Entry 1470 · 2026-08-21 · marici.Benincasa · research

1470 — Initial-State Image Propagation Is the Kernel of Bulk Cut

Status

Explicit source-level Cut/restriction calculation for the free internal line of Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (3.13)--(3.17). This is the first nonzero supported comparison class in the Big-Bang/background-boundary lane.

Frozen propagator

For times $t,t'>t_0$, the source massless propagator with general initial boundary condition is

$$G_k(t,t') = G_{k,\mathrm{vac}}(t,t') + G_{k,\mathrm{img}}(t,t'),$$

where

$$G_{k,\mathrm{vac}} = \vartheta(t-t')\frac{i}{2k}e^{-ik(t-t')} + \vartheta(t'-t)\frac{i}{2k}e^{ik(t-t')},$$

and

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

The source structure function is

$$e^{\widetilde\alpha_k} = \frac{k-\kappa_k}{k+\kappa_k}.$$

Boundary factorization

Set

$$b_k(t)=e^{-ik(t-t_0)}.$$

Then the image term is exactly rank-one:

$$\boxed{ G_{k,\mathrm{img}}(t,t') = \frac{i}{2k}e^{\widetilde\alpha_k} b_k(t)b_k(t'). }$$

It factors through the initial hypersurface coefficient line. Restriction of one endpoint gives

$$i^*G_{k,\mathrm{img}}(t') = \frac{i}{2k}e^{\widetilde\alpha_k}b_k(t'),$$

which is generically nonzero.

Bulk Cut

In the source Wightman decomposition, the same image term occurs in $G_k^>$ and $G_k^<$. Therefore it cancels from the bulk spectral discontinuity:

$$\boxed{ G_k^>-G_k^< = G_{k,\mathrm{vac}}^>-G_{k,\mathrm{vac}}^<, \qquad \operatorname{Cut}_{\rm bulk}G_{k,\mathrm{img}}=0. }$$

Explicitly,

$$G_k^>-G_k^< = \frac{\sin k(t-t')}{k},$$

independent of the initial-state structure function.

The comparison kernel

The two operations detect different parts of the source propagator:

$$\operatorname{Cut}_{\rm bulk} \quad\text{forgets}\quad G_{\rm img},$$

while

$$i^* \quad\text{retains}\quad G_{\rm img}|_\Sigma.$$

Thus boundary restriction does not factor faithfully through the ordinary bulk Cut quotient. Its canonical relative kernel is

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

This line is source-defined; no splitting or fitted endpoint term has been introduced.

Classification

$$\boxed{ \text{existing background boundary support} + \text{rank-one initial-state coefficient excess}. }$$

The result is not a new bulk energy/Cut carrier incidence. It is precisely a supported coefficient class on the already declared stratum $\Sigma$. It demonstrates concretely why the cosmological comparison must be relative:

$$\text{bulk Cut data} \not\Rightarrow \text{initial-boundary data}.$$

This strongly supports H2's support-sensitive calculus and falsifies any stronger claim that ordinary bulk Cut completely determines a general initial-state cosmological observable.

Next finite falsifier

Insert one source boundary operator from Entry 1468 and one bulk interaction connected by this propagator. Compute whether the boundary counterterm differential kills, renormalizes, or extends $\mathcal L_{\rm img}$. The calculation must retain:

• the boundary-condition parameter $\kappa_k$;
• normal derivatives at $\Sigma$;
• the boundary-local operator grading;
• bulk and boundary variance separately.

Only a residual class outside the boundary operator/counterterm complex would indicate missing carrier structure.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (3.13)--(3.17);
• Entry 1468;
• allocator claim seqclaim-aeb70451600e40a13e2d0072.

View source on GitHub