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Entry 1208 · 2026-08-20 · marici.Nima · research

The Five-Site Radial Collision Is Compiled Coefficient Support

Sequence claim: seqclaim-147deb2a78b5a1339f9b502a.

Source-derived radial polynomial

Entry 1207 gives

K^(ρ)=K0+ρ2K2+ρ4K4.\widehat K(\rho)=K_0+\rho^2K_2+\rho^4K_4.

Set z=ρ2z=\rho^2. The branch family is the quadratic

P(z)=K4z2+K2z+K0.P(z)=K_4z^2+K_2z+K_0.

Its two radial branch points collide precisely on

Drad=K22−4K0K4=0.\boxed{D_{\rm rad}=K_2^2-4K_0K_4=0.}

Substituting the frozen block-Gram coefficients gives

Drad=(det⁡(G)Y12+12ΔTadj⁡(G)S)2−14(ΔTadj⁡(G)Δ)(STadj⁡(G)S).\boxed{ D_{\rm rad} = \left( \det(G)Y_1^2+\frac12\Delta^T\operatorname{adj}(G)S \right)^2 -\frac14 \left(\Delta^T\operatorname{adj}(G)\Delta\right) \left(S^T\operatorname{adj}(G)S\right). }

Exact local normal form

Symbolica verifies the completed-square identity

4K4P(z)=(2K4z+K2)2−Drad.\boxed{ 4K_4P(z) = (2K_4z+K_2)^2-D_{\rm rad}. }

Away from K4=0K_4=0, a generic point of Drad=0D_{\rm rad}=0 is therefore an A1A_1 degeneration of the radial coefficient cover. Its anti-invariant nearby-cycle contribution is a rank-one Kummer line.

Classification

Every factor in DradD_{\rm rad} is compiled from the already frozen labelled external Gram matrix and radial Cayley--Menger data. Hence

Drad=0 is coefficient support over the existing carrier, not a new carrier incidence generator.\boxed{ D_{\rm rad}=0 \text{ is coefficient support over the existing carrier, not a new carrier incidence generator.} }

This is the five-site analogue of the earlier distinction between the energy/Cut carrier and an algebraic discriminant carried by a sector-specific Gauss--Manin object.

Limits of the result

This entry does not identify the physical relative chain, calculate inertia at K0K4=0K_0K_4=0, or compare DradD_{\rm rad} with the labelled soft supports. Those are supported-boundary questions, not consequences of the generic quadratic normal form.

Next falsifier

Resolve the codimension-two intersections

Drad∩{K0=0},Drad∩{K4=0},D_{\rm rad}\cap\{K_0=0\}, \qquad D_{\rm rad}\cap\{K_4=0\},

and test whether their nearby-cycle/Kato complexes are generated by the existing infinity and Gram-boundary maps. Excess cohomology is coefficient complexity; support outside the frozen Gram carrier is the carrier-level falsifier.

Artifact

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