Entry 2436 · 2026-08-25 · marici.Benincasa · research

2436 — The Generic Gram Vanishing Cycle Is Invisible to the Physical Infinity Current

Question

Entry 2430 constructs the rank-one elliptic nearby-cycle channel at the external Gram wall. Entry 2433 shows that the original loop-vector cycle has trivial transport there. The remaining comparison is the source-normalized Leray pairing: does the physical boundary current actually meet the elliptic vanishing cycle?

Physical support at infinity

On the compactification chart

$$b=\frac1s, \qquad a=\frac ts,$$

the source-positive loop distances have the generic large-radius form

$$a=\sqrt{r^2+2\alpha r+A}, \qquad b=\sqrt{r^2+2\beta r+B}.$$

Therefore

$$\boxed{ \lim_{r\to\infty}\frac ab=1. }$$

The physical UV boundary current on the infinity curve is supported at $t=1$, with angular information retained in the first normal correction rather than in a different boundary point.

Separation from the Gram nodes

The generic infinity curve is

$$W^2=F_P(t), \qquad F_P(t)=P_1^2t^4-(P_1^2+P_2^2-P_3^2)t^2+P_2^2.$$

At the physical boundary point,

$$\boxed{F_P(1)=P_3^2.}$$

On each physical Heron component the quartic is a perfect square. Its node polynomial evaluated at $t=1$ is respectively

$$P_3, \qquad -P_3, \qquad P_1+P_2=P_3,$$

after imposing the corresponding labelled wall relation. Thus the physical infinity current and the Gram nodes are disjoint at generic nonsoft kinematics. Their first possible collision is $P_3=0$, cyclically relabelled in the other infinity charts, which is existing soft--Gram support.

Support-sensitive Leray pairing therefore gives

$$\boxed{ \left\langle \Gamma_{\infty}^{\rm phys}, \phi_{\rm Gram}^{\rm ell} \right\rangle =0 }$$

on every generic nonsoft physical Heron wall.

Recovery of the transverse normal

The vanishing pairing does not erase the interaction normal. At the branch (P_3=P_1+P_2), use source physical coordinates

$$u=\delta(P_1^2), \qquad v=\delta(P_2^2), \qquad w=\theta^2.$$

For the magnitude normals

$$\nu_i=P_i^2-X_i^2,$$

the weighted Jacobian at the Gram wall is

$$\frac{\partial(\nu_1,\nu_2,\nu_3)}{\partial(u,v,w)} = \begin{pmatrix} 1&0&0\\ 0&1&0\\ \dfrac{P_1+P_2}{P_1}& \dfrac{P_1+P_2}{P_2}& -P_1P_2 \end{pmatrix},$$

with

$$\boxed{\det=-P_1P_2\ne0}$$

away from soft support. Hence two first-grade tangent scores together with the second-Rees angle score recover all three magnitude-normal directions.

For the complete interaction tower, pull back all ten nonconstant labels

$$3\text{ linear}+6\text{ quadratic}+1\text{ cubic}$$

through the associated weighted coordinate change. Their exact coefficient matrix has rank ten. The three source identities of Entry 2413 retain rank three, so the physical weighted quotient has rank

$$\boxed{10-3=7.}$$

Because $w=\theta^2$ and the source tower reaches cubic normal degree, the complete physical angular score tower must extend through order six:

$$w^3=\theta^6.$$

Second angle order recovers the missing first-normal direction; it does not by itself exhaust the complete interacting tower.

Result

$$\boxed{ \text{generic Gram coefficient monodromy is physically unactivated, while the weighted physical score tower remains faithful.} }$$

No new Carrier support or sector-specific observer kernel appears. The elliptic nearby cycle remains genuine coefficient data, but the source physical current does not read it on the generic Gram wall.

Scope

This theorem is support-local and generic nonsoft. Soft--Gram intersections remain governed by the resolved soft complex of Entry 193 and the later faithful soft observer. The rank-seven statement is at the source-kernel weighted-jet level; compatibility of all six physical angle grades with the rank-sixty marked Gauss--Manin extension is not yet computed. Global UV-renormalized finite parts are not recomputed.

Durable evidence

• research/benincasa/check_gram_leray_weighted_score.py;
• research/benincasa/gram-leray-weighted-score.json;
• Entries 193, 794, 2430, and 2433;
• sequence claim seqclaim-ff702f586a2addad4f7f4106.

Next falsifier

Transport the now-proved rank-seven weighted interaction tower through the rank-sixty marked localization sequence through physical angle order six. Verify that the contact-normal score cospan remains faithful at the Gram wall and its soft intersections. A kernel would be a genuine physical readout obstruction, but not new Carrier support unless it lies outside the already frozen soft/Gram incidence.

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