20260817-352 — The Rank-Twelve Marked Extension Adds No Elliptic Soft Class
Question
Entry 351 closed the site-soft supported extension in the homogeneous nine-master (q_{mathcal G_{12}})-only module. The physical three-denominator sector contains three additional conductor/top quotient classes, so that result did not yet test the canonical marked localization extension
[ 0 o H^2(S_E) o H^2(S_Esetminus W_E) o H^1(W_E)(-1) o0 ]
of ranks (9 o12 o3).
The hard-to-vary claim tested here is:
[ \boxed{ ext{one of the three marked extension columns acquires a nonzero elliptic principal part on }X_1X_2=0. } ]
Frozen extension columns
At (E=0), put (x=X_1), (y=X_2), and (s=x+y). Retain the source-derived algebraic vector
[ v_0=x^2y^2\bigl((x^2-y^2)e_7+2e_8-2e_9\bigr). ]
The complete marked algebraic block from Entry 300 is
[ Theta_{101}^{ m fix} =-\frac{e_4}{4xy} -\frac{v_0}{4x^3y^3s}, ]
[ Theta_{110}^{ m fix} =-\frac{e_2}{4xy} +\frac{v_0}{4x^3y^3s}, ]
[ Theta_{111}^{ m filt} =\frac{e_6}{8s}. ]
No lift, support summand, or carrier cell is changed.
Exact Gysin identity
At (E=0), the infinity-Gysin map on ((e_7,e_8,e_9)) is
[ e_7mapstoomega_0, ]
[ e_8mapsto\frac{y^2}{2}omega_0-\frac{x^2}{2}omega_2, ]
[ e_9mapsto\frac{x^2}{2}omega_0-\frac{x^2}{2}omega_2. ]
Therefore
[ R_infty\bigl((x^2-y^2)e_7+2e_8-2e_9\bigr)=0 ]
as a polynomial identity. The other coordinates (e_2,e_4,e_6) are already in the algebraic Gysin kernel.
Soft principal parts
At (y=0), multiply the two wall columns by (y) before specializing. Their (v_0)-tails become finite multiples of
[ x^2e_7+2e_8-2e_9, ]
whose Gysin image remains zero on the specialized fiber.
At (x=0), exact source involution gives the corresponding vector
[ -y^2e_7+2e_8-2e_9, ]
again with zero Gysin image.
The primitive top column has denominator (s=x+y), so it is regular at either individual site-soft branch when the other site energy is nonzero.
Consequently,
[ \boxed{ operatorname{rank} R_infty!left( operatorname{PP}{X_1X_2=0}C{ m alg}
ight)=0. } ]
Executable replication
The polynomial identity, generic wall-column images, and both soft principal limits were checked at 128 samples over each of
[ p_1=2305843009213693951, \qquad p_2=2305843009213693921, ]
with zero mismatches.
Narrow result
Adding the three conductor/top classes of the canonical rank-twelve marked extension introduces simple Tate/Kummer soft poles internal to the algebraic kernel, but no new elliptic supported extension:
[ \boxed{ ext{rank-twelve soft nearby data}
ext{existing algebraic soft poles}
- ext{the previously known elliptic degeneration}. } ]
No new carrier datum is required.
Scope
This establishes a de Rham/Gysin statement for the frozen marked total-energy residue. It does not yet:
- pair the marked extension with the global physical relative chain;
- construct the chain-level Cut--nearby commutator;
- prove an integral normal form through the soft collision;
- include simultaneous (x=y=0);
- sew the three cyclic (q_{mathcal G_{ij}}) sectors.
Evidence
research/benincasa/marked-soft-support-certificate.json;research/benincasa/marked-soft-support-replication-certificate.json;research/benincasa/marici-gm/src/bin/marked_soft_support.rs.
Next falsifier
Pair the canonical physical Leray chain with the rank-twelve localization sequence at the corner
[ E_T=q_{mathcal G_{12}}=y_{12}=0. ]
Compute both orders of specialization—Cut then nearby, and nearby then Cut—using the frozen positive Cayley--Menger sheet and signed-minor boundary. The first nonzero commutator outside existing soft support or graph homology would falsify the current shared-calculus claim.