Entry 1477 · 2026-08-21 · marici.Benincasa · research

1477 — Boundary Running Is an Affine Extension from the Initial-State Line

Status

Leading-order renormalization-group typing of Entry 1476's dimension-four boundary coefficient block. The frozen source is Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (6.39)--(6.43).

Source running

Let

$$Z_\Sigma^{(4)} = \mathbb C\langle z_2,z_3,z_4\rangle$$

denote the three labelled counterterm coefficients associated respectively with the covariantly completed second normal jet, curvature coupling, and cubic boundary interaction. The source derives

$$\widehat\beta_2 = \frac{\lambda_Rc_1}{32i\pi^2}+\cdots,$$

$$\widehat\beta_3 = \frac{\lambda_Rc_1}{16i\pi^2}+\cdots,$$

$$\widehat\beta_4 = \frac{\lambda_R^2c_1}{16i\pi^2}+\cdots.$$

At this order, the displayed running is not an autonomous matrix acting on $(z_2,z_3,z_4)$. It is a source map from the leading initial-state coefficient line:

$$\boxed{ \beta_\Sigma^{(4)}: \mathbb C\langle c_1\rangle \longrightarrow Z_\Sigma^{(4)}, \qquad c_1\longmapsto \begin{pmatrix} \lambda_R/(32i\pi^2)\\ \lambda_R/(16i\pi^2)\\ \lambda_R^2/(16i\pi^2) \end{pmatrix}c_1. }$$

Triangular coefficient object

The renormalized boundary sector is therefore an affine/triangular extension

$$0 \longrightarrow Z_\Sigma^{(4)} \longrightarrow \mathcal E_{c_1} \longrightarrow \mathbb C\langle c_1\rangle \longrightarrow0,$$

with the beta vector supplying the extension data at the calculated order. It should not be replaced by three unrelated running constants, nor by one scalar second-normal coefficient.

The ratios of the first two components are fixed:

$$\widehat\beta_3=2\widehat\beta_2+\cdots,$$

while the cubic component carries one additional bulk coupling. This records the distinct field-degree variance of $\mathcal O_4$.

Support and filtration

The map preserves the initial hypersurface support and the boundary operator dimension. It mixes normal order with curvature and field degree inside the same dimension-four coefficient grade, but it does not mix into a bulk or new incidence support.

Thus the source yields

$$\boxed{ \text{boundary filtration preserved} + \text{nontrivial affine coefficient extension}. }$$

This is stronger than mere locality: it exhibits the explicit mechanism by which ultraviolet initial-state data is converted into a local boundary coefficient module.

Scope

The source gives the displayed leading terms only. It does not establish that higher-loop running remains triangular or that no $z_i$-to-$z_j$ mixing appears later.

Next falsifier

Compare this affine RG extension with Cut sewing. For a labelled internal edge ending on $\Sigma$, test whether applying the beta map before Cut and after resolved endpoint restriction gives the same three-component vector. Failure by a local boundary homotopy would be coefficient extension data; failure requiring new support would threaten the shared carrier calculus.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (6.39)--(6.43);
• Entries 1475--1476;
• allocator claim seqclaim-4e4b0768027cefabb4d3c5d3.
• epistemic event ev-000000001589-a6a8bf92-026c-4c34-b046-70d5066314c9.

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