS-bar counterterm $S_{ct}=-(\text{h.h}+\text{c.K}+\text{c.cb poles})$ is ${\sigma }_0$-closed in every ghost $+1$ sector (h.c, c.b, c.cbs).

• Without the c.cb term, or with $\Sigma$ unconverted (the p19 error), ${\sigma }_0(S_{ct})$ has 88 entries in the c.b sector.

• Tree-level controls: $V_{S_0}[\mathrm{cb},\mathrm{c}]=2iD{V}_{S_0}[K,c]$, and ${\sigma }_0({S_0}^{Kc}+{S_0}^{\bar{c}c})=0$ (the ${S_0}^{Kc}$ part alone has a nonzero image).

• In quotient coordinates, the c.K-only data have c.cb components $(-3/8,1/4)$, which cancel against $\Sigma =(+3/8,-1/4)$. The full data including $\Sigma$ have quotient coordinates $0$.

redefined. The p19 flag (renormalised $P_{ch}{\sigma }_0(S_{ct})=0$, $x=0$) follows from the identity and locality, so it is not independent. In p20, projected_restoration_verified means full ${\sigma }_0$ closure of the complete one-loop ${\kappa }^2$ MS-bar $S_{ct}$ (including c.cb, $\Sigma$ held fixed) in all ghost $+1$ sectors, together with the projected identity. Scope: that counterterm only; not bulk symmetry, not O.1. The corrected computation closes, so the flag is earned. The p19 check is kept as a labelled route-consistency control, and it holds.

spots. The full-closure test acts as a mutation detector, with $\Sigma$ held fixed:

• drop D13 and flip K.c are detected (c.b, 88 entries);

• drop D01, drop D03, flip D03 and D03 $\times 2$ are detected (h.c, 280 entries);

• the p19 orientation error is detected;

• drop D11 is not detected: D11 is transverse, so its pole is closed by itself.

{D11 normalisation pin (P20_D11_pin.json).}

• Derived from package data. The 4D UV pole of D11 (P18_D11.json, package orbit convention; transverse; only spin-2 and spin-0s parts) has Barnes–Rivers coefficients $c_2=-i/240$ and $c_0=-i/24$, so $c_0:c_2=10$. The quadratic forms on $g=\eta +h$ are derived here from linearised curvature: $C^2\to \frac{1}{2}{p}^{4}h{P}_{2}h$, $R^2\to 3p^{4}h{P}_0^{s}h$, and Gauss–Bonnet $\to 0$.

• External literature. The $a_2$ coefficient of a minimally coupled real scalar is $\frac{1}{120}{C}^2+\frac{1}{72}{R}^2$ (Birrell–Davies 6.2; 't Hooft–Veltman 1974). This predicts $\frac{1}{240}:\frac{1}{24}=1:10$.

• Ratio pin: PASS. Drop D11 gives $c_2=c_0=0$ and is detected, and so is a swap of the two $gg$ structures.

• Absolute pin: OPEN [superseded in p21: the absolute pin was derived and verified, sign and magnitude (P21, section 1; certificates/P20_D11_pin.json, absolute_pin_status PASS); the text below is the p20 statement, kept for the record]. The magnitudes agree up to a common factor $-i$, but the map from $\hat{G}$ to $a_2$ (phase, $N_{ϵ}$, $\kappa$ normalisation, sign of $\Gamma$) is not derived. An overall $\times 2$ error is caught only by this open pin and by M2 (P18).

1 wording. The $\mathbb{Q}(i)$ counts are the real dimensions under a compatible real structure. The earlier wording, “the physical field”, is replaced.

and gate. The static scan is still a pattern check, not a proof. It now also flags:

• Add(*listcomp over e.args);

• an aliased Poly import;

• Poly.__call__;

• as_coefficients_dict;

• a functools.reduce rebuild;

• Symbol(name=);

• an explicit-constructor rebuild.

A standalone reject of the p19 or p20 gate no longer leaves a stale RELEASE_STATUS.json.

. independent_rebuild_passed_p19 is true (bound to the p19 tree and zip; same box). The p20 independent rebuild is pending.

.{p19 keeps the p14r contract and the p15–p18 calculation layers byte-identical (release_p19.py recomputes their sha256). It assembles the ch projection (P.3, frozen) of the $\hslash {\kappa }^2$ part of $(S_{\Psi },{\Gamma }_{\Psi })$ from every evaluated input, exactly in $d$. It then solves for a symmetry-restoring counterterm in the declared domain. Only the ch projection is covered (one $c$, one $h$; one loop; ${\kappa }^2$; off shell). bulk_symmetry_restored is false. O.1 stays NOT_CLOSED; the insertion stays frozen; XV-B and XIV-J are unopened; Route B is not discharged; no $c\delta$ is added to ${u^R}_{\mu }$.}

The ledger P13_ch_dependency_rows.json has three surviving rows:

• $R_{hh}$, $(S_0,{{\Gamma }_1}^{[2]})$$s_{2}K:K.c$, inputs D01+D02+D03+D11+D12, with $X_{hh}=\int (s_{0}h)G_{hh}h$;

• $R_{Kc}$, $(S_0,{{\Gamma }_1}^{[2]})$$s_{1}h:h^2$, input D13, with $X_{Kc}=\int G_{Kc}cEh$, where $E$ is the $h^2$-monomial operator of $S_0$;

• $R_{1pt}$, $(S_1,{{\Gamma }_1}^{[1]})$, inputs D14a–c.

Every input is evaluated; none is Missing, and a Missing input is refused rather than set to zero (control). The defect is computed for seven external assignments, including “c hat, h tilde”. For each it is given as the exact $d$-dependent value times MB, its pole, its coefficient of $L$ and its MS-bar local part.

