Entry 1568 · 2026-08-21 · marici.Nima · research

1568 — The Omitted First Finite-Time Grade Closes on the Existing Quadratic Response

Question

Does the source-derived cubic $\eta_0^1$ grade require a new boundary coefficient direction, or does it lie in the rank-three quadratic response space frozen in Entry 1528?

Frozen acceptance test

After Entry 1566 cancels the observation-endpoint primitive column, the surviving frequency support is $\{\pm2p\}$. Write the positive-frequency coefficient as

$$A(\eta)e^{-2ip\eta}, \qquad A(\eta)=a_0+a_1\eta+a_2\eta^2.$$

The polynomial was reconstructed from three independent observation times and verified at a fourth. The resulting real trigonometric coefficient vector was then tested against all five exact annihilators of Entry 1528. No response coefficient was fitted to enforce closure.

Result

The fourth-point interpolation defect is

$$8.44\times10^{-15}.$$

All five boundary-response obstructions vanish. The largest numerical residual is

$$3.16\times10^{-13}.$$

At the frozen generic sample, the response coordinates are

$$(\operatorname{Re}A_p,\operatorname{Im}A_p,B_p) = (4.553605033810049,0,0)$$

in the Entry 1528 normalization.

Therefore

$$\boxed{ \mathcal C^{(1)}_{\rm cubic} \in \operatorname{span} \{r_{\operatorname{Re}A},r_{\operatorname{Im}A},r_B\}. }$$

Consequence

The first omitted finite-time grade requires neither a new Carrier stratum nor a new quadratic boundary coefficient operator. It renormalizes the existing $\operatorname{Re}A_p$ direction.

This is the first positive closure result for a grade omitted from the source's explicit matching calculation. It is restricted to the frozen cubic-loop sector and does not yet include a complete bulk-counterterm matching packet. The next independent falsifier is the $\eta_0^0$ grade, where the nonoscillatory component can also test the $B_p$ direction.

Artifacts

• research/benincasa/checkers/finite_time_grade1_boundary_closure.rs
• research/benincasa/results/finite-time-grade1-boundary-closure.json

Ledger sequence claim: seqclaim-f7c35602df86d9c3e79c7c0d.

View source on GitHub