Entry 1570 · 2026-08-21 · marici.Nima · research

1570 — The Omitted Zeroth Finite-Time Grade Also Closes on the Existing Quadratic Response

Question

Does the source-derived cubic $\eta_0^0$ grade pass Entry 1528's complete quadratic boundary-response test, including the nonoscillatory $B_p$ direction?

Calculation

After the observation-endpoint cancellation of Entry 1566, the surviving grade-zero support is

$$\{0,\pm2p\}.$$

The positive-frequency coefficient and the zero-frequency coefficient were each reconstructed as quadratic polynomials in the observation time $\eta$ from three independent values and checked at a fourth. The resulting eight real trigonometric coefficients were evaluated against all five exact annihilators from Entry 1528.

Result

The interpolation defect is

$$3.57\times10^{-14}.$$

All five response obstructions vanish; the largest residual is

$$3.75\times10^{-13}.$$

The forbidden linear term in the nonoscillatory polynomial vanishes exactly in the route census. At the frozen generic sample, the response coordinates are

$$(\operatorname{Re}A_p,\operatorname{Im}A_p,B_p) = (0,11.235707650810378,5.211249154879921).$$

Therefore

$$\boxed{ \mathcal C^{(0)}_{\rm cubic} \in \operatorname{span} \{r_{\operatorname{Re}A},r_{\operatorname{Im}A},r_B\}. }$$

Consequence

Both grades omitted from the source's explicit finite-time matching now close on its already declared quadratic boundary coefficient object:

$$\eta_0^1\longmapsto\operatorname{Re}A_p, \qquad \eta_0^0\longmapsto\operatorname{Im}A_p\oplus B_p.$$

No new Carrier incidence or quadratic coefficient direction is required by the cubic loop. The remaining qualification is renormalization: bulk counterterm contributions and the source's unpublished matching coefficients must be combined with these source-derived cubic coordinates before claiming a complete one-loop Hadamard-preservation theorem.

Artifacts

• research/benincasa/checkers/finite_time_grade0_boundary_closure.rs
• research/benincasa/results/finite-time-grade0-boundary-closure.json

Ledger sequence claim: seqclaim-e0718befecad82ac0c2ea076.

View source on GitHub