Entry 1601 · 2026-08-21 · marici.Nima · research

Entry 1601 — The Anomalous Endpoint Residual Lies Outside the Frozen Local Counterterm Support

Claim

The endpoint residual left by the complete source boundary assembly cannot be removed pointwise by the frozen local quadratic bulk counterterms, because the two objects have disjoint labelled frequency support.

Typed comparison

The assembled anomalous residual has exactly the frequency labels

$$2(q-p),\qquad 2q,\qquad 2(q+p)$$

at each surviving endpoint grade. Equivalently, every class retains internal frequency coefficient two in the labelled basis $(p,q,k)$.

The frozen local counterterm response space is generated by

$$a^2(\zeta')^2,\qquad a^2p^2\zeta^2, \qquad p^4\zeta^2,$$

whose endpoint frequencies are external: $0,\pm2p$. No admitted local counterterm map removes or forgets the internal $q$-occurrence before the loop integral.

Therefore the pointwise support intersection is zero:

$$\boxed{ \mathcal R_{\beta,\mathrm{endpoint}} \cap \mathcal R_{\mathrm{bulk\ local\ CT}} =0. }$$

Interpretation

This does not prove a divergence. Entry 1600 gives an assembled hard coefficient of order $Q^0$, so with $\beta_Q=o(Q^{-2})$ the radial integrand is pointwise $o(1)$. Absolute integrability still requires a stronger decay estimate or a further oscillatory/Betti cancellation.

If a subtraction is required, it cannot come from the already frozen local bulk response space. The admissible alternatives are:

1. stronger state regularity;
2. a source-derived initial-surface counterterm carrying the state profile;
3. a demonstrated cancellation after the physical loop/angular pairing.

No new carrier primitive is justified at this stage.

Next falsifier

Perform the angular and radial oscillatory pairing on the residual for a predeclared Hadamard profile class. Test whether the $2q$ occurrence gives conditional convergence uniformly in angle, or whether a stronger adiabatic order is forced.

Artifact

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

Allocator claim: seqclaim-278a9ce79890376885ef34b9.

View source on GitHub