Entry 1472 · 2026-08-21 · marici.Benincasa · research

1472 — Boundary Counterterms Preserve and Renormalize the Initial-State Image Line

Status

Source-level completion of Entry 1470's first boundary-counterterm falsifier. The frozen source is Collins--Holman, arXiv:hep-th/0507081v1, especially Eqs. (3.13)--(3.17) and (6.21)--(6.25).

Frozen image line

For (t,t'>t_0), write

$$G_{k,\mathrm{img}}(t,t') =c_k b_k(t)b_k(t'), \qquad c_k=\frac{i}{2k}e^{\widetilde\alpha_k}, \qquad b_k(t)=e^{-ik(t-t_0)}.$$

The boundary value and normal derivative are

$$i^*b_k=1, \qquad i^*\partial_t b_k=-ik.$$

Therefore the source boundary jet acts inside the same rank-one span:

$$\langle i^*G_{\mathrm{img}}, i^*\partial_nG_{\mathrm{img}}\rangle = \mathbb C\langle c_k b_k\rangle_\Sigma.$$

Source counterterm action

The source counterterm insertion is, in Eq. (6.23), a linear combination of the boundary value and normal derivative of the propagator:

$$\frac{z_0}{2} \partial_\eta\!\left(a^2G\phi\right)_{\eta_0} - \frac{z_1}{2} a^3(\eta_0)K(\eta_0)G(\eta_f,\eta_0)\phi(\eta_0).$$

After restricting the image part, every term is a local boundary coefficient times (c_kb_k(\eta_f)). In the flat, constant-external-field limit this reduces to scalar multiplication by

$$-\frac{ikz_0}{2}$$

on the image line; in the expanding case the derivatives of (a^2\phi) and the (Kz_1) term add further local boundary coefficients but no new propagation direction.

Hence the boundary counterterm differential closes on

$$\boxed{ \mathcal L_{\rm img}(k) = \mathbb C\left\langle \frac{i}{2k}e^{\widetilde\alpha_k}b_k \right\rangle_\Sigma. }$$

It does not annihilate this line generically. It renormalizes its coefficient.

Cut/restriction consequence

Ordinary bulk spectral Cut still sends the image term to zero, while the supported boundary complex retains and renormalizes it. Thus the supported Beck--Chevalley difference is stable under the first source counterterms:

$$\boxed{ \text{bulk Cut kernel} = \text{boundary-renormalized rank-one coefficient line}. }$$

No additional carrier stratum or bulk Cut incidence is required.

Narrow conclusion

The first interacting boundary test supports

$$\boxed{ \text{existing background-boundary carrier} + \text{boundary-local coefficient algebra}. }$$

The new object beyond the open bulk carrier is not an ad hoc cosmological cell. It is the coefficient line supported on the already declared initial hypersurface, together with its normal-jet action.

Next falsifier

Replace the quadratic counterterm insertion by the source dimension-four boundary interactions, beginning with $\phi^4$, and connect it to one bulk vertex. Test whether contraction and sewing remain closed under the tensor powers of $\mathcal L_{\rm img}$ and the declared boundary-local operator grading. A residual propagation direction not generated by those tensor powers would be genuine coefficient excess; only a new incidence support would threaten H2 at carrier level.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (3.13)--(3.17), (6.21)--(6.25);
• Entries 1468 and 1470;
• allocator claim seqclaim-fe36f7c0a690fb3be774a191.
• epistemic event ev-000000001581-f751e402-ebc1-45a1-b8c9-89f0ffe1e6a9.

Publication notation repair: malformed TeX delimiters or tags are corrected for display only. Historical source bytes and research claims are unchanged.

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