{Result: the defect is zero, exactly in $d$, in every part and every assignment.} Why it is zero. Each surviving row vanishes separately in the projected slot. No counterterm cancellation is involved, and no cancellation between rows.

• (i)

  hh row: full-row transverse decoupling. D01 and D03 are each nonzero in the slot and cancel each other (the p17/p18 transverse sum). D11 is transverse by itself, and D02 = D12 = 0.

• (ii)

  K.c row: D13 reaches the ch slot only through $T_1$ and $E_1$ ($T_2$, $T_3$ and $E_2$ are longitudinal; $E$ is transverse), and $A_{T_1}=A_{E_1}=0$ exactly in $d$.

• (iii)

  One-point row: the tadpoles D14a–c vanish under the declared prescription.

This rests on:

• M2 / the D11 matching, verified on the 4D components only (the D11 matching for the evanescent components is not applicable and not verified);

• the evanescent decoupling proofs (P16 E2; P17/P18 hh evanescent and full-row decoupling);

• the frozen P.3 projection and the P13 ledger;

• the declared prescription.

Evanescent decoupling in the assembly. The evanescent parts of both rows were assembled explicitly and are zero for all seven assignments. The evanescent poles themselves are nonzero; they are regulated and not subtracted.

Route G applies ${\sigma }_0=(S_0,\cdot )$ to the 11 p14r domain densities: h.h#0–4, c.cb#0–1, c.K#0–2, $\varphi .\varphi$. Route C computes the covariant ch images exactly in $d$. Both give rank $P_{ch}{\sigma }_0=3$.

• Over $\mathbb{Q}(i)$ (these counts are the real dimensions under a compatible real structure; wording corrected in p20): rank ${\sigma }_0{|}_D=5$, $\dim D\cap \ker {\sigma }_0=6=\dim D\cap \mathrm{Im}{\sigma }_0$, cohomology 0, and the quotient has dimension 5 (complement h.h#0–2, c.cb#0–1).

• The h.h + c.K ${\Gamma }_1$ data, without the ghost self-energy and in route-G (c,cb) labels, have quotient coordinates pole $(0,0,0,-3/8,1/4)$, coefficient of $L$$(0,0,0,3/8,-1/4)$ and local part $(0,0,0,-2/3,5/12)$. The h.h pole is transverse, i.e.\ ${\sigma }_0$-exact. These coordinates equal $-[\Sigma ]$: $\Sigma$ itself has $(+3/8,-1/4)$, and the data including $\Sigma$ have quotient $0$ [clarified in p20].

• The linear solve $\Delta +P_{ch}{\sigma }_0(x)=0$ gives $x=0$ (outcome ZERO); no residual and no obstruction.

• The MS-bar counterterm has zero ch image. The h.h + c.K part alone has a nonzero full ${\sigma }_0$ image in the c.b sector; adding the c.cb ghost self-energy counterterm in the correct orientation closes it [corrected in p20, see the p20 erratum note].

and their definitions.

• projected_defect_computed: $\Delta$ is assembled from all evaluated inputs, exactly in $d$, in all parts and all seven assignments.

• projected_identity_verified: $\Delta =0$ for the regulated ${\Gamma }_1$.

• projected_restoration_verified [REDEFINED in p20]: full ${\sigma }_0$ closure, in every ghost $+1$ sector (h.c, c.b, c.cbs), of the complete one-loop ${\kappa }^2$ MS-bar counterterm $S_{ct}$ (h.h + c.K + c.cb, ghost self-energy held fixed, derived relative factor $+1$, orientation converted). The p19 definition ($P_{ch}{\sigma }_0(S_{ct})=0$, $x=0$) follows from the identity and locality; it is kept only as a labelled route-consistency control.

Scope of the restoration flag: the one-loop ${\kappa }^2$ counterterm only; not bulk symmetry, not O.1. The release gate re-checks each definition from the certificate content. bulk_symmetry_restored is false regardless.

1 [corrected in p20; the p19 text is retracted]. p19 reported that full closure of the MS-bar counterterm fails with the derived ghost-equation factor $+1$ and needs $-1$. That was a Grassmann-orientation bookkeeping error: the route-G densities are built in the order (c,cb), whereas the P15 $\Sigma$ coordinates are (cb,c)-oriented, and $V[c.cb\#k]=-V[(\mathrm{cb},\mathrm{c})\#k]$. With the measured ratio $-1$ applied, the derived factor $+1$ gives full closure (p20 erratum note). The relations below are stated in the (cb,c) orientation:

• The ghost-equation operator $2iD$, verified by P15 as $\Sigma =2iD[G_{Kc}]$, gives $2iDV[c.K\#1]=-V[(\mathrm{cb},\mathrm{c})\#0]$ and $2iDV[c.K\#2]=-V[(\mathrm{cb},\mathrm{c})\#1]$.

• Applied to the c.K pole, this reproduces the P15 $\Sigma$ pole $(-3/8,1/4)$ in (cb,c) coordinates, i.e.\ $(+3/8,-1/4)$ in the route-G (c,cb) coordinates.

1: finding about the p14r contract (contract not modified). The p14r counterterm-domain certificate counts Im ${\sigma }_0$ 3, quotient 8, $D\cap \ker$ 5 and rank $P_{ch}{\sigma }_0$ 4. These are artefacts of a real/imaginary split over $\mathbb{Q}$: the densities carry an explicit $i$ for odd derivative counts, so an $i$-multiple counts as independent. Over $\mathbb{Q}(i)$ the counts are 6, 5, 6 and 3 (the real dimensions under a compatible real structure), and Route C agrees. With the split, p19 reproduces the p14r counts (control).

and mutations. All controls are true. Eight mutations each change the outcome:

• drop D03; drop D01; flip the D03 phase; double D03; trace operator instead of $s_{0}h$; de Donder $q^{2}P$ instead of $E$; drop the D03 evanescent part (naive 4D). Each gives OBSTRUCTION (nonlocal residual).

