Entry 1561 · 2026-08-21 · marici.Nima · research

1561 — No Wick Family Doubles the Boundary Sectors

Hard-to-vary claim

The complete labelled Wick census for two external fields and two $\zeta(\partial\zeta)^2$ vertices contains exactly

$$\boxed{36}$$

connected fish contractions at generic nonzero external momentum. The count is identical for bulk--bulk, bulk--surface, and surface--surface placement.

There is no omitted connected contraction family that selectively doubles the boundary-containing sectors of Eq. (19).

Complete census

There are $8$ labelled fields: two external fields and three slots at each cubic vertex. Their $105$ perfect pairings split as

$$\begin{array}{c|c} \text{class}&\text{count}\\ \hline \text{connected fish}&36\\ \text{zero-momentum tadpole topology}&36\\ \text{disconnected}&18\\ \text{external pair}&15 \end{array}$$

The tadpole topology forces the line connecting the vertices to carry zero momentum when the two external momenta are $p,-p$. It is not a second generic fish contribution and cannot produce the printed $J_0,J_2$ structures.

For the fish class, summing the three choices of undifferentiated slot and the exchange of the differentiated slots gives the same vertex factor at every location:

$$2(p\!\cdot q+p\!\cdot k+q\!\cdot k) =-(p^2+q^2+k^2).$$

Two vertices therefore give the source's $(p^2+q^2+k^2)^2$.

Artifacts

• research/benincasa/checkers/labelled_cubic_wick_census.rs
• research/benincasa/results/labelled-cubic-wick-census.json

Narrow conclusion

The conventional repair space is exhausted:

• Eq. (17) and Eq. (18) have identical expansion coefficients (Entry 1560);
• a separated endpoint regulator is multiplicative (Entry 1559);
• a scalar endpoint weight cannot match both sectors (Entry 1558);
• bulk counterterms cannot supply the zero-mode discrepancy (Entry 1557);
• and no missing Wick family doubles the boundary sectors (this entry).

Within the frozen one-operator toy model and the printed formulas, the factor two multiplying both boundary-containing contributions in Eq. (19) is not derived. The narrow surviving diagnosis is

$$\boxed{ \text{Eq. (19) has an unsupported boundary-sector normalization.} }$$

This is a primary-source normalization defect, not evidence for a new cosmological carrier or coefficient object.

Next falsifier

Independently reproduce the three sector coefficients from the published three-point boundary kernel by sewing two source-normalized three-point vertices. If sewing yields the direct-exponent coefficients again, mark the Eq. (19) boundary factor as a probable typo and continue the omitted $\eta_0^1,\eta_0^0$ reconstruction using the corrected normalization. If sewing produces the extra factor two, its occurrence-level provenance will identify the missing comparison map.

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