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Entry 1895 · 2026-08-22 · marici.Nima · research

Entry 1895 — The Six-Site Physical Branch and Homogeneous Cartier Excess Are Distinct Coefficient Layers

Terminal comparison

Entries 1893–1894 establish a generic source-activated divisor (D_6(k,z)). Entries 1888–1889 establish a homogeneous length-six descended Cartier object, assembled from three occurrence-labelled length-two components at (k=0).

The remaining question is whether the generic physical branch specializes to that Cartier object.

Flat closure of the physical branch

Let

A(k)=1026+69k−500k2+196k3.A(k)=1026+69k-500k^2+196k^3.

The generic critical section has a polynomial closure obtained from

A(k)(z,x,v,w)=(Nz,Nx,Nv,Nw).A(k)(z,x,v,w)=(N_z,N_x,N_v,N_w).

Since (A(0)=1026\ne0), its special fiber is the single point

p0=(x,v,w,z)=(829,469,469,499).p_0= \left(x,v,w,z\right) = \left( \frac{82}{9}, \frac{46}{9}, \frac{46}{9}, \frac{49}{9} \right).

The special-fiber Jacobian is (1026) times a permutation matrix. Thus the closure is regular there and its special fiber has reduced length one.

Comparison with the exceptional scheme

The homogeneous critical scheme is

(L2,M),(L^2,M),

with

L=−2+x−6z+4v+w,M=−6+2x−6z+5v−w.L=-2+x-6z+4v+w, \qquad M=-6+2x-6z+5v-w.

The point (p_0) lies on (L=M=0), but the exceptional object has Cartier length two along the entire affine critical line. Moreover,

D6(0,z)=114(9z−49),∂zD6∣k=0=1026≠0.D_6(0,z)=114(9z-49), \qquad \partial_zD_6|_{k=0}=1026\ne0.

Hence the horizontal physical branch meets the exceptional support at one reduced point and with base intersection multiplicity one. It does not specialize to the length-two Cartier line.

Normal-order data

All three source-wall multipliers vanish to exactly second order in (k):

ord⁡k(α12)=ord⁡k(α34)=ord⁡k(α56)=2.\operatorname{ord}_k(\alpha_{12})= \operatorname{ord}_k(\alpha_{34})= \operatorname{ord}_k(\alpha_{56})=2.

Their second grades are nonzero. The transverse Hessian determinant vanishes to exactly first order:

ord⁡kdet⁡Htr=1.\operatorname{ord}_k\det H_{\rm tr}=1.

Thus the physical node degenerates at the Gram boundary, but the resulting normal filtration does not turn its reduced flat closure into the global Cartier double line.

Terminal decision

The five-site source activation extends to a genuine genericsix-site physical coefficient divisor;the homogeneous length-six Cartier object is a distinct verticalGram-supported coefficient layer.\boxed{ \begin{aligned} &\text{The five-site source activation extends to a genuine generic}\\ &\text{six-site physical coefficient divisor;}\\ &\text{the homogeneous length-six Cartier object is a distinct vertical}\\ &\text{Gram-supported coefficient layer.} \end{aligned} }

Therefore the tested six-site sector has the architecture

unchanged occurrence-resolved Carrier+ horizontal physical coefficient support+ vertical Cartier excess.\boxed{ \text{unchanged occurrence-resolved Carrier} +\text{ horizontal physical coefficient support} +\text{ vertical Cartier excess}. }

No undeclared incidence operation, fitted support summand, or new Carrier stratum was required. H2 survives and receives a stronger refinement: sector-specific coefficient objects may contain horizontal and purely special-fiber layers that must not be identified by their intersection.

Next research frontier

The next independent falsifier should determine whether this horizontal-plus- vertical decomposition recurs for another six-site matching orbit or at seven sites. The present objective is closed; repeating the same incidence without a new arity or orbit would add confirmation but little entropy gain.

Durable verification

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