• D13 with $G_t$ dropped gives NONZERO but CANCELLABLE (rank 3).

Recorded blind spots of the projected identity: dropping D11 (transverse by itself), dropping D13 ($A_{T_1}=A_{E_1}=0$) and flipping the K.c row sign (each row vanishes separately). In p20, the full-closure test detects the last two, and the D11 normalisation pin covers the first. Removing D11 from the inventory makes the inventory incomplete, and projected_defect_computed is then not earned.

Route G over Q(i): rank ${\sigma }_0{|}_D$ = 5, dim $D\cap \ker {\sigma }_0$ = 6, dim $D\cap \mathrm{Im}{\sigma }_0$ = 6, cohomology 0, quotient 5 (complement h.h#0, h.h#1, h.h#2, c.cb#0, c.cb#1); rank $P_{ch}{\sigma }_0$ = 3 (Route C: 3). Over Q with the re/im split (p14r convention): dim $D\cap \mathrm{Im}$ = 3, quotient 8, dim $D\cap \ker$ = 5.

Quotient coordinates of the ${\Gamma }_1$ local data: pole ['0', '0', '0', '-3/8', '1/4'], coeff. of L ['0', '0', '0', '3/8', '-1/4'], local ['0', '0', '0', '-2/3', '5/12'].

Counterterm solve: ZERO; x = 0 (no counterterm needed beyond the MS-bar poles); the identity-preserving finite renormalisations are ker(P_ch sigma_0) cap D (dim 8), of which 2 modulo D cap Im sigma_0.

Why zero. each surviving ch row vanishes separately in the projected slot; no counterterm cancellation and no cancellation between rows is involved: (i) hh row (D01+D02+D03+D11+D12): full-row transverse decoupling – D01 and D03 are individually nonzero and cancel against each other in the slot (p17/p18 transverse sum), D11 is transverse by itself, D02 = D12 = 0; (ii) K.c row (D13): D13 reaches the ch slot only through T1 and E1 (T2, T3, E2 are longitudinal, E transverse) and A_T1 = A_E1 = 0 exactly in d; (iii) one-point row: the tadpoles D14a-c vanish under the declared prescription. The defect is therefore zero without any counterterm (solve x = 0).

Flag definitions. projected_defect_computed: Delta = P_ch[(S_Psi, Gamma_Psi)]_{hbar kappa\^{}2} assembled from every evaluated input of the three surviving rows (none Missing), exact in d, pole + nonlocal + MS-bar local parts, all seven external assignments. projected_identity_verified: Delta = 0 for the REGULATED one-loop functional Gamma1: pole, nonlocal and local parts vanish for all seven assignments, exactly in d, with all controls true and every mutation changing the outcome. projected_restoration_verified: p20 definition (replaces p19's, which follows from the identity + locality): FULL sigma_0 closure of the COMPLETE one-loop kappa\^{}2 MS-bar counterterm S_ct = -(pole) in the declared domain, h.h + c.K + c.cb (ghost self-energy Sigma held fixed at its P15 value, orientation converted, derived relative factor +1), in every ghost +1 sector (h.c, c.b, c.cbs), together with projected_identity_verified. Scope: the one-loop kappa\^{}2 counterterm only; NOT bulk symmetry, NOT O.1. bulk_symmetry_restored: hard stop: False regardless (not claimed; O.1 NOT_CLOSED).

K1 (corrected in p20; erratum to p19). derived relative factor 1; orientation ratio (route-G (c,cb) over P15 (cb,c)) ['-1', '-1']; Sigma pole in (c,cb) coordinates ['3/8', '-1/4']; full sigma_0 image of the complete MS-bar counterterm: {'closed': True, 'entries': 0, 'sectors': []}. ERRATUM to p19: the p19 statement 'closure fails with the derived factor, needs -1, unresolved' was a Grassmann-orientation bookkeeping error (P15 (cb,c) coordinates placed in route-G (c,cb) slots). Corrected: derived +1, full closure holds (finding K1 of the p19 independent rebuild).

J1 (finding about the p14r contract). FINDING ABOUT THE p14r CONTRACT (contract NOT modified): the p14r counterterm-domain counts (Im sigma_0 3, quotient 8, D cap ker 5, P_ch sigma_0 rank 4) are artefacts of a re/im split over Q (the densities carry explicit i for odd derivative counts, so i-multiples count as independent). The counts over Q(i) – sigma_0 rank 5, D cap ker 6, D cap Im 6, cohomology 0, quotient 5, P_ch sigma_0 rank 3 (= Route C) – equal the real dimensions under a compatible real structure. The p14r certificate stays as released; both are reported

.{p18 keeps the p14r contract and the p15, p16 and p17 calculation layers byte-identical (release_p18.py recomputes their sha256). It discharges the D11 matching premise M2 on the 4D components, evaluates D11 under the declared prescription, and rechecks the ch-projected hh slot with D11 included. quantum_inputs_evaluated is earned (every graph of the inventory is evaluated in one prescription). projected_identity_verified, projected_restoration_verified and bulk_symmetry_restored are NOT earned. O.1 stays NOT_CLOSED; the insertion stays frozen; XV-B and XIV-J are unopened; Route B is not discharged; no $c\delta$ is added to ${u^R}_{\mu }$.}

M2 states that on $U={\mathbb{R}}^4\setminus \{0\}$ the inverse Fourier transform of ${P^{DR}}_{AB}$ equals $R_A{R}_B{{\Pi }^R}_{\mathrm{tr}}$, with ${{\Pi }^R}_{\mathrm{tr}}=(i{\kappa }^2/4)K{u^R}_{\mu }$, $K=\alpha D_2+\beta D_0$, $\alpha =A^2/60$, $\beta =A^2/18$ and $A=1/4{\pi }^2$.

