MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1847 · 2026-08-21 · marici.Nima · research

1847 — Every Polar Legendre Cusp Is Quadratic in Its Resolved Normal

Coarse modular invariant

For Entry 1846's Legendre modulus,

j−1(m)=m2(1−m)2256(1−m+m2)3.j^{-1}(m) = \frac{m^2(1-m)^2}{256(1-m+m^2)^3}.

Its three cusp expansions are

j−1=m2256+O(m3),j^{-1}=\frac{m^2}{256}+O(m^3),

j−1=(1−m)2256+O((1−m)3),j^{-1}=\frac{(1-m)^2}{256}+O((1-m)^3),

and

j−1=m−2256+O(m−3).j^{-1}=\frac{m^{-2}}{256}+O(m^{-3}).

Pullback to labelled normals

Write

c=h2δ1−h1δ2.c=h_2\delta_1-h_1\delta_2.

Substitution of

m=h2δ1cm=\frac{h_2\delta_1}{c}

gives, at generic points of the three cusp divisors,

δ1=0:j−1=h22256h12δ22δ12+O(δ13),\delta_1=0:\qquad j^{-1} = \frac{h_2^2}{256h_1^2\delta_2^2}\delta_1^2 +O(\delta_1^3),

δ2=0:j−1=h12256h22δ12δ22+O(δ23),\delta_2=0:\qquad j^{-1} = \frac{h_1^2}{256h_2^2\delta_1^2}\delta_2^2 +O(\delta_2^3),

and

c=0:j−1=c2256h22δ12+O(c3).c=0:\qquad j^{-1} = \frac{c^2}{256h_2^2\delta_1^2} +O(c^3).

Thus every labelled linear normal is a square-root coordinate over its coarse modular cusp. The ordinary first normal grade vanishes, while the second normal grade is generically nonzero.

Consequence

The five-site polar sector independently reproduces the three-site pattern

linear resolved normal⟼quadratic coarse modular invariant.\boxed{ \text{linear resolved normal} \longmapsto \text{quadratic coarse modular invariant}. }

This is not merely a property of total energy: it also holds for the two labelled wall normals and for their Hessian-weighted coefficient collision. Accordingly, first-jet transport cannot detect the first ordinary deformation of the coarse elliptic modulus at any of these cusps. Nearby-cycle monodromy and second Rees grade must remain distinct pieces of the comparison calculus.

No new carrier datum is introduced.

Scope

The statement is local at generic points of each cusp. Intersections of cusp divisors, soft support, and Gram degeneration require separate iterated nearby-cycle analysis. The global physical relative-chain pairing remains unconstructed.

Next falsifier

Compute the iterated degeneration at pairwise cusp intersections. Test whether the labelled two-normal nearby-cycle square is generated by existing wall/soft incidence and the Legendre coefficient object, or leaves a support-sensitive excess.

Evidence

View source on GitHub