20260817-355 — Double-Soft Physical Comparison Is a Two-Axis Rees Module
Question
Entry 353 extended the physical Cut--nearby comparison through either individual site-soft branch. The remaining corner is
[ x=X_1=0,\qquad y=X_2=0, ]
where both site-soft normals vanish simultaneously.
The hard-to-vary claim is:
[ \boxed{ ext{the double-soft corner produces a new mixed class, torsion prime, or support generator not obtained from the two soft axes and occurrence identifications.} } ]
Frozen maps
Retain the source occurrence matrices
[ Phi_{ m exc}
\begin{pmatrix}
1&-1&1&-1
1&1&-1&-1
1&-1&-1&1
end{pmatrix},
]
[
J=
\begin{pmatrix}
2&0&1
0&2&1
0&0&1
end{pmatrix},
\qquad
K=
\begin{pmatrix}
0&0&1&-1
0&1&0&-1
1&-1&-1&1
end{pmatrix},
]
with
[ Phi_{ m exc}=JK, \qquad ker K=mathbb Z(1,1,1,1), \qquad operatorname{coker}K=0. ]
The physical enhanced coordinates are
[ (y e_3,;x e_5,;e_6). ]
Thus the two-normal Rees operator is
[ D(x,y)=operatorname{diag}(y,x,1). ]
Exact comparison
Over
[ R=mathbb Z[x,y], ]
exact multiplication gives
[ \boxed{ D(x,y)Phi_{ m exc}
(D(x,y)J)K. } ]
Unimodular row operations
[ R_1mapsto R_1-yR_3, \qquad R_2mapsto R_2-xR_3 ]
reduce (D(x,y)J) to
[ operatorname{diag}(2y,2x,1). ]
After permutation, the exact presentation is
[ \boxed{operatorname{diag}(1,2x,2y).} ]
Why there is no ordinary Smith form
The ring (mathbb Z[x,y]) is not a principal ideal domain, and the second determinantal ideal is not principal. Therefore a one-variable Smith triple is not a valid invariant at the double corner.
The correct Fitting ideals are
[ \boxed{ I_1=(1), \qquad I_2=(2x,2y), \qquad I_3=(4xy). } ]
The nonprincipal ideal ((x,y)) records the already frozen transverse intersection of the two site-soft Cartier divisors. It is not evidence for a third carrier generator.
Multi-Rees grades
At the ordinary origin (x=y=0), only the unit direction survives:
[ operatorname{rank}_{(0,0)}=1. ]
The other two directions occur separately in the first normal grades:
[ operatorname{rank}{(1,0)}=1, \qquad operatorname{rank}{(0,1)}=1. ]
There is no additional primitive direction in grade ((1,1)). Hence
[ \boxed{ 1+1+1=3, } ]
recovering the generic conductor/enhanced rank as a two-axis Rees object.
Result
The tested claim is falsified.
- no new torsion prime appears;
- the only integer saturation remains the existing two occurrence factors of (2);
- support is exactly (x=0), (y=0), and their ordinary intersection;
- no product-only (xy) class appears;
- the diagonal occurrence relation remains the sole relation;
- the filtered Cut--nearby comparison remains exact.
Thus
[ \boxed{ ext{double-soft physical comparison}
ext{one central grade}
oplus ext{one }x ext{-grade} oplus ext{one }y ext{-grade}. } ]
No new carrier datum is required.
Classification
| Datum | Home |
|---|---|
| unit direction | ordinary double-soft fiber |
| (2x) direction | (X_1)-Rees grade plus occurrence saturation |
| (2y) direction | (X_2)-Rees grade plus occurrence saturation |
| ((2x,2y)) | existing two-axis soft incidence |
| (4xy) determinant | product of frozen soft normals and occurrence factors |
| new carrier structure | none |
Scope
This is the integral occurrence/Rees comparison after the source orientation twist and common Leray normalization. It does not yet:
- sew all three cyclic marked-Cut sectors at the double-soft point;
- include simultaneous collision with an independent conductor or elliptic discriminant;
- construct a global integral basis over the entire energy arrangement;
- establish the full wavefunction normalization omitted by the source.
Evidence
research/benincasa/double-soft-rees-certificate.json;research/benincasa/marici-gm/src/bin/double_soft_rees.rs;- entries 301, 308, 312, 353.
Next falsifier
Transport the two-axis presentation cyclically to all three (q_{mathcal G_{ij}}) sectors while retaining the six lower-denominator occurrences. Compute the global occurrence/Cech gluing at pairwise and triple site-soft intersections. A non-Cech relation, new torsion prime, or divisor outside the frozen signed-energy arrangement would falsify global shared-carrier assembly.