20260817-350 — Saturated Soft-Support Extension Test at (X_2=0)
Question
Does the final four-master algebraic--elliptic extension acquire a class supported at the soft intersection
[ E_T=X_2=0 ]
that is invisible on the generic total-energy divisor?
Frozen input
Work in the homogeneous chart
[ X_1=1,\qquad u=E_T,\qquad y=X_2=\frac{u+v}{2}-1. ]
The source-defined algebraic kernel generator degenerates at (u=0,v=2) as
[ v_{ m alg}=y^2\bigl(0,1-y^2,2,-2\bigr). ]
The supported test therefore uses the saturated generator
[ widetilde v_{ m alg}=\frac{v_{ m alg}}{X_2^2} ]
before specializing to (X_2=0). No carrier cell, support summand, or fitted normalization is added.
Finite falsifier
- Reconstruct the exact rational (u)-connection of the final four-master block.
- Extract its logarithmic residue along (u=0).
- Conjugate that residue into the regular saturated frame [ (e_6,widetilde v_{ m alg},e_7,widetilde e_8). ]
- Expand at the soft normal (v-2=0).
- Read the kernel-to-quotient principal part. A nonzero principal part would be a soft-supported extension class on the existing soft carrier.
Result
At both independent finite fields,
[ p_1=2305843009213693951, \qquad p_2=2305843009213693921, ]
with disjoint deterministic reconstruction streams, every rational fit closes and the Laurent orders agree.
The only soft pole in the saturated total-energy residue lies in the elliptic quotient block:
[ operatorname{PP}{X_2=0} R{E_T}
\begin{pmatrix}
0&0&0&0
0&0&0&0
0&0&0&-\frac12
0&0&0&0
end{pmatrix}.
]
The algebraic-to-elliptic extension principal part is
[ \boxed{ operatorname{PP}{X_2=0} \bigl(R{E_T}^{ m ext}\bigr)=0. } ]
A regular finite off-diagonal coefficient survives:
[ operatorname{FP}{X_2=0} \bigl(R{E_T}^{ m ext}\bigr)
\begin{pmatrix}
0&-\frac14
0&0
end{pmatrix}.
]
Thus the test distinguishes vanishing of a supported logarithmic class from vanishing of all finite mixing: only the former is established.
Narrow conclusion
For the final four-master block, after the source-forced (X_2^2)-saturation of (v_{ m alg}), no kernel-to-elliptic extension class is supported at the generic (X_2=0) soft intersection of the total-energy divisor.
The remaining pole is the expected quotient degeneration, and the regular (-1/4) mixing is coefficient-level connection data. This calculation requires no new carrier datum.
Classification:
[ \boxed{ ext{existing soft carrier} + ext{elliptic quotient degeneration} + ext{regular coefficient mixing}; quad ext{no new supported extension detected}. } ]
Scope
This is a modular, generic fiberwise de Rham calculation in the final four-master sector. It does not establish:
- extension through every discriminant component;
- an integral-lattice statement;
- compatibility with the physical relative integration chain;
- vanishing at the other soft corner (X_1=0);
- vanishing of supported classes in the remaining five algebraic masters;
- the full Cut--nearby comparison.
Evidence
research/benincasa/soft-support-saturated-certificate.jsonresearch/benincasa/soft-support-saturated-replication-certificate.json- executable command:
cargo run --release --bin marici-gm -- soft-support-test <output.json> - replication:
add
--features replication-prime
Next falsifier
Repeat the saturation analysis at the other site-soft branch (X_1=0), then test the complete nine-master algebraic kernel. Any nonzero supported kernel-to-elliptic principal part must be classified as soft-support coefficient data unless it requires an incidence divisor absent from the frozen energy carrier.