• Route B (exact; primary). Write $P^{DR}={\kappa }^2{N}_{\epsilon }\hat{G}$ with $\hat{G}={\sum }_s{A}_s(d)S_s\mathrm{MB}$ and $\mathrm{MB}=1/\epsilon +2-L$. At $d\to 4$ its nonlocal part is $-(i{\kappa }^2/16{\pi }^2)L{\sum }_s{A}_s(4)S_s(p)$; everything else is polynomial. On the other side, $\mathcal{F}[u^R]=c_{\mathrm{FT}}\log (-p^2-i0)+$const, and $K\to \hat{K}(p)$ with ${\hat{P}}_{mn}=p_m{p}_n-{\eta }_{mn}{p}^2$. This operator is fourth order, so the sign convention of the transform drops out. M2 then holds iff, for each of the 100 component pairs, $X_{AB}={\sum }_s{A}_s(4)S_s{|}_{AB}=-4{\pi }^2{c}_{\mathrm{FT}}{A}^2{Y}_{AB}=-(i/4)Y_{AB}$, where $Y=R_A{R}_B({\hat{D}}_2/60+{\hat{D}}_0/18)$. This holds exactly on all 100 pairs. The difference then has a polynomial Fourier transform, so it is supported at $\{0\}$, and $\mu$ enters only through polynomials.

• {$c_{\mathrm{FT}}$, calibrated twice.} (a) From the library DR bubble, $\mathcal{F}[{D_F}^2]=-N_{\epsilon }\mathrm{MB}$ and ${D_F}^2=A^{2}u$. (b) As $-r$, with the XVI-O residue $r=-i{\pi }^2$. Both give $c_{\mathrm{FT}}=i{\pi }^2$.

• Route A (structural corroboration). M1 is re-run: the P integrand equals the Fourier convolution of the XVI-O product $T=VV(CC+CC)$ (100/100). The frozen XVI-O code, imported read-only, re-derives $\alpha ,\beta$ and verifies $T=Ku$ on $U$ for all 256 components. A regularised convolution agrees with the pointwise product on $U$ (cited: XVI-O lem:BV_rules_on_U, Constructed).

• Controls and mutations. D11 is transverse exactly in $d$; the FP and PV loop routes agree. Six mutations are each rejected: $\alpha \leftrightarrow \beta$, off-diagonal weight 1, flipped FT sign, doubled symmetry factor, D01 in place of D11, and $\beta =0$.

• Scope. The flag is qualified as D11_matching_verified_scope = “4D components”. M2 is shown on the 4D components only. The evanescent components are the 8 $\tilde{g}$-carrying structures, i.e.\ components with an index pair in the $\epsilon$-scalar directions. Their status is not_applicable_not_verified, because XVI-O is strictly 4D and has no counterpart. The evanescent structures of D11 have nonzero $L$-coefficients (8, recorded); they are $\epsilon$-scalar components outside the 4D statement.

• Consequence (M3/M4). M3 is now unconditional on the 4D components: $L_{\mathrm{match}}$ is supported at 0. It is transverse and homogeneous of degree four, so it lies in the 2-dimensional transverse h$\cdot$h subspace. Its coordinates need the XVI-O finite constant of $u^R$ and are not computed.

• Prescription. Operational DRED R1–R6 with opaque $d$, MS-bar, Feynman $+i0$, N4 contact exclusions, M3 bounds 4/8/4, exactly-degree-four normalisation. Both points are checked per structure: every structure has momentum degree 4, so the order-0/2 contacts excluded by N4 do not arise.

• Inventory. quantum_inputs_evaluated is earned. Every graph of the three surviving ch rows is evaluated, and none is Missing: hh D01, D02, D03, D11, D12; K$\cdot$c D13; one-point D14a–c.

• Evaluation. 14 structures, 6 of them 4D. The covariant form is exact and unique; the FP and PV routes agree exactly in $d$; the result is exchange symmetric. The nonlocal coefficient equals minus the pole. Premature $d=4$ is refused and would shift the local parts. Each of the four mutations is rejected.

• 4D pole. $z_{D11}=i(-1/160,-1/480,1/80,1/240,-1/120)=i(\frac{1}{240}{k}_1-\frac{1}{120}{k}_2)$, which is transverse.

• Full row. $p_m{G_{D11}}^{(mn)(rs)}=0$ holds identically in $d$, for the 4D and the evanescent parts. D11 therefore decouples from the slot by itself; D01 and D03 decouple only as a sum.

  • $X_{D01+D03+D11}=0$ for all seven external assignments. This includes $\hat{c},\tilde{h}$ (H2), where $X_{D01}=-X_{D03}=p^4\tilde{a}\cdot \tilde{b}\hat{c}\cdot p/16$.

  • No evanescent structure survives, so D11 does not reopen the evanescent question and no counterterm revision is needed.

  • The summed 4D pole is $i(\frac{41}{80}{k}_1-\frac{43}{120}{k}_2)$, which is transverse.

  • Caveat: D11 phase and symmetry-factor mutations are invisible to the slot. They are tested by M2 instead.

  • The summed evanescent poles of the hh Green function (8 structures) are nonzero but transverse. Renormalising that Green function itself is outside the projection and not claimed.

M2: DISCHARGED (4D components); route B: 100 component pairs, 0 mismatches, constant -I/4; c_FT = I*pi\^{}2 (DR bubble) = I*pi\^{}2 (-r).

