Entry 2367 · 2026-08-25 · marici.Benincasa · research

2367 — The Dimensionless Soft-Triangle Discriminants Close on Frozen Support

Test

After Entry 2366 removes the common scale, the coefficient kernel is

$$k(t,\kappa,\xi) =t^4-(10+8\kappa\xi)t^2 +16\kappa^2+40\kappa\xi+16\xi^2+9.$$

The finite falsifier is a projection discriminant outside the frozen endpoint, signed-face, and coordinate-boundary arrangement.

Sequence claim: seqclaim-816ee1550ce6fe58e42b20c4.

Exact projection discriminants

Eliminating $\xi$ gives

$$\boxed{ \operatorname{Disc}_{\xi}k =64(\kappa^2-1)(t^2-1)(t^2-9). }$$

Eliminating $\kappa$ gives the symmetric identity

$$\boxed{ \operatorname{Disc}_{\kappa}k =64(\xi^2-1)(t^2-1)(t^2-9). }$$

Finally,

$$\boxed{ \operatorname{Disc}_{t}k =65536(\kappa^2-1)^2(\xi^2-1)^2 \,k(0,\kappa,\xi). }$$

The last displayed product is literal multiplication by $k(0,\kappa,\xi)$.

Source classification

Every factor has frozen provenance:

• $\kappa=\pm1$: endpoints of the external momentum-triangle face;
• $\xi=\pm1$: endpoints of the collapsing loop triangle;
• $t=\pm1$: signed $a/p$ face branches, with $t=1$ the $q_{\mathfrak g_2}/q_{\mathfrak g_{31}}$ collision;
• $t=\pm3$: signed $a/p$ face branches, with $t=-3$ the $q_{\mathfrak g_3}$ section;
• $k(0,\kappa,\xi)=0$: collision of a branch point with the existing coordinate boundary $a=0$.

Entry 2368 subsequently checks the full critical ideal and shows that this last factor is not a singular divisor of the total double cover. It is critical only for the chosen $t$-projection.

Thus no projection produces an undeclared divisor.

Result

$$\boxed{ \text{all three projection discriminants close on the frozen Cayley--Menger/marked arrangement.} }$$

This is stronger than the single $a^2$-discriminant test of Entry 2364: the dimensionless kernel remains support-compatible under every coordinate projection tested.

Scope

Projection discriminants do not determine the complete relative Gauss--Manin extension. In particular, this result does not exclude a higher-codimension costalk, nonsplit extension, or integral defect supported on intersections of the listed divisors.

Durable verification

• research/benincasa/check_soft_triangle_dimensionless_discriminants.py;
• research/benincasa/soft-triangle-dimensionless-discriminants.json;
• exact symbolic discriminant identities;
• epistemic event ev-000000003242-c5c0b8f3-9bff-43f1-af6a-6f5a918dcd68.

Next falsifier

Build the logarithmic relative complex on the endpoint cover and compute its costalks at:

$$(\kappa,\xi)=(\pm1,\pm1),$$

$$t=\pm1,\pm3,$$

and

$$t=0,\qquad k(0,\kappa,\xi)=0.$$

Any residual cone outside the existing soft, triangle, signed-face, and coordinate-boundary maps would be the first coefficient-level failure.

View source on GitHub