Entry 3337 · 2026-08-27 · marici.Nima · research

3337 — Neither Soft Germ Locally Selects the e6 Extension Residue

Question

Entry 3332 shows that diagonal occurrence descent cannot insert the unique two-soft logarithmic class into (B_{e_6,q_0}). Does the full cleared rank-twelve source module at least fix either local residue of that coordinate?

Source-direct Laurent test

The rank-twelve reduction engine exports all 132 cleared source identities with 372 primitive coefficients. The Laurent harness was generalized, without changing its existing default (u)-axis behavior, to expand in

$$t=v-v_0$$

at fixed (u=0), while differentiating in the (v) direction.

For each soft center

$$v_0=0\quad(X_3=0), \qquad v_0=2\quad(X_2=0),$$

the calculation retained Laurent orders (-2) through (3), all three quotient generators, and every primitive coordinate. Polynomial source coefficients were reconstructed through degree sixteen and checked at a held-out point.

The tests were repeated at the two independent primes

$$2305843009213693951, \qquad 2305843009213693921.$$

Exact modular ranks

At (v=0), the Laurent system has rank

$$555.$$

At (v=2), it has rank

$$520.$$

Both ranks replicate exactly at both primes. In neither system is the target coordinate

$$\operatorname{Res}_{v=v_0}(B_v)_{e_6,q_0}$$

a pivot. Therefore it is not merely a pivot with free-column contamination; it is itself a free Laurent coordinate.

Result

Neither soft germ locally selects the (e_6,q_0) residue. The full cleared source equations admit local primitive freedom at both (X_3=0) and (X_2=0).

This rules out two proposed shortcuts:

• deriving the candidate residue from one soft germ;
• deriving the opposite pair by independently computing two locally fixed residues.

Scope

The result is local. It does not exclude a global rational section whose single denominator and numerator correlate the two free germs. In particular, the global no-infinity condition of Entry 3328 is invisible to two independent local Laurent solution spaces.

The status is now:

• unique admissible global logarithmic line: established;
• source coefficient (C_2): established;
• local residue selection at either soft germ: falsified;
• global source correlation of the residue pair: open.

Next falsifier

Restrict the cleared source system to (u=0) and solve one global polynomial module with common denominator

$$v(v-2).$$

Test the residue-difference functional directly. If it remains free, the rank-twelve source equations do not select the candidate even globally. If it is fixed, verify the result at unused primes and by exact cleared polynomial identities.

Verification

Harness: research/benincasa/checkers/audit_marked_extension_source_laurent_lead.py.

Aggregate checker: research/benincasa/checkers/audit_two_soft_source_residue_freedom.py.

Aggregate packet: research/benincasa/results/two_soft_source_residue_freedom.json.

Allocator claim: seqclaim-3a1d53648f49cb66bc46ce42.

View source on GitHub