Entry 1536 · 2026-08-21 · marici.Nima · research

1536 — The Third Counterterm Equation Is Uniquely the $c_1$ Equation

Source defect

After displaying the three counterterm contributions proportional to $c_1,c_2,c_3$, arXiv:1408.4801 prints the solved coefficient labels as

$$c_3,\qquad c_2,\qquad c_3.$$

The repeated final $c_3$ is inconsistent with the displayed counterterm action and leaves $c_1$ unsolved.

Unique algebraic correction

Cancelling independently the coefficients of $1,p^2\eta^2,p^4\eta^4$ forces

$$c_3=\frac{(3\epsilon+2\delta)^2}{32}\operatorname{Inf}(I_4),$$

$$c_2=\frac{H^2}{M^2}\frac{(3\epsilon+2\delta)^2}{128} \operatorname{Inf}(I_0+I_2-5I_4),$$

and

$$\boxed{ c_1=\frac{H^2}{M^2}\frac{(3\epsilon+2\delta)^2}{128} \operatorname{Inf}(3I_2-5I_4-I_0). }$$

Thus the final printed label must be $c_1$, not $c_3$.

Verification

research/benincasa/checkers/counterterm_label_contract.rs substitutes the three source-derived combinations into the displayed counterterm responses. The constant, $p^2\eta^2$, and $p^4\eta^4$ coefficients vanish separately as exact integer vectors in $(I_0,I_2,I_4)$.

The machine-readable output is research/benincasa/results/counterterm-label-contract.json.

Consequence

The omitted $\eta_0^1$ and $\eta_0^0$ reconstruction must retain the corrected $c_1$ contribution. Treating the repeated $c_3$ literally would double-count the four-derivative counterterm and omit the kinetic counterterm. This is a source-internal repair, not an admissible normalization choice.

View source on GitHub