Entry 1557 · 2026-08-21 · marici.Nima · research

1557 — Bulk Counterterms Cannot Supply the Missing Second-Grade Zero Mode

Hard-to-vary claim

For the three quadratic bulk counterterms printed in the primary source, the maximal finite-lower-end grades are

$$\boxed{ \deg_{\eta_0}(c_1,c_2,c_3)=(0,0,2). }$$

At grade two, $c_3$ has only frequencies $\pm2p$. None of the three counterterms has a nonoscillatory $\eta_0^2$ component.

Derivation

In conformal time:

• $c_1$ contributes $a^2(\zeta')^2$. Each differentiated mode is proportional to $\eta$, so the scale factor cancels the two powers and the maximal integrand power is zero.
• $c_2$ contributes $a^2p^2\zeta^2$. The two undifferentiated modes have maximal power two, again canceled by $a^2$.
• $c_3$ contributes $p^4\zeta^2$ without a conformal scale factor, so its maximal power is two.

All quadratic commutators carry nonzero frequency $\pm2p$; their endpoint primitives preserve the maximal Laurent power. The corrected counterterm labels of Entry 1536 are used.

Artifacts

• research/benincasa/checkers/finite_time_counterterm_endpoint_grades.rs
• research/benincasa/results/finite-time-counterterm-endpoint-grades.json

Narrow conclusion

Entry 1556's corrected cubic cancellation cannot be completed to the printed $-(1+p^2\eta^2)J_0$ by the three displayed quadratic bulk counterterms. Thus one convention in the cubic-sector assembly remains wrong or incomplete.

The obstruction is narrow: frequency support is correct, the bulk--bulk coefficient is independently selected by large-$|\eta_0|$ numerical asymptotics, and the remaining uncertainty lies in the mixed and boundary--boundary endpoint normalization/Wick factors.

Next falsifier

Derive the mixed and boundary--boundary contraction multiplicities directly from Eq. (18), retaining:

• the two ordered cross terms;
• the formal endpoint delta normalization from Entry 1539;
• the three choices of the undifferentiated leg at each cubic vertex;
• identical-line automorphisms;
• the relative $+iS_0^{(3)}$ contour phase.

No coefficient may be normalized by fitting Eq. (19). The derived census must either restore $-J_0$ or expose a source inconsistency.

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