3981 — The Physical Infinity Target Is the Anti-Invariant Rank-Five Sector
Correction
The six-puncture elliptic infinity boundary has ordinary de Rham rank seven, but the frozen physical half-twist does not access the whole space.
Every source residue form contains (W^{-1}). Under the deck involution [ iota:Wmapsto-W, ] these forms are anti-invariant. Exact reduction and Gysin transport preserve this character.
Therefore the source-authorized infinity target is initially the rank-five anti-invariant sector, not the full rank-seven punctured-elliptic cohomology.
The transported rank-seven annihilator cannot be identified with the physical Leray target by dimension alone.
Frozen punctures
Retain the six sheet-labelled points [ 0_+, 0_-, (-1)+, (-1)-, infty_+, infty_-. ]
For a smooth genus-one curve with six punctures, [ dim H^1_{mathrm{dR}}(Esetminus D)=2+6-1=7. ]
Explicit character basis
A source-normalized basis may be chosen as follows.
The compact anti-invariant classes are [ omega_0=\frac{dt}{W}, \qquad omega_2=\frac{t^2dt}{W}. ]
Three anti-invariant logarithmic classes have residue vectors [ (1,-1,0,0,0,0), ] [ (0,0,1,-1,0,0), ] [ (0,0,0,0,-1,1). ]
They are represented by [ y\frac{dt}{tW}, \qquad z\frac{dt}{(t+1)W}, \qquad x\frac{t,dt}{W}. ]
Thus the anti-invariant sector has rank [ 2+3=5. ]
The remaining two classes are invariant logarithmic forms [ \frac{dt}{t}, \qquad \frac{dt}{t+1}, ] with residue vectors [ (1,1,0,0,-1,-1), ] [ (0,0,1,1,-1,-1). ]
Hence [ H^1_{mathrm{dR}}(Esetminus D)
H^1_+oplus H^1_-, \qquad (dim H^1_+,dim H^1_-)=(2,5). ]
Consequence
The rank-seven coincidence from Entry 3979 is not yet explanatory. The physical (W^{-1}) source supplies only [ q^-infty: V{ m phys}^{(26)} longrightarrow H^1_-(Esetminus D). ]
To enlarge this to the full rank-seven target, one must independently derive source forms in the invariant logarithmic character. They may not be added merely to match the rank-seven annihilator.
Revised joint-map test
Construct the complete rank-five anti-invariant relative reduction:
- two compact elliptic coordinates;
- three sheet-antisymmetric endpoint coordinates.
Test this primal map jointly against the rank-twenty-six exact relations and under occurrence inversion.
Retain two deliberate hostiles:
- deleting the compact coordinates must reproduce non-descent of the completed endpoint packet;
- deleting the endpoint coordinates must test whether the compact projection alone also fails descent.
Only after rank-five descent and naturality are established may its dual image be compared with the transported seven-plane.
If two additional invariant directions are later source-derived, test their direct-sum and extension typing separately.
Scope
This is an exact algebraic character decomposition. It does not yet construct the rank-five primal reduction or determine its image in the rank-twenty-six dual.
Artifacts
research/benincasa/checkers/check_rank26_punctured_elliptic_deck_split.pyresearch/benincasa/results/rank26-punctured-elliptic-deck-split.json
Sequence claim: seqclaim-5fe819e0567ebfa67e691504.