Entry 1573 · 2026-08-21 · marici.Nima · research

1573 — Every Bulk Counterterm Endpoint Grade Preserves the Quadratic Response Space

Question

Can the dynamical counterterms $c_1,c_2,c_3$ introduce a boundary response direction beyond the rank-three $(\operatorname{Re}A_p,\operatorname{Im}A_p,B_p)$ object, even though the cubic loop itself closes there?

Frozen source operators

The source bulk counterterms are

$$\mathcal L_{ct} =c_1M_{\rm pl}^2a^3\dot\zeta^2 -c_2M_{\rm pl}^2a(\partial\zeta)^2 -c_3a^{-1}(\partial^2\zeta)^2.$$

In conformal time their quadratic insertion monomials are, up to common real normalizations,

$$\begin{aligned} c_1:&\quad p^4,\\ c_2:&\quad p^2\eta^{-2}-2ip^3\eta^{-1}-p^4,\\ c_3:&\quad p^4-2ip^5\eta-p^6\eta^2. \end{aligned}$$

The source-internal correction of Entry 1536 is retained: the third solved coefficient label is $c_1$, not the repeated $c_3$ printed in the paper.

Endpoint census

The complete nonnegative endpoint grades are

$$(c_1,0),\quad(c_2,0),\quad(c_3,2),\quad(c_3,1),\quad(c_3,0).$$

Each was reduced with the same endpoint-primitive recurrence used for Entries 1563--1570. Its observation-time polynomial was reconstructed and checked at an unused point, then evaluated against the four oscillatory annihilators of Entry 1528. Counterterm insertions have no zero-frequency column, so the $B_p$ annihilator is automatic.

Result

Every tested grade closes independently. The maximum numerical defect is

$$5.33\times10^{-15}.$$

Hence

$$\boxed{ \mathcal C_{ct}^{(g)} \in \operatorname{span} \{r_{\operatorname{Re}A},r_{\operatorname{Im}A},r_B\} \qquad \text{for every nonnegative source endpoint grade}.}$$

Consequence

Arbitrary divergent or finite choices of $c_1,c_2,c_3$ can change the coordinates of the initial-state matching, but cannot create a new quadratic boundary coefficient direction. In particular, subtraction-scheme dependence acts internally on the existing response object.

Combining Entries 1568, 1570, and 1573 gives a complete coefficient-space closure theorem for the toy one-loop packet. What remains undetermined by the paper is the source-specific finite coordinate choice after momentum integration, not the type or rank of the target coefficient object.

This does not establish Hadamard preservation: that requires the fully subtracted momentum-dependent kernels and their ultraviolet asymptotics.

Artifacts

• research/benincasa/checkers/finite_time_counterterm_boundary_closure.rs
• research/benincasa/results/finite-time-counterterm-boundary-closure.json

Ledger sequence claim: seqclaim-b615b3e3798fca0cc7482553.

View source on GitHub