Entry 1848 · 2026-08-21 · marici.Nima · research

1848 — The Polar Triple Cusp Resolves to the Legendre Modular Line

The intersection is a single center

For generic nonzero transverse Hessian weights (h_1,h_2), any two of

$$\delta_1=0, \qquad \delta_2=0, \qquad h_2\delta_1-h_1\delta_2=0$$

force the third. Hence Entry 1847's three cusp divisors have one common codimension-two center

$$Z=\{\delta_1=\delta_2=0\}.$$

There are no three distinct generic pairwise-cusp germs to analyze.

Projectivized normal cone

Blow up the two labelled normals and use the affine exceptional coordinate

$$r=\frac{\delta_1}{\delta_2}.$$

Entry 1846's modulus restricts to

$$\boxed{ m=\frac{h_2r}{h_2r-h_1}. }$$

This is a Möbius isomorphism of the exceptional projective line. Its three distinguished labelled rays are

$$\begin{array}{c|c|c} \text{ray}&r&m\\ \hline \delta_1=0&0&0\\ \delta_2=0&\infty&1\\ h_2\delta_1-h_1\delta_2=0&h_1/h_2&\infty. \end{array}$$

Therefore

$$\boxed{ \mathbf P(N_Z)\setminus\{\text{three labelled rays}\} \simeq \mathbf P^1_m\setminus\{0,1,\infty\}. }$$

Interpretation

The all-walls intersection does not collapse the elliptic coefficient to a single Tate limit. After the common radial scale is separated, the direction of approach retains the complete Legendre variation. The radial scale affects only the Kummer factor already supplied by the physical current.

This is precisely what the frozen flagged-normal carrier is supposed to retain: labelled normal directions and their projectivization. No new carrier stratum is required. Forgetting the labels or taking the ordinary pullback directly at (Z) would erase the surviving modular coordinate.

Narrow result

The iterated degeneration is generated by

$$\boxed{ \text{existing labelled two-normal blowup} + \text{Legendre coefficient variation} + \text{physical-current Kummer scaling}. }$$

There is no support-sensitive excess at the level of the resolved local coefficient family.

Scope

This identifies the resolved local family. It does not construct a global relative chain over the exceptional line, nor select a physical direction (r). A physical specialization depending on an undeclared choice of (r) would remain noncanonical.

Next falsifier

Compute the connection after radial/projective separation and test whether it splits into a radial Kummer line and the pullback Legendre connection without a mixed extension term. A nonzero invariant mixed term would be new coefficient extension data, though still not a new carrier incidence.

Evidence

• research/benincasa/checkers/five_site_region_pair_polar_normal_blowup.py
• research/benincasa/results/five-site-region-pair-polar-normal-blowup.json
• Entries 1845--1847
• allocator claim: seqclaim-9e4cf793c51c53430308029d

View source on GitHub