20260817-353 — Physical Cut–Nearby Comparison Extends Through Site-Soft Support as a Rees Isomorphism
Question
The source-normalized physical Cut--nearby commutator is already nonzero on the generic nonsoft locus:
[ [psi_{E=0},operatorname{Res}{q{mathcal G_{12}}=0}]
-\frac{2pi^2}{xy} left(y e_3+x e_5+e_6 ight) ]
in the positive occurrence chart. Entries 301, 305, 308, and 312 identify its four occurrence/deck companions with the primitive rank-three conductor lattice.
Entries 351--352 closed the de Rham infinity-Gysin extension at (xy=0), but did not yet determine whether the integral physical comparison survives there.
The hard-to-vary claim tested here is:
[ \boxed{ ext{the physical enhanced/conductor comparison acquires new torsion, support, or an unrepresented vanishing class at }xy=0. } ]
Frozen integral maps
The enhanced-point realization is
[ Phi_{ m exc}
\begin{pmatrix}
1&-1&1&-1
1&1&-1&-1
1&-1&-1&1
end{pmatrix}.
]
The horizontal conductor embedding is
[
J=
\begin{pmatrix}
2&0&1
0&2&1
0&0&1
end{pmatrix},
]
and the occurrence quotient is
[
K=
\begin{pmatrix}
0&0&1&-1
0&1&0&-1
1&-1&-1&1
end{pmatrix}.
]
They obey the exact generic identity
[ Phi_{ m exc}=JK, ]
where
[ ker K=mathbb Z(1,1,1,1), \qquad operatorname{coker}K=0. ]
Soft Rees scaling
The physical exceptional functional has weighted target coordinates
[ (y e_3,;x e_5,;e_6) ]
after stripping its common source Leray scalar and invertible generic monomials.
Near (y=0), retain the soft normal (t=y) and set
[ D_y(t)=operatorname{diag}(t,1,1). ]
Near (x=0), use
[ D_x(t)=operatorname{diag}(1,t,1). ]
Do not set (t=0) before forming the comparison. Exact multiplication gives, on both branches,
[ \boxed{ D_{ m soft}Phi_{ m exc}
(D_{ m soft}J)K. } ]
Thus Cut then nearby and nearby then Cut agree as filtered occurrence lattices.
Smith/Rees calculation
For either branch, unimodular polynomial row and column operations reduce
[ D_{ m soft}J ]
to
[ \boxed{operatorname{diag}(1,2,2t).} ]
Equivalently, its determinantal ideals are
[ I_1=(1), \qquad I_2=(2), \qquad I_3=(4t). ]
Because (K) is a unimodular presentation of the augmentation quotient, the same local Smith/Rees data govern (D_{ m soft}Phi_{ m exc}).
At the ordinary special fiber (t=0),
[ operatorname{rank}=2. ]
The missing third direction is not destroyed: it occurs at the next Rees grade with weight (t). Therefore the filtered ranks are
[ \boxed{2+1=3.} ]
Result
The generic Smith type
[ (1,2,2) ]
extends to the soft Rees type
[ \boxed{(1,2,2t).} ]
Consequently:
- no new torsion prime appears;
- the existing ((mathbb Z/2)^2) occurrence saturation remains the only integral defect;
- the only new factor is the predeclared site-soft normal (t);
- the diagonal occurrence relation remains the sole source relation;
- no physical Cut--nearby class is lost after retaining the Rees filtration.
The hard-to-vary claim is falsified.
Narrow conclusion
At either individual site-soft branch,
[ \boxed{ psi^{ m Rees}{xy=0}operatorname{Cut}{12} simeq operatorname{Cut}{12}psi^{ m Rees}{xy=0} } ]
on the source occurrence quotient, after the already established orientation twist and universal Leray normalization.
The comparison is not strictly rank-preserving on the ordinary special fiber. It is rank-preserving only as a filtered/Rees object. This is another concrete instance of
[ ext{ordinary pullback}
otRightarrow ext{filtered nearby object}. ]
Classification
- leading soft rank two: ordinary special-fiber occurrence data;
- shifted rank one: higher Rees grade on existing soft support;
- factors (2,2): existing occurrence-identification saturation;
- (t): existing site-soft Cartier normal;
- elliptic image: none for the commutator;
- graph homology: none;
- new carrier datum: none.
Scope
This result uses the previously derived source-oriented Leray comparison and strips its common nonzero generic scalar. It does not yet cover:
- the simultaneous double-soft point (x=y=0);
- cyclic sewing of all six literal top-sector occurrences through soft support;
- intersections with conductor and elliptic discriminants beyond the individual site-soft branches;
- a global integral basis over the full signed-energy arrangement.
Evidence
research/benincasa/soft-rees-smith-certificate.json;research/benincasa/marici-gm/src/bin/soft_rees_smith.rs;- entries 226, 301, 305, 308, 312, and 352.
Next falsifier
Compute the two-parameter Rees/Smith form at the simultaneous double-soft point
[ x=y=0. ]
The finite test must retain both normals independently and determine whether the comparison is generated by (x), (y), their product, and the two existing occurrence factors of two. A new irreducible factor, torsion prime, or extra incidence generator would be the first failure of the present filtered physical comparison.