Entry 2364 · 2026-08-25 · marici.Benincasa · research

2364 — The Soft-Triangle Strict Transform Has Only Endpoint Branch Support

Frozen chart

Entry 2363 derives the physical weighted chart

$$X_1=x,\qquad X_2=p+\frac{x\kappa}{2},\qquad X_3=p-\frac{x\kappa}{2},\qquad b=E+x\xi.$$

The physical exceptional ranges are

$$-1\leq\kappa\leq1,\qquad -1\leq\xi\leq1.$$

Sequence claim: seqclaim-092bd48470bc3b1c2426222a.

Exceptional coefficient kernel

The complete $q_{\mathcal G_{12}}$-residue Cayley--Menger polynomial has the source-derived expansion

$$K_0=x^2K_{\rm exc}+O(x^3),$$

with

$$\begin{aligned} K_{\rm exc}={}&a^4-(10+8\kappa\xi)p^2a^2\\ &+\left( 16\kappa^2+40\kappa\xi+16\xi^2+9 \right)p^4. \end{aligned}$$

Treating this as a quadratic in $A=a^2$ gives the exact discriminant

$$\boxed{ \operatorname{Disc}_{A}(K_{\rm exc}) =64p^4(1-\kappa^2)(1-\xi^2). }$$

Thus every branch collision on the generic $p\ne0$ exceptional surface lies on one of the four already frozen endpoint divisors

$$\kappa=\pm1,\qquad \xi=\pm1.$$

The strict transform introduces no independent branch divisor.

Marked arrangement and source valuation

The five source walls become

$$\begin{aligned} q_{\mathfrak g_1}&=x(\xi+1),\\ q_{\mathfrak g_2}&=a-p+x\left(\frac{\kappa}{2}-1\right),\\ q_{\mathfrak g_3}&=a+3p+x\left(1+\xi-\frac{\kappa}{2}\right),\\ q_{\mathfrak g_{23}}&=2p+x\xi,\\ q_{\mathfrak g_{31}}&=a-p-\frac{x\kappa}{2}. \end{aligned}$$

The two colliding marked walls retain their divided difference:

$$\boxed{ \frac{q_{\mathfrak g_2}-q_{\mathfrak g_{31}}}{x} =\kappa-1. }$$

Hence their collision is supported on the existing positive base-triangle endpoint $\kappa=1$.

After the normal Jacobian cancels the square-root factor, the source form has exceptional valuation $-1$. Multiplying by $x$ gives the exact leading rational coefficient

$$\boxed{ \frac{a+p} {2p(a-p)^2(a+3p)(\xi+1)}. }$$

The pole at $\xi=-1$ is the already identified $q_{\mathfrak g_1}$ endpoint. The double pole at $a=p$ is the collision of the two existing labelled walls, not an undeclared support component.

Result

$$\boxed{ \text{the physical soft--triangle strict transform closes on the frozen endpoint and marked-incidence carrier.} }$$

The new structure is coefficient-theoretic: a sector-specific exceptional Cayley--Menger kernel and a nontrivial Rees valuation of the source form. No new Carrier divisor is required.

Scope

This is an exact associated-grade calculation. It does not determine:

• the rank of the full logarithmic relative cohomology of $K_{\rm exc}$;
• the Cartier length induced by the source's $x^{-1}$ valuation;
• monodromy at the four endpoint divisors;
• the physical period rank of the four candidate quotient numerators;
• the all-soft specialization $p=0$.

Durable verification

• research/benincasa/check_x1_soft_physical_strict_transform.py;
• research/benincasa/x1-soft-physical-strict-transform.json;
• exact symbolic kernel, discriminant, wall, and source-normalization identities;
• epistemic event ev-000000003239-1e7eb88c-8420-4dc9-a63e-428b3337e16f.

Next falsifier

Construct the logarithmic relative de Rham complex of

$$w^2=K_{\rm exc}$$

with the retained labelled walls and endpoint divisors. Compute its cohomology, inertia, and source-generated score closure. A rank or monodromy class supported away from

$$p(1-\kappa^2)(1-\xi^2)(a-p)(a+3p)=0$$

would be the first new-support obstruction at the physical soft corner.

View source on GitHub