Entry 1849 · 2026-08-21 · marici.Nima · research

1849 — The Polar Normal-Cone Coefficient Splits as Kummer Tensor Legendre

Radial/projective coordinates

On the normal-blowup chart of Entry 1848, write

$$\delta_2=\rho, \qquad \delta_1=\rho r.$$

The polar elliptic curve becomes

$$w^2 = \rho\, x(1-x) \bigl(xh_1+(1-x)h_2\bigr) \bigl(xr+(1-x)\bigr).$$

After the root-cover rescaling

$$W=\frac{w}{\sqrt\rho},$$

the strict-transform equation is independent of the radial coordinate:

$$W^2 = x(1-x) \bigl(xh_1+(1-x)h_2\bigr) \bigl(xr+(1-x)\bigr).$$

Connection splitting

For the elliptic de Rham basis, every period therefore factors as

$$\Pi(\rho,r)=\rho^{-1/2}\Pi_{\rm Leg}(r).$$

Consequently the resolved local connection is

$$\boxed{ \nabla = d -\frac12d\log\rho\,\operatorname{Id} +A_{\rm Leg}(r)\,dr. }$$

The radial Kummer term is scalar, so its commutator with the Legendre block vanishes. There is no mixed radial--projective curvature and no invariant mixed extension term in this local coefficient family.

Chart compatibility

On the second normal chart,

$$\delta_1=\rho', \qquad s=\frac{\delta_2}{\delta_1},$$

the overlap is

$$\rho'=\rho r, \qquad s=r^{-1}.$$

The corresponding square-root factor is exactly the Kummer/root-cover transition already typed by the oriented physical current in Entry 1843. It is not a new coefficient extension or carrier datum.

Narrow result

On the resolved two-normal cone,

$$\boxed{ \mathbb V_{\rm polar} \simeq \mathcal K_{\rho^{-1/2}} \otimes m(r)^*\mathbb H_{\rm Leg}. }$$

Thus the local five-site polar block has the same structural form as the homogeneous three-site elliptic block: a source-derived Kummer character tensor a Legendre Gauss--Manin variation compiled from labelled normal data.

Scope

The splitting is a theorem for the local polar double-Morse coefficient family. It does not construct or normalize the global physical integration chain, and it does not exclude extensions involving other five-site master blocks away from this local quotient.

Next falsifier

Test whether the local polar elliptic quotient embeds canonically into the full five-site marked-relative Gauss--Manin system. The required map must be derived from source residues/Gysin operations and must intertwine the cyclic occurrence action; matching local differential equations is insufficient.

Evidence

• research/benincasa/checkers/five_site_region_pair_polar_radial_legendre_split.py
• research/benincasa/results/five-site-region-pair-polar-radial-legendre-split.json
• Entries 1843--1848
• allocator claim: seqclaim-0e8c52d3701de7ab497127f8

View source on GitHub