Entry 1862 · 2026-08-21 · marici.Nima · research

1862 — The Active-Soft Zero Rays Are Tangentially Simple but Normally Unresolved

Tangential derivatives

For the two zero-bearing endpoint types of Entry 1861, differentiate the source numerators in their shape coordinate (x).

At

$$x_2=\frac{7-\sqrt5}{2}, \qquad N_2(x)=x^2-7x+11,$$

one has

$$\boxed{N_2'(x_2)=-\sqrt5\neq0.}$$

At

$$x_5=\frac{\sqrt5-1}{2}, \qquad N_5(x)=x^2+x-1,$$

one has

$$\boxed{N_5'(x_5)=\sqrt5\neq0.}$$

Thus all ten cyclic numerator-zero occurrences are simple zeros transverse to their coefficient ray inside endpoint shape space.

Normal-order distinction

Tangential simplicity does not determine the next ordinary/Rees normal grade. The known endpoint expansion has the form

$$A_0(x,a)\,\delta^2\log\delta+\cdots.$$

On (A_0=0), deciding whether the next nonanalytic term is (delta^3\log\delta) requires the radial derivative of the complete source coefficient and denominator, followed by the physical angular integration. Neither follows from (dA_0/dx\neq0) alone.

Therefore

$$\boxed{ \text{simple tangential coefficient zero} \not\Rightarrow \text{third Rees normal leading term}. }$$

Narrow result

The zero divisor is reduced as a coefficient locus. Its normal/Rees specialization remains unresolved. No new carrier datum appears.

Next falsifier

Expand the full active-soft local integrand one radial order beyond Entry 1836, impose one numerator-zero equation, and perform the angular coefficient integral. A nonzero result would establish a third-Rees logarithmic term; angular cancellation would force a higher grade.

Evidence

• research/benincasa/checkers/five_site_region_pair_active_soft_zero_transversality.py
• research/benincasa/results/five-site-region-pair-active-soft-zero-transversality.json
• Entries 1836 and 1860--1861
• allocator claim: seqclaim-580ed22a71d585631383b600

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