Entry 1556 · 2026-08-21 · marici.Nima · research

1556 — The Cubic Loop Zero Mode Cancels Before Bulk Counterterms

Correction status

This entry replaces the original, incorrect claim that the three cubic sectors reconstruct $J_0$ by themselves. The error was omission of the first subleading inner primitive on the combined-zero-frequency bulk route.

Hard-to-vary claim

After restoring that route, the three source-normalized cubic sectors have nonoscillatory second-grade ratio

$$\boxed{ \text{bulk--bulk}:\text{mixed}:\text{boundary--boundary}=1:-2:1. }$$

Therefore

$$\boxed{c_{0,\rm cubic}^{(2)}=0.}$$

The printed $-(1+p^2\eta^2)J_0$ in Eq. (19) cannot come from these three cubic sectors alone. Its remaining source-defined location is the bulk counterterm contribution included through $\bar H_I=H_I+H_{ct}$ in Eq. (18).

Finite falsifier

The identity was checked at four independent generic $(q,k)$ samples. Each sector is a common nonzero unit times $1,-2,1$, and the sum vanishes below relative tolerance $2\times10^{-14}$.

Artifacts

• research/benincasa/checkers/finite_time_zero_mode_assembly.rs
• research/benincasa/results/finite-time-zero-mode-assembly.json
• research/benincasa/checkers/finite_time_bulk_bulk_route_census.rs

Narrow conclusion

The cubic boundary completion behaves as a discrete second difference in the nonoscillatory second grade. Eq. (19) compares the completed cubic loop plus bulk counterterms, not the cubic sectors alone.

Next falsifier

Expand the finite-lower-end contribution of the three printed quadratic bulk counterterms $c_1,c_2,c_3$ at order $\eta_0^2$, using the corrected labels from Entry 1536. Require their zero mode to supply exactly

$$-(1+p^2\eta^2)J_0$$

while their oscillatory contribution reduces to the printed $c_3$ term.

View source on GitHub