The extension–descent problem

A distributional construction on a presentation space need not depend only on the reconstructed distribution. This distinction is particularly important when different presentation degrees can cancel after reconstruction. Extending each displayed term separately does not, by itself, preserve those cancellations.

The problem has three logically distinct parts:

1. extend prescribed canonical values from their proved domain;

2. annihilate every reconstruction-null presentation;

3. satisfy the specified support, normalization, continuity, and microlocal requirements.

The first two parts are already different before any complete-graph wavefront theorem is considered.

Cheung's XIV-J makes the descent condition explicit and records a null relation mixing presentation degrees CheungXIVJ. The present paper develops an exact relative criterion for correcting such defects. It also incorporates the finite normal-jet reconstruction test supplied in XV-C CheungXVC and develops the annihilator and point-puncture interfaces discussed in the supplied account of XV-E CheungXVE. All abstract statements used below are proved here; no unavailable proof from a companion manuscript is required.

The distributional background is standard Hormander. Extension with prescribed scaling or wavefront control requires further hypotheses, as in the frameworks of Brunetti–Fredenhagen and Dang BF,Dang. Our algebraic lifting criterion is not a substitute for those analytic hypotheses.