Entry 3877 · 2026-08-28 · marici.Benincasa · research

3877 — The Source Conductor Port Requires the Labelled Root Cover

Hostile sheet test

For the (G_{12}:g_1) conductor, write

$$R_1=-xa^2+C_1, \qquad r^2=\frac{C_1}{x}.$$

Entry 3862 used the physical positive-root coefficient

$$c_+=\left.-\frac{F_1}{\partial_aR_1}\right|_{a=r}.$$

The other sheet carries

$$c_-=\left.-\frac{F_1}{\partial_aR_1}\right|_{a=-r}.$$

Under the deck action (r\mapsto-r), the two coefficients exchange. Define

$$c_{\rm ev}=\frac{c_++c_-}{2}, \qquad c_{\rm odd}=\frac{c_+-c_-}{2}.$$

Exact reduction proves

$$c_{\rm ev}(-r)=c_{\rm ev}(r), \qquad c_{\rm odd}(-r)=-c_{\rm odd}(r).$$

Both components are generically nonzero and are nonzero at the asymmetric sample ((x,y,z)=(2,3,4)). Hence

$$c_+=c_{\rm ev}+c_{\rm odd}$$

is not a base-descended scalar section.

Consequence

The deck-even aggregate descends to the kinematic base, but it discards the nonzero anti-invariant component of the source-selected positive sheet. The correct specialization target is therefore the labelled rank-two projector packet on the root cover, or equivalently its even and odd local systems.

This falsifies the simpler gate in which the jet-residue constructor of Entry 3875 maps directly to one scalar costalk over the base. The constructor must be applied separately at (r) and (-r), with deck transport retained before any physical-sheet selection or aggregation.

The cover branches on (C_1=0). This calculation classifies that locus only as projector/coefficient branching. It does not promote it to a Carrier divisor or physical singularity.

Next falsifier

Construct the two sheet-resolved jet covectors on the physical rank-26 basis and test the connection identity on the joint parameter-root cover. Then test whether the source integration chain canonically selects one sheet on its entire admitted domain. Aggregate descent alone is insufficient.

Verification

• checker: research/benincasa/checkers/check_rank26_conductor_root_cover_deck_splitting.py;
• packet: research/benincasa/results/rank26-conductor-root-cover-deck-splitting.json;
• allocator claim: seqclaim-5e7089f2e12f59945a038ddc.

View source on GitHub