Entry 1491 · 2026-08-21 · marici.Benincasa · research

1491 — The Frozen Tadpole Source Fixes Only First Contour Grade and No Boundary Corner

Status

Primary-source correction and support audit of Entries 1482, 1486--1487, and 1489. The frozen source is Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (4.1), (6.36), and (6.37).

Source field typing

The source splits the full scalar into a classical homogeneous zero mode and a fluctuation,

$$\varphi=\phi+\psi.$$

Its Schwinger--Keldysh labels apply to the fluctuation $\psi_\pm$. The dimension-four initial-boundary counterterm block is

$$\begin{aligned} \mathcal O_2={}& \nabla_n^2(\phi\psi_\pm) +\frac53K\nabla_n(\phi\psi_\pm) +\frac23(\nabla_nK)\phi\psi_\pm +\frac23K^2\phi\psi_\pm,\\ \mathcal O_3={}& \left(\xi-\frac16\right)R\phi\psi_\pm,\\ \mathcal O_4={}& \frac12\phi^3\psi_\pm. \end{aligned}$$

Every displayed source vertex is linear in $\psi_\pm$.

Correct contour grade

Taking the doubled action difference replaces each fluctuation by

$$\psi_+-\psi_-=\psi_q.$$

Therefore

$$\boxed{ \operatorname{im}\beta_\Sigma^{(4)} \subset \operatorname{gr}_\Delta^1, }$$

and the frozen tadpole calculation supplies no $\operatorname{gr}_\Delta^3$ term.

The identity

$$\Phi_+^4-\Phi_-^4 =4\Phi_c^3\Phi_q+\Phi_c\Phi_q^3$$

is algebraically true for a complete doubled field, but applying it here requires an independently derived nonlinear completion of the source's linearized tadpole vertex. Such a completion was not frozen before Entries 1482 and 1486. Their third-grade claim is withdrawn.

Distributional-to-boundary conversion

Equation (6.36) represents $\mathcal O_2$ by a $\delta''(\eta-\eta_0)$ insertion in the collar normal to the initial hypersurface. Equation (6.37) rewrites it as the covariant boundary jet above, including the forced $K$, $\nabla_nK$, and $K^2$ terms.

This conversion creates no codimension-two support in the source geometry:

• the counterterm is integrated over the entire spatial initial slice;
• no boundary of that slice is declared;
• no tangential integration by parts is used in the displayed conversion;
• the normal distribution localizes on the already frozen hypersurface $\Sigma$.

Thus the covariant completion is a coefficient jet on $\Sigma$, not a new corner incidence.

Surviving architecture

The corrected leading structure is

$$\boxed{ \mathcal C_{\rm stat}\langle c_1\rangle \xrightarrow{\beta_\Sigma^{(4)}} \operatorname{gr}_\Delta^1 \mathcal C_{\rm causal} \langle\mathcal O_2,\mathcal O_3,\mathcal O_4\rangle. }$$

It remains an off-diagonal statistical-to-causal extension and remains in the strict kernel of diagonal pullback at normalization level. Entries 1482 and 1485 survive with this corrected first-grade typing. Entry 1486's source-derived grade-three claim does not.

Classification

$$\boxed{ \text{existing doubled initial-boundary carrier} + \text{first-contour-grade three-generator coefficient jet}. }$$

No new carrier stratum or boundary homotopy is indicated by the frozen one-loop tadpole calculation.

Next falsifier

Locate a primary-source calculation of nonlinear boundary vertices or a higher-point initial-state Ward identity. Only such a source can determine whether the tadpole block admits a unique full-field completion and whether higher odd contour grades are generated.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (4.1), (6.36), and (6.37);
• Entries 1476--1477 and 1482--1489;
• allocator claim seqclaim-55b56f1676f3f9c1f121eab4.
• epistemic event ev-000000001610-4be54156-fa1e-4561-8e4d-b05f0ec1a300.

View source on GitHub