20260817-351 — Both Soft Branches and the Nine-Master Supported Extension Close
Hard-to-vary claim
For the frozen homogeneous nine-master (q_{mathcal G_{12}})-residue system, the total-energy nearby object acquires no algebraic-kernel-to-elliptic logarithmic extension supported on
[ X_1X_2=0. ]
This claim concerns the supported principal part. It does not assert that all soft-local connection coefficients vanish.
Frozen structure
The source decomposition is
[ 1+2+2+4 ]
under (C_2^{(a)} imes C_2^{(b)}). The elliptic quotient occurs only in the final four-master character. The other three blocks belong to distinct algebraic Tate/Kummer characters.
The source involution is
[ xleftrightarrow y,\qquad aleftrightarrow b,\qquad e_8leftrightarrow e_9, ]
with the compensating sign from reversal of (dawedge db).
Direct branch
Entry 350 directly computed the (X_2=0) branch in the (X_1=1) chart. After the source-forced saturation
[ widetilde v_{ m alg}=v_{ m alg}/X_2^2, ]
the kernel-to-elliptic soft principal part vanished.
Transported branch
The point (X_1=0) is not a finite point of the (X_1=1) chart. It was therefore not evaluated by substituting an affine infinity.
Instead, the frozen source involution transports the calculation to the (X_2=1) chart. It sends
[ v_{ m alg}/X_2^2 longmapsto v_{ m alg}/X_1^2 ]
and exchanges (e_8,e_9). In the involuted adapted basis the residue matrix is identical, up to the source orientation sign. Consequently,
[ \boxed{ operatorname{PP}{X_1=0} \bigl(R{E_T}^{ m ext}\bigr)=0. } ]
This branch is source-derived by exact involutive transport; it is not an independent affine specialization.
Remaining algebraic blocks
The exact other-block connections give soft-principal ranks
[ (0,0,1) ]
at (X_2=0), and after site exchange,
[ (0,1,0) ]
at (X_1=0).
These rank-one poles remain internal to the corresponding algebraic rank-two character blocks. Exact character decomposition gives zero off-character entries into the elliptic block.
Therefore the full nine-master kernel-to-elliptic supported principal rank is
[ \boxed{0}. ]
Replication
The direct final-block reconstruction and the resulting nine-master synthesis agree over
[ p_1=2305843009213693951, \qquad p_2=2305843009213693921, ]
using disjoint deterministic streams.
Narrow result
[ \boxed{ ext{On }E_T=0, ext{ no algebraic-to-elliptic logarithmic extension is supported on }X_1X_2=0 } ]
in the frozen homogeneous nine-master de Rham system.
What remains at the soft branches consists of:
- the elliptic quotient degeneration;
- Tate/Kummer poles internal to algebraic character blocks;
- regular finite coefficient mixing.
All are carried by the existing site-soft divisor. No new cosmological carrier incidence is required by this test.
Scope boundary
This does not prove:
- compatibility with the physical relative integration chain;
- integral-lattice or integral-monodromy closure;
- extension across every nonsoft discriminant component;
- the global Cut--nearby comparison;
- absence of supported classes after character symmetry is broken by additional marked data.
Evidence
research/benincasa/soft-support-both-sites-certificate.jsonresearch/benincasa/soft-support-both-sites-replication-certificate.jsonresearch/benincasa/soft-support-nine-master-certificate.jsonresearch/benincasa/soft-support-nine-master-replication-certificate.json- executable commands:
soft-support-both-sites-testsoft-support-nine-master-test
Prior update
This closes the finite homogeneous soft-support loophole left by generic localization. It strengthens H2:
[ ext{shared carrier and calculus} + ext{sector-specific coefficient objects}. ]
It does not distinguish H2 from H3 globally, because the physical relative chain and Cut--nearby square remain unconstructed.
Next falsifier
Construct the physical relative-chain realization of the nine-master nearby object and test whether the chain boundary map commutes with the frozen infinity-Gysin quotient at both soft branches. A failure that cannot be represented by the existing marked denominator boundary or site-soft support would be the first evidence here for new carrier data.