Entry 1600 · 2026-08-21 · marici.Nima · research

Entry 1600 — Boundary Completion Lowers the Anomalous Hard Grade by One

Claim

For one labelled internal Bogoliubov variation, the complete source-normalized bulk--bulk, mixed, and boundary--boundary endpoint assembly does not vanish. It does lower the surviving hard coefficient by exactly one power relative to the bulk sector.

Frozen assembly

The calculation retains the source boundary kernel

$$C(K,\eta_0)=\frac1K\left(\frac1{K\eta_0}-i\right), \qquad K=p+q+k,$$

the two mixed placements, the boundary--boundary corner, all contour signs, and the labelled anomalous $q$-occurrence. Coefficients are compared by their endpoint grade and frequency label before evaluation at $\eta_0$.

Result

The bulk map, mixed map, and boundary corner each occupy the same nine labelled endpoint classes. Their sum retains all nine. On hard rays

$$q=Q,\qquad k=Q-c$$

with three generic fixed values of $c$, the ratio

$$\frac{\|BB+BS+SB+SS\|_1}{\|BB\|_1}$$

scales as $Q^{-1}$ under every doubling from $Q=40$ through $320$. Since the bulk coefficient scales as $Q$, the complete endpoint residual is of order $Q^0$.

Narrow conclusion

$$\boxed{ \text{source boundary completion gives one extra hard power but not closure} }$$

After the radial measure and the Hadamard condition,

$$Q^2\beta_Q\,O(1)=o(1).$$

This is pointwise hard decay, not a proof of absolute radial integrability. The remaining class is still occurrence-labelled and oscillatory.

Next falsifier

Project the residual onto the frozen local counterterm response space. Local counterterms have only external frequency support $0,\pm2p$; the test must decide whether the internal-frequency labels cancel after angular integration or define a genuinely nonlocal state-dependent residual.

Artifact

research/benincasa/marici-gm/src/bin/gaussian_anomalous_endpoint_recurrence.rs

Allocator claim: seqclaim-926d4760f435edea6397858f.

View source on GitHub