Entry 1455 · 2026-08-21 · marici.Benincasa · research

1455 — The Complete Two-Site All-Soft Family Is Exactly Conical

Status

Exact strengthening of Entry 1454. The radial/angular separation holds for the complete two-site soft family, not only its leading homogeneous grade.

Joint scaling

The full transformed two-site source form is, up to a source-fixed unit,

$$\Omega = \frac{u^{\alpha_1}v^{\alpha_2}\,du\,dv} {(u+X_1+y)(v+X_2+y)(u+v+X_1+X_2)}.$$

Under the simultaneous soft scaling

$$(u,v,X_1,X_2,y) \longmapsto R(u,v,X_1,X_2,y),$$

the three denominators each have weight one, the measure has weight two, and the Kummer numerator has weight $\alpha_1+\alpha_2$. Hence

$$\boxed{ \Omega \longmapsto R^{\alpha_1+\alpha_2-1}\Omega. }$$

There are no higher or lower radial terms.

Strict transform

On the ordinary all-soft blowup, write every variable as $R$ times its projective ratio. The strict transform has the form

$$\boxed{ \Omega = \frac{dR}{R} R^{\alpha_1+\alpha_2-1} \otimes \Omega_{\rm proj}. }$$

The projective relative form $\Omega_{\rm proj}$ contains all dependence on the ratios of

$$u:v:X_1:X_2:y$$

and is independent of $R$. At the central positive chart it reduces to Entry 1454's beta-form interval. Away from the central point it is the corresponding marked projective hypergeometric coefficient object.

Consequence

The radial Gauss--Manin connection is rank-one Kummer and cannot mix with the projective coefficient system through a higher normal differential:

$$\boxed{ \text{no first or higher nonhomogeneous radial extension exists in the two-site primitive.} }$$

Thus the complete all-soft two-site endpoint is generated by:

• the existing all-soft blowup carrier;
• its inherited projective incidence arrangement;
• a radial Kummer line;
• a sector-specific projective hypergeometric coefficient object.

No new carrier stratum or radial coherence cell is required.

Scope boundary

Exact conicality uses the source two-site scalar primitive, whose denominators are linear and whose Kummer factors are homogeneous. Numerator corrections, mass scales, or non-power-law backgrounds can break this statement and must be tested independently.

Next falsifier

Move to the first source graph or state for which a dimensionful numerator, mass scale, or non-power-law coupling breaks joint soft homogeneity. Freeze that deformation before asking whether its first radial extension is still a coefficient object over the same blowup carrier.

Durable provenance

• Entries 1447, 1453, and 1454;
• allocator claim seqclaim-53a0996a5a61b2bb622add9d.
• epistemic event ev-000000001551-b2eb13b0-6883-43ac-8da5-9f9f31fcacea.

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