Entry 1474 · 2026-08-21 · marici.Benincasa · research

1474 — Boundary Quartic Sewing Is Generated by Labelled Image-Line Tensor Products

Status

First mixed boundary-interaction sewing test following Entries 1468, 1470, and 1472. The source boundary operator is the dimension-four $\phi^4$ operator listed in Collins--Holman, arXiv:hep-th/0507081v1, Eq. (6.8).

Boundary-to-bulk line

For the source propagator and a physical endpoint $t>t_0$, restriction of the second endpoint to the initial hypersurface gives

$$\boxed{ G_k(t,t_0) = \frac{i}{2k} \left(1+e^{\widetilde\alpha_k}\right) b_k(t), \qquad b_k(t)=e^{-ik(t-t_0)}. }$$

The vacuum and image contributions therefore become proportional after boundary restriction. They do not define two independent outgoing states. The boundary-to-bulk propagator is one labelled coefficient line per edge.

Quartic boundary vertex

For four labelled momenta $k_1,\ldots,k_4$, the tree contribution from one boundary $\phi^4$ insertion is proportional to

$$\delta^3\!\left(\sum_{i=1}^4\mathbf k_i\right) \prod_{i=1}^4 \left[ \frac{i}{2k_i} \left(1+e^{\widetilde\alpha_{k_i}}\right)b_{k_i}(t_i) \right].$$

Thus its coefficient object is generated by

$$\boxed{ \bigotimes_{i=1}^4\mathcal L_{\Sigma,k_i}. }$$

The four edge occurrences must remain labelled, but no additional nesting, contact, or propagation primitive is needed. Expanding each factor into vacuum plus image produces a Boolean occurrence expansion; restriction maps all terms for a fixed edge to the same one-dimensional line.

Mixed bulk sewing

Connecting one labelled boundary leg to a bulk vertex inserts the same mode $b_k(t)$ into the bulk time integral. Consequently boundary multiplication and bulk sewing preserve the tensor category generated by the labelled lines $\mathcal L_{\Sigma,k}$. Normal derivatives multiply a line by its boundary jet and remain inside the boundary-local coefficient algebra derived in Entry 1472.

Cut comparison

Bulk spectral Cut annihilates an image contribution on any selected internal occurrence, while boundary restriction retains its coefficient. For a quartic boundary vertex, the supported kernel is therefore generated by the tensor ideal containing at least one labelled image factor. This ideal is intrinsic to the boundary coefficient algebra:

$$\boxed{ \ker(\operatorname{Cut}_{\rm bulk})\big|_\Sigma = \left\langle \text{labelled tensors with an image-line factor} \right\rangle. }$$

Narrow conclusion

The first interacting boundary contact closes under

$$\boxed{ \text{existing background-boundary incidence} + \text{labelled tensor products of boundary coefficient lines}. }$$

No new carrier stratum is produced. The additional cosmological information is coefficient-theoretic and support-sensitive.

Next falsifier

Test a boundary loop or a boundary-to-bulk-to-boundary return path. Unlike the tree quartic, it integrates products of boundary lines over a loop momentum and can generate new period or renormalization structure. Freeze the source boundary operator grading and ask whether every singularity remains on the declared boundary/soft/Cayley--Menger support. A new integrated period is allowed as coefficient complexity; only an undeclared incidence support would falsify H2 at carrier level.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (3.13)--(3.17), (6.8);
• Entries 1468, 1470, and 1472;
• allocator claim seqclaim-231e56e19ef17f0146cc21a7.
• epistemic event ev-000000001583-a4b8b0df-fdef-4b4c-bfdf-790b07ce390f.

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