Entry 2363 · 2026-08-25 · marici.Benincasa · research

2363 — The Physical Soft Current Lies on the Triangle Rees Corner

Typing question

Entries 2360--2362 describe the algebraic $X_1=0$ fiber at generic $(X_2,X_3)$. They establish a maximal-parabolic coefficient extension

$$0\longrightarrow M_{16}^{\rm del} \longrightarrow M_{20} \longrightarrow R_4^{\rm res} \longrightarrow0$$

and contextual faithfulness of its marked score ports. They do not imply that a generic point such as $(0,3,5)$ lies in the closure of the physical Euclidean routing chain.

The finite test is to derive the complete physical $X_1$-soft chart from the frozen Cayley--Menger faces before importing that rank decomposition.

Sequence claim: seqclaim-c0f6063560fe4de262b2e203.

The base triangle forces a second normal

Set $x=X_1$. The fixed-base triangle face is

$$F_{123} =((X_2+X_3)^2-x^2)((X_2-X_3)^2-x^2).$$

At $x=0$, Euclidean incidence forces $X_2=X_3$. The lossless weighted chart is therefore

$$X_2=p+\frac{x\kappa}{2}, \qquad X_3=p-\frac{x\kappa}{2},$$

for which

$$F_{123} =x^2(\kappa^2-1)(4p^2-x^2).$$

Thus the physical exceptional base retains the labelled normal

$$-1\leq\kappa\leq1.$$

The earlier generic-soft rank-sixteen object cannot be called the literal physical soft fiber. Event 3235's initial physical rank-sixteen interpretation is withdrawn by correction event 3237.

The loop face and its marked endpoint

On the $q_{\mathcal G_{12}}$-residue, write

$$c=y_{12}=-E,\qquad b=y_{31},\qquad E=x+2p.$$

The second forced coordinate is

$$b=E+x\xi.$$

The corresponding triangle face factors exactly as

$$F_{cb} =x^2(\xi^2-1) (4p+x\xi+x) (4p+x\xi+3x).$$

Hence $-1\leq\xi\leq1$ on the exceptional physical interval. The labelled walls obey

$$q_{\mathfrak g_1}=x(\xi+1), \qquad q_{\mathfrak g_{23}}=2p+x\xi.$$

At generic $p\ne0$, $q_{\mathfrak g_{23}}$ is a unit on the exceptional current, whereas $\xi=-1$ is the marked $q_{\mathfrak g_1}$-endpoint after removing the common soft normal. Therefore the physical boundary has no $q_{\mathfrak g_{23}}$-supported Gysin component.

This does not imply that the bulk relative period annihilates $R_4^{\rm res}$.

Finite strict transform

The complete residue-surface Cayley--Menger polynomial has exact order two:

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

where

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

At fixed external parameters,

$$da\wedge db=x\,da\wedge d\xi.$$

Consequently the normal Jacobian cancels the $x$-factor in $\sqrt{K_0}$, and

$$\frac{da\wedge db}{\sqrt{K_0}} \longrightarrow \frac{da\wedge d\xi}{\sqrt{K_{\rm exc}}}$$

on the source positive sheet. The weighted physical bulk period therefore exists without adding a support summand.

The four raw quotient numerators

$$1,\qquad a,\qquad a^2,\qquad a^3$$

remain polynomially independent on this strict transform. This is only a candidate-provenance statement; independence or nonvanishing of their periods has not been established.

Ordinary specialization ranks

At the exact physical point $(X_1,X_2,X_3)=(0,3,3)$, the source closure has:

$$\operatorname{rank}=1$$

under the physical tangent $(0,1,1)$, and

$$\operatorname{rank}=4$$

under the two ordinary soft-plane tangents $(0,1,0),(0,0,1)$.

These are ordinary associated-fiber closures. They omit the weighted $\kappa$-normal and cannot be promoted to the nearby/Rees object.

Classification

• carrier support: existing site-soft plus momentum-triangle support;
• marked exceptional endpoint: existing $q_{\mathfrak g_1}$ incidence;
• $q_{\mathfrak g_{23}}$-supported boundary activation: absent for $p\ne0$;
• exceptional coefficient geometry: finite strict-transform $K_{\rm exc}$;
• ordinary physical-tangent closure: rank one;
• ordinary two-tangent closure: rank four;
• weighted $\kappa$-Rees recovery: uncomputed;
• bulk pairing with $R_4^{\rm res}$: uncomputed;
• new Carrier datum: none.

The refined H2 architecture survives. The correction is that physical soft specialization is an iterated soft--triangle Rees problem, not the generic algebraic soft divisor.

Scope

The theorem is generic in $p\ne0$. It does not cover the all-soft point, prove period independence, construct the weighted nearby-cycle module, or identify the resulting object with the separate rank-twelve soft Tate line.

Durable verification

• research/benincasa/check_x1_soft_physical_current_factorization.py;
• research/benincasa/x1-soft-physical-current-factorization.json;
• research/benincasa/check_x1_soft_physical_strict_transform.py;
• research/benincasa/x1-soft-physical-strict-transform.json;
• research/benincasa/check_rank26_physical_soft_triangle_closure.py;
• research/benincasa/rank26-physical-soft-triangle-closure.json;
• correction event ev-000000003237-e5448700-4666-4271-82f5-7e35a5441276.

Next falsifier

Construct the labelled weighted Rees module in the simultaneous normals

$$x=X_1,\qquad \kappa=\frac{X_2-X_3}{X_1}.$$

Derive the strict transforms of every marked denominator and exact differential, then determine whether the four ordinary soft-plane directions lift to a flat rank-four nearby object or acquire additional Cartier grades. Only after that module descends may its physical bulk period and score closure be compared with $R_4^{\rm res}$.

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