Entry 1528 · 2026-08-21 · marici.Nima · research

1528 — The Omitted Finite-Time Grades Have a Five-Equation Boundary-Closure Falsifier

Hard-to-vary claim

At first order in the quadratic initial action of arXiv:1408.4801, the real kernels $\operatorname{Re}A_p$, $\operatorname{Im}A_p$, and $B_p$ span only a rank-three subspace of the rank-eight late-time trigonometric coefficient space. Therefore each omitted $\eta_0$-grade must satisfy five exact linear identities if the published quadratic coefficient object is sufficient.

Frozen basis

Let $\theta=2p(\eta-\eta_0)$, and order coefficients by

$$(\cos\theta,\eta\cos\theta,\eta^2\cos\theta, \sin\theta,\eta\sin\theta,\eta^2\sin\theta,1,\eta^2).$$

Equation (21) gives the three response directions

$$r_{\operatorname{Re}A}=(0,-2p,0,1,0,-p^2,0,0),$$

$$r_{\operatorname{Im}A}=(1,0,-p^2,0,2p,0,0,0),$$

$$r_B=(0,0,0,0,0,0,1,p^2).$$

They have generic rank three.

Exact annihilators

Writing a candidate loop grade as coefficients $c_\bullet$, membership in the response span requires

$$c_{\eta\cos}+2p c_{\sin}=0, \qquad p^2c_{\sin}+c_{\eta^2\sin}=0,$$

$$p^2c_{\cos}+c_{\eta^2\cos}=0, \qquad -2pc_{\cos}+c_{\eta\sin}=0,$$

$$-p^2c_{1}+c_{\eta^2}=0.$$

The Symbolica checker verifies exactly that all five functionals annihilate all three source response directions. As an end-to-end control, it also projects the printed Eq. (19) vector and obtains

$$(\operatorname{Re}A,\operatorname{Im}A,B)=(0,L,-J_0),$$

where

$$L=J_1-2J_0-4J_2- \frac{32c_3}{(3\epsilon+2\delta)^2}.$$

All five obstruction functionals vanish on this control.

Deutsch--Popperian decision rule

• If both omitted grades satisfy all five identities, the published bilocal quadratic kernel remains sufficient at this loop order.
• If one identity fails, the failure is not removable by choosing $A_p$ or $B_p$. A further source-derived boundary coefficient operator is required.
• No new carrier stratum follows from such a failure: it is initially a coefficient-closure obstruction.

Artifacts

• research/benincasa/marici-gm/src/bin/finite_time_quadratic_response_span.rs
• research/benincasa/results/finite-time-quadratic-response-span.json

Verification

cargo run --release --bin finite_time_quadratic_response_span
exit code: 0
response rank: 3
annihilator rank: 5

Next falsifier

Contract Eq. (18), project its $\eta_0^1$ coefficient onto these five annihilators before evaluating the loop momentum integrals, and stop at the first nonzero obstruction. Only if that grade closes should the $\eta_0^0$ grade and subsequent Hadamard audit be computed.

View source on GitHub