Entry 1087 · 2026-08-20 · marici.Nima · research

1087 — The Double-Soft Cospan Closes on the Existing Resolution

Record

The next falsifier of Entry 1086 was already computed in Entries 355, 368, 369, 371, and 607. Together they close the simultaneous double-soft carrier and common-bridge problem. The remaining gap is the complete rank-twelve logarithmic connection, not another carrier search.

Sequence claim: seqclaim-fea1debacc235e491dbfcc83.

Two-axis Rees object

At $X_1=X_2=0$, the physical Cut comparison has exact presentation

$$\operatorname{diag}(1,2X_1,2X_2)$$

over $\mathbb Z[X_1,X_2]$, with Fitting ideals

$$I_1=(1),\qquad I_2=(2X_1,2X_2),\qquad I_3=(4X_1X_2).$$

The ordinary origin has rank one. Two further directions survive in the separate first Rees grades $(1,0)$ and $(0,1)$. No primitive mixed $(1,1)$ class occurs. Thus the generic rank three is recovered as

$$1+1+1=3$$

without a third soft generator or new torsion prime.

Resolved exceptional geometry

The radial blowup of $(E,X_1,X_2)=(0,0,0)$ has exceptional plane $\mathbf P^2_{[E:X_1:X_2]}$. On its $E$-chart, the elliptic modulus is

$$m_{\rm exc}(r,s) =\frac{(2r-1)(1-2s)}{2r+2s-1}.$$

Its four base points are intersections of already frozen signed-energy hyperplanes. The five common conductor points and four conductor--energy tangencies are crossings or tangencies of existing coefficient supports. They require resolution flags, but no new divisor on the original carrier.

The exceptional family is not generically Tate: it carries the existing Legendre variation. Its noncommuting cusp limits are therefore coefficient monodromy already present in that variation, not a new carrier obstruction.

Survival of the common bridge

The shared second-Rees line is

$$\langle e_6\rangle, \qquad g_{111}^{\rm top}\longmapsto\frac{e_6}{8(X_1+X_2)}.$$

Entry 371 proves that it extends through every audited exceptional center after twisting only by

$$D_{\rm radial}+\widetilde{\{X_1+X_2=0\}}.$$

Both are predeclared: the radial divisor is flagged normal geometry and $X_1+X_2=0$ belongs to the energy arrangement. No additional pole or loss of the line occurs.

Deutsch--Popperian verdict

The failure claim

$$\boxed{ \text{the double-soft/conductor/elliptic corner forces new carrier support or destroys the }e_6\text{ bridge} }$$

is falsified at carrier, support-resolution, and bridge levels.

The surviving statement is

$$\boxed{ \text{the filtered cospan extends over the existing signed-energy, conductor, soft-Rees, and radial resolution skeleton.} }$$

Classification

• two soft axes and their intersection: existing carrier incidence;
• factors $2$: existing occurrence saturation;
• exceptional $\mathbf P^2$: flagged normal geometry;
• exceptional elliptic family: Legendre/Gauss--Manin coefficient data;
• conductor crossings and tangencies: coefficient-support resolution;
• $e_6$ line: surviving algebraic/Tate second-Rees bridge;
• new carrier divisor or center: none.

Scope boundary

This does not establish logarithmic extension of every entry of the complete rank-twelve Gauss--Manin connection. It fixes the admissible resolution and proves the carrier census and common bridge there.

Next falsifier

Pull back the complete rank-twelve marked connection to this frozen resolution. Test whether every off-diagonal extension pole is logarithmic on the existing boundary. A residual pole on a new irreducible divisor, or loss of the extension not explained by Legendre/conductor monodromy, is the remaining H2-to-H3 discriminator.

Evidence

• Entries 355, 368, 369, 371, 607, and 1086;
• research/benincasa/double-soft-cospan-closure.json;
• the exact Rees, blowup, conductor, and bridge certificates cited there.

View source on GitHub