{p2.6cmp2.6cmp2.6cmp1.2cmp1.6cm} assignment $(c,h)$ & $X_{D01}$ & $X_{D03}$ & $X_{D11}$ & sum
Q4S (declared, R2) & p.p\^{}2*(d*a.b*c\^{}.p - 3*a.b*c\^{}.p + a.c*b\^{}.p + a\^{}.p*b.c - a\^{}.p*b .c  + 2*a .b *c\^{}.p - a .c *b\^{}.p)/(16*(d - 1)) & -p.p\^{}2*(d*a.b*c\^{}.p - 3*a.b*c\^{}.p + a.c*b\^{}.p + a\^{}.p*b.c - a\^{}.p*b .c  + 2*a .b *c\^{}.p - a .c *b\^{}.p)/(16*(d - 1)) & 0 & 0
QdS (hat) & {p.p\

^{}2*(d*a\^{}.b\^{}*c\^{}.p - 3*a\^{}.b\^{}*c\^{}.p + a\^{}.c\^{}*b\^{}.p + a\^{}.p*b\^{}.c\^{})/(16*(d - 1))} & -p.p\^{}2*(d*a\^{}.b\^{}*c\^{}.p - 3*a\^{}.b\^{}*c\^{}.p + a\^{}.c\^{}*b\^{}.p + a\^{}.p*b\^{}.c\^{})/(16*(d - 1)) & 0 & 0
c tilde, h hat & 0 & 0 & 0 & 0
c tilde, h mixed & 0 & 0 & 0 & 0
c hat, h mixed & 0 & 0 & 0 & 0
all tilde & 0 & 0 & 0 & 0
c hat, h tilde & a .b *c\^{}.p*p.p\^{}2/16 & -a .b *c\^{}.p*p.p\^{}2/16 & 0 & 0
Full-row verdict: DECOUPLED (proof, full row): p_m G = 0 exactly in d for D11 by itself and for D01 + D03 as a sum; no evanescent structure survives in the slot.

17 = P.5 calculation, third step; D11 is NOT evaluated; no slot is complete.{ p17 keeps the p14r contract and the p15 and p16 calculation layers (byte-identical) and adds: (1) a deterministic analysis of whether the evanescent ($\tilde{g}$) UV poles of D01 and D03 reach the ch-projected slot, with controls and mutations; (2) the explicit coordinates of the 4D poles in the p14r $h\cdot h$ basis and the constraint they satisfy; (3) the fixes G1–G4 and G6 of the p16 independent review (one disclosed dred.py revision: the guarded loader parse_d). D11 stays NOT evaluated (M2 undischarged), D11_matching_verified=false and quantum_inputs_evaluated stays not earned. O.1 is NOT_CLOSED, symmetry_restored=bulk_symmetry_restored=false, the insertion stays frozen, XV-B and XIV-J stay closed, Route B is not discharged, XVI-P is unregistered, and no $c\delta$ term is added to ${u^R}_{\mu }$.}

Reading the ledger ledgers/P13_ch_dependency_rows.json (a computed control, not an assertion), the graphs D01, D03 occur in exactly one row: $(S_0,{{\Gamma }_1}^{[2]})s_{2}K:K\cdot c$, with ${\Gamma }_1$ pattern $h^2$. The same row also holds D02, D12 (zero, p15) and D11 (not evaluated). The slot functional is

The operator $sh$ is read off the $K\cdot c$ monomial of the frozen ${S_{\Psi }}^{(0)}$ (16 nonzero components). Its covariant fit is exact and unique: $(sh{)}_{(mn)\nu }=\frac{i}{2}({\eta }_{m\nu }{p}_n+{\eta }_{n\nu }{p}_m)$. It is therefore LONGITUDINAL in $(mn)$.

• H1. $sh$ is longitudinal (above), so $X_{hh}$ depends on $G$ only through $p_m{G}^{(mn)(rs)}$. Because $p\in$ QdS, $p\cdot \tilde{g}=0$, so $\tilde{g}$ on the pair $(mn)$ drops out.

• H2/H3. $p_m(G_{D01}+G_{D03}{)}^{(mn)(rs)}=0$ identically in $d$. This holds for the 4D part and for the evanescent part separately: each of the six structures of $p_{m}G$ (for example ${\tilde{g}}_{rs}{p}_n$: $-\frac{i{p}^4}{8(d-1)}$ from D01 and $+\frac{i{p}^4}{8(d-1)}$ from D03) cancels exactly. Hence $X_{D01+D03}=0$ for all six external assignments (Q4S, QdS, mixed $\hat{}/\stackrel{~}{}$). With $D02=D12=0$, the hh slot receives nothing from the evaluated graphs.

• H4 (controls). $X_{D01}$ and $X_{D03}$ are each nonzero, and so are their evanescent parts alone. The cancellation is therefore a property of the SUM (the ghost loop cancels the longitudinal part of the bubble), not of each graph. The map is not trivially zero.

• Order. At one loop the ch projection is linear in ${\Gamma }_1$, so no product of evanescent poles occurs. The cancellation is exact in $d$ and not merely $O(ϵ)$.

Mutations, each rejected (summed slot becomes nonzero): wrong relative sign of D03; D03 doubled (symmetry factor); D03 with its evanescent part dropped (naive 4D); the trace operator ${\eta }_{mn}{p}_{\nu }$ in place of $sh$.

Verdict: DECOUPLED (proof), as the sum D01 + D03 (+ D02 = D12 = 0), and not graph by graph. No evanescent structure survives in the slot. No evanescent $h\cdot h$ counterterm is added, so the counterterm quotient is unchanged and no contract file changes for this. The summed evanescent poles of the hh Green function are nonzero (eight structures, recorded in the certificate) but transverse. Renormalizing the hh Green function itself would need evanescent $h\cdot h$ counterterms; that matters only outside this projection and is not claimed. D11, in the same row, is open.

