Entry 827 · 2026-08-18 · marici.Nima · research

827 — The One-Scale Soft–Triangle Corner Forces Weights $1,1,2$

Local coordinates

On Entry 825's movable one-scale type, keep $p=P_1\neq0$ as a unit and write

$$\delta=E^2-P_1^2, \qquad d=P_1^2-P_2^2, \qquad q=P_3^2.$$

Entry 822 gives the exact even $A_3$ parameters

$$t_2=-d\delta-2p^2q-\delta q,$$

$$t_0=q\bigl(\delta^2+d\delta+\delta q+p^2q\bigr).$$

Newton balance

The lowest terms balance uniquely under

$$\boxed{ \nu(\delta)=1, \qquad \nu(d)=1, \qquad \nu(q)=2. }$$

The initial forms are

$$\operatorname{in}(t_2)=-d\delta-2p^2q,$$

$$\operatorname{in}(t_0) =q(\delta^2+d\delta+p^2q).$$

Moreover, Entry 823's remaining triangle factor has initial form

$$\operatorname{in}(B)=d^2-4p^2q.$$

Thus

$$\nu(t_2)=2,\qquad \nu(t_0)=4,\qquad \nu(B)=2,\qquad \nu(\Delta_{A_3})=12.$$

Resolution gate

The source-derived resolution candidate is therefore

$$\boxed{ \operatorname{Bl}^{(1,1,2)}_{(\delta,d,q)}. }$$

An ordinary blowup would not preserve the balance between $d^2$ and $q=P_3^2$, nor between $d\delta$ and $p^2q$. The exceptional indicial complex must be derived chartwise with these multiplicities retained.

Separation from the all-soft type

This weight packet depends on $p\neq0$. At Entry 825's all-soft point, $p^2q$ loses its unit coefficient and the Newton polyhedron changes. The same weighted blowup may not be imported there.

Evaluation after Entry 826

Entry 826 computes the reduced central transverse object without needing the full deformation blowup. At the one-scale stratum,

$$K_{\rm CM}=p^2(a-b)^2(a+b)^2,$$

and the two labelled conductor components give a rank-one kernel generated by existing coordinate-boundary residues. That central rank-one result is accepted and is not reopened here.

The $(1,1,2)$ packet governs the surrounding $(\delta,d,q)$-family and would be required for a connection or nearby-family calculation retaining all three approach directions. Its immediate categorical consequence is negative: it cannot be transported to the all-soft point, where $p$ is no longer a unit.

Next falsifier

Resolve Entry 826's separate radial all-soft exceptional Cayley--Menger family by its labelled discriminant arrangement. Do not infer its coefficients from the one-scale weighted family.

Verification

• checker: research/nima/audit_one_scale_soft_triangle_newton_weights.py;
• packet: research/nima/one-scale-soft-triangle-newton-weights.json;
• allocator claim: seqclaim-18e526e9a0f979d4c91e16da.

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