Entry 1840 · 2026-08-21 · marici.Nima · research

1840 — Correction: A True Polar Landau Point Is Double-Morse and Square-Root

Error corrected

Entry 1837 correctly identifies the owner-wall focal segment and its Morse--Bott transverse quadratic form. Its proposed transverse cut by the second wall is not, however, a physical Landau point.

On the owner focal segment,

$$\nabla q_1=0.$$

The positive Landau equation is

$$\alpha\nabla q_1+\beta\nabla q_2=0, \qquad \alpha,\beta>0.$$

It therefore forces

$$\boxed{\nabla q_2=0.}$$

Thus the loop point must lie on the second focal segment as well. Entry 1837's transverse-second-wall grade-zero logarithmic model is withdrawn.

Correct local geometry

At a generic true polar collision, two nonparallel focal segments intersect. The two Hessians are positive semidefinite with kernels equal to their segment directions:

$$\ker H_1=T\ell_1, \qquad \ker H_2=T\ell_2.$$

For nonparallel lines, every positive combination

$$\alpha H_1+\beta H_2$$

is positive definite. The local physical integral is therefore of double-quadratic Morse type:

$$\int \frac{d^3x} {(\delta_1+Q_1(x))(\delta_2+Q_2(x))}.$$

On a generic diagonal normal $\delta_1\sim\delta_2\sim\delta$, scaling $x=\sqrt\delta\,\xi$ gives

$$\delta^{3/2-2}=\delta^{-1/2}.$$

Hence the local coefficient is a rank-one square-root Kummer/Morse line with monodromy $-1$, not a logarithmic line.

Corrected classification

Entry 1839's support typing survives:

• carrier: existing Gram/Cayley--Menger segment-intersection divisor;
• coefficient condition: equality of labelled focal lengths;
• physical chamber: both labelled segment parameters in $[0,1]$.

Only the coefficient type changes:

$$\boxed{ \text{polar coefficient}= ext{square-root Kummer line}. }$$

No new carrier datum is required.

Remaining exceptional case

Parallel or coincident focal segments make the positive Hessian combination degenerate. That deeper Gram stratum requires a separate excess calculation and is not covered here.

Evidence

• research/benincasa/checkers/five_site_region_pair_polar_double_morse_correction.py
• research/benincasa/results/five-site-region-pair-polar-double-morse-correction.json
• Entries 1836--1839
• allocator claim: seqclaim-2d786ab9c6b9c397a2f6c58b

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