The five p14r $h\cdot h$ directions span the five pair-symmetric 4-derivative 4D structures (rank 5), so span membership is automatic. The sharper statement is a constraint. Longitudinality $p_m{G}^{(mn)(rs)}=0$ imposes rank-3 linear conditions on the coordinates $z$, which leaves a 2-dimensional transverse subspace with basis $k_1=(\frac{1}{2},-\frac{1}{2},-1,1,0)$ and $k_2=(1,0,-2,0,1)$. The individual pole coordinates are not transverse:

Their sum is transverse:

This is the Ward-type statement the ch identity needs at this order. It is incomplete without D11.

Every graph is exported as $G={\kappa }^2{N}_{ϵ}\hat{G}$ with $N_{ϵ}=i{S}_{ϵ}/16{\pi }^2$. For D01 and D03, $\hat{G}=-i\mathrm{pref}/{\kappa }^2\int$ with real export prefactors (${\kappa }^2/2$, $-{\kappa }^2$), so $A(d)$ carries an overall $i$. D13's exported prefactor is $i{\kappa }^2$, so its $A(d)$ is real. Because $sh=i(\dots )$, $X_{hh}$ is real like the D13 slot. The D01/D03 relative phase is tested by H3 (mutation M1 breaks it). The D13-versus-hh relative phase is not tested by a nonzero identity, since both slot contributions vanish. No physical overall phase is asserted.

G2: dred.parse_d (locals $\{d:$ opaque $d\}$) and dred.assert_opaque_d close the plain-Symbol('d') hazard of re-parsed strings (disclosed contract revision; no value affected). G1/G3: the guard docstring and the disclosure now state exactly what is refused at runtime and what is excluded by the extended static AST scan: replace with any function query, sympify/S/parse_expr of non-sympy data without the guarded locals, builtin eval/exec, srepr round-trips, .func(*args) rebuilds, Poly(..).eval, lambdify. The scan is a pattern check, not a proof. G4: the change log lists the p17 reference-values file as added.

Verdict: DECOUPLED (proof): as the sum D01 + D03 (+ D02 = D12 = 0); not graph by graph. Surviving evanescent structures in the slot: none. Transverse kernel (p14r coordinates): (1/2, -1/2, -1, 1, 0); (1, 0, -2, 0, 1). D11: NOT evaluated.

16 = P.5 calculation, second step; D11 is NOT evaluated; no slot is complete.{ p16 keeps the p14r contract and the p15 calculation layer (byte-identical) and adds: (1) a deterministic proof that the evanescent D13 pole (structure E2) decouples from the ch-projected slot, with mutation controls; (2) Batch C, the graphs D01 and D03 of the ${\kappa }^{2}hh$ slot, under the same frozen prescription (DRED R1–R6, Feynman $+i0$, MS-bar, S0, $d$ opaque); (3) the fixes F3–F5 of the p15 independent review. D11 stays NOT evaluated (M2 undischarged), D11_matching_verified=false and quantum_inputs_evaluated stays not earned. O.1 is NOT_CLOSED, symmetry_restored=bulk_symmetry_restored=false, the insertion stays frozen, XV-B and XIV-J stay closed, Route B is not discharged, XVI-P is unregistered, and no $c\delta$ term is added to ${u^R}_{\mu }$.}

In p15 D13 has an evanescent structure $E{2}_{(mn)\nu }=({\tilde{g}}_{m\nu }{p}_n+{\tilde{g}}_{n\nu }{p}_m)p^2$ with $A_{E2}(d)=3/(16(d-1))$ and UV pole $1/16$; p14r declares no evanescent counterterm. D13 enters the ch projection only through the row $(S_0,{{\Gamma }_1}^{[2]})s_{1}h:h^2$:

where $E$ is the operator of the $h^2$ monomials of ${S_{\Psi }}^{(0)}$ (linearized Einstein–Hilbert). Its exact covariant form is fitted from the frozen density and verified on all components; the fit is unique.

• L1. $p_m{E}^{(mn)(rs)}=0$ as a covariant identity. Only $g_4\cdot p=p$ is used; no trace is taken, so the identity holds exactly in $d$.

• L2. $E2$ is longitudinal in $(mn)$.

• L3. Therefore $E2\cdot E=0$ identically. The slot value is $0$ for every external assignment (Q4S, QdS, mixed $\hat{}/\stackrel{~}{}$ components), exactly in $d$ and not merely $O(ϵ)$.

• L4 (control). The non-longitudinal $T1$ and $E1$ give nonzero slot values, so the map is not trivially zero. $A_{T1}=A_{E1}=0$ for D13, so D13's whole ch-slot contribution is $0$.

Mutation controls: the gauge-fixed operator $q^{2}P$ in place of $E$, $E1$ in place of $E2$, and a wrong index slot each make the $E2$ slot nonzero and are rejected. At one loop the ch projection is linear in ${\Gamma }_1$, so no product of evanescent poles arises.

Outcome: DECOUPLED (proof). No evanescent counterterm is added and no contract file changes for E2. The $E2$ pole remains a UV pole of the $ϵ$-scalar component of the $K^{\prime }\cdot c$ Green function (R6: retained and recorded). Its renormalization matters only outside this projection, and it is not claimed here. D13 stays regulated, with a ch-slot contribution of $0$.

Finding C1. For rank-4 numerators, the p14r per-component R3 lift produces component sums quadratic in $\tilde{g}$ (D03: 10 of 100 components). After the component sums, their abstract value cannot be recovered, because the linear realization $\tilde{g}=(4-d)M$ is not faithful there. A finite component basis cannot carry a projector of non-integer trace.

Batch C therefore uses a covariant route that applies the same rules R1–R6 with free external indices only:

1. The exact covariant 4D numerator is fitted from the frozen export and verified as a polynomial identity on all 100 components. It is unique (upper external indices, off-diagonal weight 2).

2. The lift $\eta \to g_4$ is made.

3. Two independent loop routes are used: Feynman parameters with symmetric integration, and the frozen dred.reduce_tensor with the J library and S0. They agree exactly in $d$.

