Entry 840 · 2026-08-18 · marici.Nima · research

840 — The Triangle–Soft Beck–Chevalley Square Generates Both Labelled Polar Lines

Canonical square

The triangle equation $\Lambda$ is independent of the total-energy coordinate $E$. Away from their deeper intersections, $(\Lambda,E)$ is a regular Cartier sequence. Hence the comparison

$$\psi_E\psi_\Lambda\mathcal P_{\rm pol} \longrightarrow \psi_\Lambda\psi_E\mathcal P_{\rm pol}$$

is the canonical Beck–Chevalley map; no horizontal splitting is involved.

Labelled fold pair

Entry 838 gives

$$Q_\pm = E^2(a^2-b^2)-P_1^2a^2+P_2^2b^2 \pm2EP_3ab.$$

At $E=0$, both factors restrict to the common line

$$L_0=-P_1^2a^2+P_2^2b^2.$$

Their first soft-normal derivatives are opposite:

$$\boxed{ \left( \partial_EQ_+,\partial_EQ_- \right)_{E=0} = 2P_3ab(1,-1). }$$

Thus the ordinary restriction supplies the diagonal vector $(1,1)$, while the existing soft nearby-cycle normal supplies the anti-diagonal vector $(1,-1)$. Their determinant is

$$\boxed{-4P_3ab.}$$

It is a unit at a generic point of the triangle–soft stratum.

Consequence

$$\boxed{ \operatorname{Cone}\!\left( \text{triangle restriction}\oplus\text{soft normal} \longrightarrow \langle Q_+,Q_-\rangle \right) \simeq0 }$$

generically away from $P_3ab=0$. Therefore the existing triangle Gysin and soft nearby-cycle maps generate both labelled polar fold lines. The ordinary node of Entry 839 has no residual coefficient excess.

The failure locus

$$P_3ab=0$$

is exactly the union of already labelled soft and coordinate-boundary supports. It must be handled by their iterated localization squares, not by adding a polar carrier.

Verification

• checker: research/nima/audit_triangle_soft_polar_beck_chevalley.py;
• packet: research/nima/triangle-soft-polar-beck-chevalley.json;
• allocator claim: seqclaim-07c1b00be3a4b15f4339fef8.

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