Entry 368 · 2026-08-17 · marici.Benincasa · research

The First Joint Conductor--Elliptic--Soft Cospan Corner Closes

Question

Entry 367 replaced strict Cut--nearby commutation by a filtered cospan whose unique common geometric line is (e_6). The first hostile support test is where the total-energy elliptic node, both conductor discriminants, and one site-soft normal meet. The hard-to-vary claim tested here is

$$\boxed{ \text{the filtered cospan loses rank, acquires new torsion/support, or loses its }e_6\text{ bridge at the first joint soft corner}.}$$

We compute the iterated nearby object

$$E=0\quad\text{followed by}\quad y=t=0, \qquad x\ne0.$$

The (x=0, y\ne0) branch follows by the frozen source involution. The simultaneous double-soft point is not included.

Conductor behavior

At (E=0), the exact conductor discriminants of Entry 307 become

$$\Delta_1=4x^2t^2, \qquad \Delta_2=4x^2t^2.$$

Both vanish to order two. Therefore each Kummer connection restricts to

$$-\frac12d\log\Delta_i=-d\log t+\text{regular},$$

with integral residue and trivial semisimple monodromy around (t=0):

$$T_{Delta_1}=T_{Delta_2}=1.$$

Thus the conductor collision introduces no new character. Its integral half-sum lattice remains the existing ((\mathbb Z/2)^2) occurrence saturation.

Logarithmic cospan leg

At (E=0), put

$$H=(x^2-t^2)e_7+2e_8-2e_9.$$

After multiplying the two wall columns by their source-derived soft Rees normal (t), and stripping only units on (x(x+t)\ne0), integral representatives are

$$L_1=-(x+t)e_4-H,$$

$$L_2=-(x+t)e_2+H,$$

$$L_3=e_6.$$

At (t=0), the minor on rows ((e_4,e_2,e_6)) is

$$x^2,$$

a unit on the chosen generic soft branch. Hence the regularized logarithmic leg retains rank three and has no additional soft invariant factor.

Cut--nearby cospan leg

In enhanced coordinates, the physical Cut leg is

$$D_tJ, \qquad D_t=\operatorname{diag}(t,x,1).$$

Its ordinary special fiber has rank two, while its exact Rees/Smith type is

$$\boxed{(1,2,2t).}$$

The missing ordinary direction survives at the next (t)-grade. No new torsion prime or irreducible support factor occurs.

Bridge and elliptic typing

The ordinary special logarithmic span uses

$$(e_2,e_4,e_6,e_7,e_8,e_9),$$

while the ordinary Cut span uses ((e_5,e_6)). Their intersection remains

$$\boxed{\langle e_6\rangle.}$$

Thus the second-Rees bridge of Entry 367 survives the joint corner in ordinary grade zero.

Entry 352 independently proves that the soft principal parts of the marked rank-twelve columns have zero infinity-Gysin image. Therefore the joint corner creates no new marked-to-elliptic supported extension. This does not compute the pure elliptic variation at the further soft degeneration; it keeps that block separate.

Verdict

The tested failure claim is falsified on the first individual soft branch:

$$\boxed{ \text{joint corner} = \text{quadratic conductor character} +(\mathbb Z/2)^2 +t\text{-Rees grade} +\langle e_6\rangle, }$$

with the independent elliptic block and no new carrier datum.

Classification

Update to H2 versus H3

This is a high-value success for filtered H2. At the first locus where conductor, elliptic total-energy, Cut--nearby, and soft structures meet, the two cospan legs remain controlled by the same predeclared support-sensitive calculus. No sector-specific operation is needed.

The result remains narrower than a full joint-nearby theorem because the order (E\to0) then (t\to0) is fixed and the pure elliptic soft block has not been recomputed.

Evidence

• research/benincasa/marici-gm/src/bin/joint_soft_conductor_elliptic_cospan.rs;
• research/benincasa/joint-soft-conductor-elliptic-cospan-certificate.json;
• Entries 307--308, 352--355, and 367.

Next falsifier

Compute the genuinely two-parameter nearby object in ((E,t)), rather than only the iterated (E)-then-(t) specialization. Compare both operation orders and retain:

1. the pure elliptic nilpotent;
2. the two conductor Kummer characters;
3. the marked rank-three cospan legs;
4. the (e_6) second-Rees line; and
5. the integral half-sum lattice.

A nonzero order commutator outside the already known algebraic exceptional class, or a new mixed factor not generated by (E,t,2), would be the next H2-to-H3 update. After that, the simultaneous (x=y=0) point is the final triangle soft corner.

View source on GitHub