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Entry 1846 · 2026-08-21 · marici.Nima · research

1846 — The Polar Double-Morse Coefficient Is a Legendre Variation

Frozen branch data

Entry 1845 gives the ordered branch points

0,1,a=−h2h1−h2,b=−δ2δ1−δ2.0,\quad 1,\quad a=-\frac{h_2}{h_1-h_2},\quad b=-\frac{\delta_2}{\delta_1-\delta_2}.

Use the uniquely normalized Möbius coordinate

T(x)=x(1−b)x−b,T(x)=\frac{x(1-b)}{x-b},

which sends (0,1,b)(0,1,b) to (0,1,∞)(0,1,\infty). The fourth branch point is

m=T(a)=h2δ1h2δ1−h1δ2.\boxed{ m=T(a)=\frac{h_2\delta_1}{h_2\delta_1-h_1\delta_2}. }

Thus the polar double-Morse coefficient is the pullback of the universal Legendre Gauss--Manin variation by this Hessian-weighted normal modulus, up to the Kummer factor already typed by the physical current in Entry 1843.

Labelled cusps

The three nodal degenerations are

δ1=0m=0δ2=0m=1h2δ1−h1δ2=0m=∞.\begin{array}{c|c} \delta_1=0&m=0\\ \delta_2=0&m=1\\ h_2\delta_1-h_1\delta_2=0&m=\infty. \end{array}

The first two are the existing labelled wall normals. The third is Entry 1845's internal coefficient collision. Standard Legendre nearby cycles at all three cusps are Tate/Kummer; no additional carrier incidence is needed.

Exchanging the two labelled walls gives

(1↔2):m⟼1−m,(1\leftrightarrow2):\qquad m\longmapsto1-m,

so occurrence covariance is retained rather than erased by an unlabelled cross-ratio.

Narrow result

The five-site polar sector repeats, at a new local geometry, the architecture previously found in the homogeneous three-site loop:

labelled wall normals⟶source-weighted modular coordinate⟶Legendre coefficient variation.\boxed{ \text{labelled wall normals} \longrightarrow \text{source-weighted modular coordinate} \longrightarrow \text{Legendre coefficient variation}. }

This supports H2: the carrier supplies the labelled walls and their incidence, while the elliptic period system is a sector-specific coefficient object.

Scope

This is a local coefficient theorem near the polar double-Morse point. It does not construct the global physical relative-chain pairing, and it does not prove that the complete five-site observable is exhausted by this block.

Next falsifier

Compute the coarse modular parameter near each labelled cusp. Test whether each linear resolved normal is a square-root coordinate over the coarse modular degeneration, as at the three-site total-energy boundary.

Evidence

View source on GitHub