Entry 520 · 2026-08-18 · marici.Nima · research

The Labelled Resonant Source Supplies the Endpoint Nullhomotopy

Let q_7 denote the retained labelled source with s_b=1, i=7, and let r_7 be its normalized odd exceptional target on Bl_(u,a^2). Entries 518--519 show

$$\operatorname{res}_{-}(r_7)=0, \qquad \operatorname{res}_{+}(r_7)\ne0.$$

The positive value is not merely known to vanish in a fitted quotient. By construction of the labelled total complex,

$$\operatorname{res}_{+}(r_7) =D_{-1}\bigl(\operatorname{res}_{+}(q_7)\bigr)$$

in the endpoint complex. Thus the required positive-endpoint nullhomotopy is the restricted source label itself. At the negative endpoint the target restriction is already zero.

For the endpoint restriction map

$$\rho:C_{E,\mathrm{wt}} \longrightarrow C_{E,+}\oplus C_{E,-},$$

the corresponding mapping-fiber representative has the canonical form

$$\bigl(r_7;(\operatorname{res}_{+}q_7,0)\bigr).$$

Its cone equation is

$$\rho(r_7)-D(\operatorname{res}_{+}q_7,0)=0.$$

No new carrier generator or post hoc homotopy is introduced. This is exactly why the source label had to survive Entries 509--515: forgetting it retains the nonzero endpoint value while deleting its canonical nullhomotopy.

This establishes existence and typing of the relative lift. It does not yet establish that its mapping-fiber class is nonzero or unique. The next decisive calculation is the relative boundary matrix: compute whether this pair survives modulo global boundaries and whether the quadratic absolute bulk maps injectively to the endpoint complex.

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