Manuscript VII-D@1 · VII-D.tex · section · ef67c89c

Integration-sector coverage

Intersection and sector audit › Integration-sector coverage · lines 1726–1777

{}1.14

Sector	Controlling calculation
All frequencies large and comparable	The normalization ledger and the central errors $E_{P}, E_{δ}, E_{ω}$ .
Only a proper subset large	Bounded $q$ belongs to Lemma lem:S. For large $q$ , a small graviton momentum forces a shifted tail, covered by (eq:shifttail).
Ratios tending to zero or infinity	The shifted-tail bounds include both limits and $P = 0, P^{'} = 0$ , with their locally integrable denominators.
A direct branch present	Time integration by parts in a frequency $P + ω (r)$ or $P^{'} + ω (r)$ , independently of $d$ .
Angular degeneracy	Uniform estimates on $S^{2}$ , not angular stationary phase. The two directions in $Θ_{r}$ are kept distinct.
A graviton cutoff argument tends to zero	The exact telescoping and mean-value identities (eq:multtelescoping)–(eq:meanvalueexact); only boundedness and Lipschitz continuity are used.
Bounded frequencies	The common distribution $S$ , with $O (ε)$ convergence.

The partitions at $∣ q ∣ = 1$ and $∣ α ∣, ∣ β ∣ = r /2$ are independent of $ε$ . No derivative of a cutoff-dependent partition is omitted.

audit. This numerical subtraction leaves the VII-C coefficient-continuity obstruction under its stated normal-topology hypotheses unchanged VIIC (thm:coefficient).