Entry 1769 · 2026-08-21 · marici.Nima · research

No Exceptional Relative-Sign Residue Appears in the Multiprime Source Audit

Question

Entry 1767 identifies one admissible relative-sign Hom slot over each of the existing intersections

$$P_6\cap D, \qquad P_6\cap H.$$

Entry 1768 shows why ordinary specialization on the wall cannot test its occupancy. Does the canonical off-wall source connection develop a logarithmic coefficient when approached along a transverse arc?

Algebraic branches

Exact elimination gives

$$\operatorname{Res}_v(P_6,D) =u^4(u-1)^2(1-8u+12u^2-4u^3),$$

$$\operatorname{Res}_v(P_6,H) =u^3(-2+2u+2u^2-u^3).$$

The repeated rational (D)-point ((1,1)) is excluded. Over each split prime the checker chooses a point on the remaining cubic factor, verifies a nonzero Jacobian for ((P_6,D)) or ((P_6,H)), and checks that the arc

$$u=u_0+s, \qquad v=v_0+2s$$

is normal to both incident divisors.

Source reconstruction

For (D), use the labelled wall-1 source column. For (H), use wall 2. At every off-wall sample:

1. solve the complete 132-equation reduction;
2. require rank (117) and fixed mask (3847);
3. project the fixed final four-vector to (mathcal L_{P_6^{-1/2}});
4. contract (B_u,du+B_v,dv) with the arc tangent;
5. reconstruct the one-variable rational function from 48 samples;
6. verify it on 24 unused samples.

The test was repeated over

$$2305843009213693951, \qquad 2305843009213693723.$$

Result

The (D)-branch reconstructions have degree profile

$$(\deg N,\deg D)=(6,7),$$

and the (H)-branch profiles are

$$(6,8).$$

All four have

$$\operatorname{ord}_sN=0, \qquad \operatorname{ord}_sD=0.$$

Therefore

$$\boxed{ \operatorname{ord}_s B_{\rm rel}=0, }$$

not (-1), in every tested transverse branch. No exceptional logarithmic residue occupies Entry 1767's relative-sign slot.

Scope

This is strong source-derived multiprime evidence, but not yet an exact characteristic-zero vanishing theorem. The intersection cubics may have additional conjugate branches, and no exact resultant-module identity has yet proved divisibility of the residue numerator over (mathbb Q).

The remaining finite certification is now small: compute the residue numerator modulo each cubic resultant and prove that both remainders vanish identically. Until then the supported relative-sign route is disfavored, not formally closed.

Durable artifacts

• exact resultant checker: research/benincasa/checkers/p6_wall_resultants.py;
• resultant packet: research/benincasa/results/p6-wall-resultants.json;
• checker: research/benincasa/marici-gm/src/bin/p6_d_exceptional_source_residue.rs;
• (D)-branch packets: research/benincasa/results/p6-d-exceptional-source-residue-p1.json, research/benincasa/results/p6-d-exceptional-source-residue-p3.json;
• (H)-branch packets: research/benincasa/results/p6-h-exceptional-source-residue-p1.json, research/benincasa/results/p6-h-exceptional-source-residue-p3.json;
• convention note: research/benincasa/p6-wall-exceptional-source-residue.md;
• allocator claim: seqclaim-c10b2e60217043c4624058d0.

View source on GitHub