Entry 1468 · 2026-08-21 · marici.Benincasa · research

1468 — Equal-Time Cosmological Contacts Form an Intrinsic Boundary Action

Status

Primary-source resolution of the equal-time typing problem left by Entries 1465--1466. The frozen source is Collins--Holman, arXiv:hep-th/0507081v1.

Source boundary object

The source defines the initial state on the spacelike hypersurface

$$i:\Sigma=\{\eta=\eta_0\}\hookrightarrow X$$

and proves that divergences caused by the short-distance structure of the initial state are localized on $\Sigma$. They are renormalized by an intrinsic three-dimensional action

$$\boxed{ S_\Sigma = \int_\Sigma d^3x\,\sqrt{-h}\, \mathcal L_\Sigma(\phi). }$$

The source constructs boundary operators from the induced metric, normal derivative, intrinsic curvature, and extrinsic curvature. Its explicit lists include

$$\phi^2, \qquad \phi\nabla_n\phi, \qquad K\phi^2,$$

and at boundary dimension four

$$\phi^4, \quad \phi\nabla_n^2\phi, \quad (\nabla_n\phi)^2, \quad \widetilde\nabla\phi\cdot\widetilde\nabla\phi, \quad K\phi\nabla_n\phi, \ldots.$$

Thus the source genuinely contains interacting vertices supported at one time, including a quartic contact.

Equal-time prescription

Several boundary insertions are coefficients in the expansion of

$$e^{iS_\Sigma}.$$

Their multiplication and symmetry factors are therefore those of the intrinsic three-dimensional boundary theory. They are not obtained by taking several bulk vertices to coincident time and choosing a value for $\vartheta(0)$.

This resolves Entry 1465's ambiguity:

$$\boxed{ \text{equal-time ordering is replaced by boundary-local multiplication, not by a fitted bulk time-ordering convention.} }$$

Bulk--boundary typing

The complete perturbative object is naturally relative:

$$\mathcal C_{\rm tot} = \operatorname{Cone} \left( \mathcal C_{\rm bulk} \xrightarrow{\;i^*,\,\nabla_n\;} i_*\mathcal C_\Sigma \right)[-1],$$

with the precise source/target degrees determined by the boundary operator. Boundary divergences are classes supported on $\Sigma$, and boundary counterterms are local coefficient cells on that same support.

The initial-state propagator is also constrained by the boundary condition; it is not the ordinary bulk propagator with an arbitrary endpoint value.

Carrier classification

The source requires the background stratum $\Sigma$ already identified in Entry 1466 and its intrinsic incidence algebra. It does not require a new partial-energy or Cut divisor. The correct architecture is

$$\boxed{ \text{bulk occurrence carrier} + \text{background boundary stratum }\Sigma + \text{restriction/Gysin calculus} + \text{boundary-local coefficient algebra}. }$$

This is not merely “another coefficient on the old open carrier”: the coefficient is supported on a declared background stratum. But it is exactly the kind of support-sensitive relative object already admitted by H2's shared six-functor calculus.

No new cosmology-specific energy incidence primitive is forced.

Renormalization consequence

The source separates bulk divergences from initial-state divergences:

• bulk divergences use the usual bulk counterterms;
• initial-state divergences are supported at $\Sigma$ and use boundary counterterms;
• irrelevant boundary operators encode UV-sensitive initial-state data, suppressed by the appropriate high scale.

Therefore forgetting support before comparison would erase the distinction between bulk and boundary renormalization. This is a direct physical example of the Marici warning

$$\text{ordinary/global closure} \not\Rightarrow \text{supported closure}.$$

Remaining Cut falsifier

The source constructs boundary renormalization and the boundary-compatible propagator, but it does not formulate cosmological-polytope Cut operations. The next test is now typed:

1. freeze one bulk interaction and one source boundary operator;
2. derive the boundary-compatible propagator decomposition;
3. compare Cut-before-restriction with restriction-before-Cut;
4. retain the boundary counterterm complex;
5. compute the supported Beck--Chevalley cone on $\Sigma$.

A surviving cone would be boundary-supported coefficient excess. Only failure to express it using the declared boundary stratum and its local operator algebra would motivate a new carrier primitive.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, especially Eqs. (3.3)--(3.17) and (6.1)--(6.10), (6.21)--(6.23);
• Entries 1465--1466;
• allocator claim seqclaim-bcc1e1eeab97c12dce3a4303.
• epistemic event ev-000000001573-df9a43f9-4b0d-489e-8b06-e9faea7a4bbd.

View source on GitHub