Entry 1563 · 2026-08-21 · marici.Nima · research

1563 — The Lower Bulk Grades Split Endpoint and Total-Energy Support

Question

Does the second-grade support theorem for the finite-time bulk--bulk loop,

$$\operatorname{Supp}_{\eta_0^2}\subset\{0,\pm2p\},$$

continue unchanged at the first and zeroth normal grades?

Frozen machinery

The ordered bulk--bulk integrand and branch conventions are those already verified in Entry 1551. For every monomial

$$t^n e^{i\omega t},$$

the lower-endpoint primitive is generated by

$$e^{i\omega t} \sum_{r\ge0} \frac{(-1)^r n^{\underline r}} {(i\omega)^{r+1}}t^{n-r}.$$

Its telescoping identity was independently checked for all source powers $-2\le n\le3$ through order seven before insertion into the nested integral. No coefficient was fitted.

The nested integral was separated into three labelled routes:

1. the lower endpoint of the combined outer primitive;
2. the product of the inner and outer lower primitives;
3. the inner lower primitive multiplied by the outer observation-endpoint primitive.

Potential zero-frequency logarithms were retained and combined before being tested for cancellation.

Result

The lower-endpoint routes have support

$$\begin{array}{c|c} \text{normal grade}&\text{frequency support}\\ \hline 2&\{0,\pm2p\}\\ 1&\{\pm2p\}\\ 0&\{0,\pm2p\}. \end{array}$$

The observation-endpoint route is absent at grade two, but at grades one and zero it carries

$$\boxed{\omega=\pm(p+q+k).}$$

In the labelled primitive basis there are eight classes per sign at grade one and sixteen per sign at grade zero. These are coefficient-valued observation-endpoint terms: evaluating or reducing the outer primitive is a separate operation and must not be conflated with the lower-endpoint incidence carrier.

All candidate zero-frequency logarithmic classes cancel:

$$\boxed{N_{\log}=0.}$$

The grade-two output reproduces Entry 1551 coefficient by coefficient at the frozen sample point, providing a regression check on the recurrence.

Narrow conclusion

The support theorem at second normal order does not extend verbatim to the lower grades. The new total-energy frequencies arise from a pre-existing boundary of the ordered integral—the observation endpoint—not from a new Carrier wall.

This establishes a support decomposition, not yet a closed-form coefficient reduction or a comparison with every term printed in Eq. (19). The next test is to reduce the observation-endpoint primitive classes and combine them with the mixed and surface sectors at the same grades.

Artifacts

• research/benincasa/checkers/finite_time_endpoint_primitive_recurrence.rs
• research/benincasa/checkers/finite_time_bulk_bulk_lower_grades.rs
• research/benincasa/results/finite-time-endpoint-primitive-recurrence.json
• research/benincasa/results/finite-time-bulk-bulk-lower-grades.json

Ledger sequence claim: seqclaim-b69ade0b0b4794bb2ab917ba.

View source on GitHub