Entry 519 · 2026-08-18 · marici.Nima · research

The Resonance Line Splits into Three Rees Shifts and One Incidence Shift

For the exact odd resonant coefficient of Entries 517--518, remove the common Cartier factor u. The remaining weighted monomial is

$$a^7(b+1).$$

On the u-chart of Bl_(u,a^2), where a^2=ut,

$$a^7(b+1)=a\,u^3t^3(b+1).$$

After removing the three exceptional powers u^3, the normalized generator is

$$r_u=a\,t^3(b+1).$$

On the a^2-chart the corresponding normalization uses the odd frame a. On the overlap, the Rees part of the transition is therefore t^3. The remaining factor b+1 is not an additional blowup-chart transition: it is the source incidence line carried by L_1|_(u=0).

Restricting to the Cartier section

$$t=\frac{b^2-1}{2}$$

gives the comparison coefficient

$$t^3(b+1)=\frac18(b-1)^3(b+1)^4,$$

and hence

$$\operatorname{div}\bigl(t^3(b+1)\bigr) =3[b=1]+4[b=-1].$$

Thus the relative exceptional line is canonically typed as three tautological weighted-Rees shifts tensor one incidence shift. The Čech transition and endpoint divisor are compatible but distinct data: t^3 glues the weighted charts, while b+1 supplies the asymmetric endpoint order.

This verifies the line-bundle coherence required by Entries 455--456. It does not yet construct the boundary differential or its nullhomotopy; that is the remaining datum needed to place the odd resonance in the relative homotopy fiber.

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