4. Then $\hat{g}=g_4-\tilde{g}$ on the free indices.

D13 is reproduced exactly by this route (control), and the rules are unchanged (no contract change).

Further controls:

• The integrands are regenerated from $S_{\Psi }$ (D01 from the ${\ell }_3$ density, D03 from the ghost vertices) and equal the export on 100 of 100 components.

• The results are exchange symmetric.

• The nonlocal $\log$ coefficient equals minus the pole.

• The 4D poles lie in the span of the five p14r $h.h$ counterterm directions.

Seven mutations are each rejected: ghost-loop sign, symmetry factor, premature $d=4$, dropped $\tilde{g}$, external index position, off-diagonal weight, and the component route with the linear realization.

{p0.7cmp2.9cmp3.7cmp1.4cmp3.3cm} graph & structure & $A(d)$ (coefficient of MB) & UV pole & MS-bar finite
D01 & g4_mn g4_rs (p.p)\^{}2 & I*(-2*d\^{}2 - 10*d - 9)/(16*(d\^{}2 - 1)) & -27*I/80 & I*(405*L - 982)/1200
D01 & g4_mn gt_rs (p.p)\^{}2 & I*(5*d + 6)/(16*(d\^{}2 - 1)) & 13*I/120 & I*(523 - 195*L)/1800
D01 & g4_mn p_r p_s (p.p)\^{}1 & I*(d\^{}2 + 12*d + 12)/(16*(d\^{}2 - 1)) & 19*I/60 & I*(724 - 285*L)/900
D01 & g4_mr g4_ns (p.p)\^{}2 & I*(-6*d\^{}2 + 4*d + 9)/(16*(d\^{}2 - 1)) & -71*I/240 & I*(1065*L - 1946)/3600
D01 & g4_mr gt_ns (p.p)\^{}2 & I*(-2*d - 1)/(16*(d\^{}2 - 1)) & -3*I/80 & I*(45*L - 118)/1200
D01 & g4_mr p_n p_s (p.p)\^{}1 & I*(6*d\^{}2 - 5*d - 10)/(16*(d\^{}2 - 1)) & 11*I/40 & I*(97 - 55*L)/200
D01 & g4_rs gt_mn (p.p)\^{}2 & I*(5*d + 6)/(16*(d\^{}2 - 1)) & 13*I/120 & I*(523 - 195*L)/1800
D01 & g4_rs p_m p_n (p.p)\^{}1 & I*(d\^{}2 + 12*d + 12)/(16*(d\^{}2 - 1)) & 19*I/60 & I*(724 - 285*L)/900
D01 & gt_mn gt_rs (p.p)\^{}2 & -I/(16*d\^{}2 - 16) & -I/240 & I*(15*L - 46)/3600
D01 & gt_mn p_r p_s (p.p)\^{}1 & I*(-7*d - 8)/(16*(d\^{}2 - 1)) & -3*I/20 & I*(90*L - 241)/600
D01 & gt_mr gt_ns (p.p)\^{}2 & -I/(16*d\^{}2 - 16) & -I/240 & I*(15*L - 46)/3600
D01 & gt_mr p_n p_s (p.p)\^{}1 & I*(3*d + 2)/(16*(d\^{}2 - 1)) & 7*I/120 & I*(277 - 105*L)/1800
D01 & gt_rs p_m p_n (p.p)\^{}1 & I*(-7*d - 8)/(16*(d\^{}2 - 1)) & -3*I/20 & I*(90*L - 241)/600
D01 & p_m p_n p_r p_s & I*(-13*d\^{}2 - 2*d + 8)/(16*(d\^{}2 - 1)) & -13*I/15 & I*(780*L - 1597)/900
D03 & g4_mn g4_rs (p.p)\^{}2 & I*(d + 3)/(16*(d - 1)) & 7*I/48 & I*(50 - 21*L)/144
D03 & g4_mn gt_rs (p.p)\^{}2 & -I/(8*d - 8) & -I/24 & I*(3*L - 8)/72
D03 & g4_mn p_r p_s (p.p)\^{}1 & -3*I/(8*d - 8) & -I/8 & I*(3*L - 8)/24
D03 & g4_mr g4_ns (p.p)\^{}2 & I*(d - 3)/(8*(d - 1)) & I/24 & I*(2 - 3*L)/72
D03 & g4_mr gt_ns (p.p)\^{}2 & I/(8*(d - 1)) & I/24 & I*(8 - 3*L)/72
D03 & g4_mr p_n p_s (p.p)\^{}1 & I*(7 - 2*d)/(16*(d - 1)) & -I/48 & I*(3*L + 4)/144
D03 & g4_rs gt_mn (p.p)\^{}2 & -I/(8*d - 8) & -I/24 & I*(3*L - 8)/72
D03 & g4_rs p_m p_n (p.p)\^{}1 & -3*I/(8*d - 8) & -I/8 & I*(3*L - 8)/24
D03 & gt_mn p_r p_s (p.p)\^{}1 & I/(4*(d - 1)) & I/12 &

I*(8 - 3*L)/36
D03 & gt_mr p_n p_s (p.p)\^{}1 & -3*I/(16*d - 16) & -I/16 & I*(3*L - 8)/48
D03 & gt_rs p_m p_n (p.p)\^{}1 & I/(4*(d - 1)) & I/12 & I*(8 - 3*L)/36
D03 & p_m p_n p_r p_s & I*(d - 2)/(4*(d - 1)) & I/6 & I*(5 - 3*L)/18
Status: D01: regulated. D03: regulated. E2 (D13): DECOUPLED (proof).

With $A(d)$ the coefficient of $\mathrm{MB}=1/ϵ+2-L$, $L=\log (-p^2/{\mu }^2-i0)$, in units of $N_{ϵ}$: the UV pole is $A(4)$, and the MS-bar finite part is $A(4)(2-L)-2A^{\prime }(4)$. Status: D01 and D03 are regulated. Their 4D sectors are renormalized by poles in the p14r $h.h$ span. Their evanescent ($\tilde{g}$) poles are NONZERO and retained, and no evanescent $h.h$ counterterm is declared. The $hh$ slot (D01+D02+D03+D11+D12) stays incomplete because D11 is not evaluated.

• F3 (one disclosed contract revision, dred.py). The opaque $d$ now refuses {expr.replace(d,$\cdot$)}, pattern matching on $d$ and the lambdify/code-printer paths outside substitute_d. Paths without a hook (callable-query replace, Poly.eval, string round trips) are excluded by a static AST scan. The P13–P15 certificate values are unchanged against the p15 reference values.

• F4. The page footer comes from one version variable. The PDF readback also covers the D13, Batch A, Batch C and E2 value blocks.

• F5. Every changed file, contract and non-contract, is listed in change_log_vs_p15 of the release status.

independent_rebuild_passed_p15=true is bound to the p15 tree 79f53752{} and zip 54325c1f{} (same box). The p16 tree's own independent rebuild is pending.

15 = P.5 calculation, first batch; D01, D03, D11 are NOT evaluated; no slot is complete.{ p15 evaluates the Batch A graphs D02, D12, D14a, D14b, D14c and the Batch B graph D13 under the frozen p14r prescription: DRED R1–R6 with the $\hat{g}/\tilde{g}$ algebra, contractions before the Laurent expansion, $d$ kept symbolic and evanescent parts retained; Feynman $+i0$; MS-bar with $N_{ϵ}=i{S}_{ϵ}/(16{\pi }^2)$, $S_{ϵ}=(4\pi {)}^{ϵ}{e}^{-{\gamma }_Eϵ}$. The declared action is massless (no $\varphi$ mass is declared). quantum_inputs_evaluated, projected_defect_computed, projected_identity_verified and projected_restoration_verified remain not earned, and bulk_symmetry_restored=false. O.1 is NOT_CLOSED, the insertion stays frozen, XV-B and XIV-J stay closed, and Route B is not discharged. The new flags batchA_evaluated and D13_evaluated each carry an execution, a claim and a scope part.}

The library (P15_integral_library.json) provides $A_0$, $B_0$ and the tensor reduction in $\hat{g}$. It is certified exactly in $d$ against Feynman parameters, and numerically at $d=2.6,3.3,3.7$.

The scaleless rule is stated, not assumed:

Its UV/IR reading is as follows. For $\omega =0$ the zero is the cancellation $1/{ϵ}_{\mathrm{UV}}-1/{ϵ}_{\mathrm{IR}}$. For $\omega \ne 0$ it discards a pure power divergence, and neither the UV part nor the IR part has a pole at $d=4$. Applying S0 to an integral that carries a scale is an error, and Missing is never replaced by zero by convention.

All five graphs are vanishing under the declared prescription: there is no UV pole, the finite part is $0$, and no p14r counterterm is needed. Every scaleless member is a pure power divergence ($\omega =2,4$), so the zero is not a $1/{ϵ}_{\mathrm{UV}}-1/{ϵ}_{\mathrm{IR}}$ cancellation.

• IR-regulator mass (analysis only). With a mass the poles are proportional to ${\kappa }^2{m}^{4}hh$ and ${\kappa }^2{m}^2{p}^{2}hh$, which are order-0/2 contacts excluded by N4, or to $\kappa m^{4}h$, for which the ${\kappa }^1$ basis is empty. All of them vanish as $m\to 0$.

• Analytic regulator. It does not separate the UV from the IR here.

• Cross-check. Two routes agree: the frozen dred lift with S0, and an independent symmetric tensor integration.

• D12. D12 (${\kappa }^{2}hh$) is kept separate from the order-$\kappa$ one-point slot (D14a–c). D12_status moves from Missing to vanishing_under_declared_prescription. This is derived under P's own prescription; XVI-O's value is not imported.

Write $G_{(mn)\nu }(p)=V_{{\Gamma }_1}[{K^{\prime }}_{(mn)}(-p),c^{\nu }(p)]={\kappa }^2{N}_{ϵ}\hat{G}$, with $\hat{G}={\sum }_i{A}_i(d)T_i{B}_0$, $B_0=1/ϵ+2-L$ and $L=\log (-p^2/{\mu }^2-i0)$. The structures are

together with the evanescent $E_1={\tilde{G}}_{mn}{p}_{\nu }{p}^2$ and $E_2=({\tilde{G}}_{m\nu }{p}_n+{\tilde{G}}_{n\nu }{p}_m)p^2$. The coefficients are

This gives:

• UV poles: $T_2$: $3/16$, $T_3$: $-1/4$, $E_2$: $1/16$.

• MS-bar finite parts: $T_2$: $1/3-\frac{3}{16}L$, $T_3$: $-5/12+\frac{1}{4}L$, $E_2$: $1/6-\frac{1}{16}L$. The nonlocal part is $-(\text{pole})\cdot L$, and the local part is a cubic polynomial.

• Counterterm map. The 4D pole lies in the span of the three p14r $c\cdot K^{\prime }$ directions, with coordinates $(0,3/8,-1/4)$. The local ambiguity is a finite renormalization along the same directions.

• {$d$-dependence.} The $(d-4)\times$pole terms shift the local part ($T_2$: $-1/24$, $T_3$: $+1/12$). Setting $d=4$ first gives a different MS-bar finite part.

• Evanescent pole and status. The evanescent pole ($E_2$) is nonzero, and the p14r contract declares no evanescent counterterm. D13's status is therefore regulated. Its 4D sector is renormalized; the evanescent pole is recorded but not absorbed, and stays an open contract item.