4039 — K-Depth Persistence Exchanges Seven Fiber Jets for Eight Occurrence Pairs
Question
Entry 4034 established that the depth-six persistent image (I) and rank-(26) low sector (L) satisfy
[ dim I=27,\qquad dim L=26,\qquad dim(I\cap L)=19. ]
Which labelled directions account for the complementary ranks seven and eight, and are they preserved across residue charts?
Frozen construction
Use the exact depth-three to depth-six source transition over (mathbf F_{32009}). Order all generators by their existing source labels. Perform provenance-preserving Gaussian reduction on the joint family of:
- the twenty-seven persistent source generators;
- the twenty-six internally reduced low generators.
Export:
- nineteen exact common representatives;
- a labelled basis for (L/(I\cap L));
- a labelled basis for (I/(I\cap L)).
Repeat independently in (G_{12}) and (G_{31}).
The bases are canonical relative to the frozen source-label order; no claim of basis independence is made.
Result
Both charts produce the same decomposition
[ 19+7+8. ]
The seven low directions outside persistence are exactly
[
\begin{aligned}
&(0,1,1,1,1,1;(0,4)),
&(0,1,1,1,1,1;(0,5)),
&(0,1,1,1,1,1;(0,6)),
&(0,1,1,1,1,1;(0,7)),
&(0,1,1,1,1,1;(1,4)),
&(0,1,1,1,1,1;(4,0)),
&(0,1,1,1,1,1;(4,1)).
end{aligned}
]
Thus the lost low quotient is concentrated in high fiber degree.
The eight persistent directions outside the low sector are exactly the square-free occurrence-pair labels
[
\begin{aligned}
&(0,1,1,1,2,2;(0,0)),
&(0,1,1,2,1,2;(0,0)),
&(0,1,1,2,2,1;(0,0)),
&(0,1,2,1,2,1;(0,0)),
&(0,1,2,2,1,1;(0,0)),
&(0,2,1,1,1,2;(0,0)),
&(0,2,1,2,1,1;(0,0)),
&(0,2,2,1,1,1;(0,0)).
end{aligned}
]
Of the ten possible square-free pairs, the two absent quotient representatives are
[ (0,2,1,1,2,1;(0,0)) ]
and
[ (0,1,2,1,1,2;(0,0)). ]
The nineteen common classes were exported as exact combinations of source-labelled persistent generators and ambient quotient labels.
Narrow conclusion
Increasing (K)-depth does not simply add or remove a homogeneous sector. It performs a chart-invariant exchange:
[ ext{seven high fiber jets} quad ightsquigarrowquad ext{eight occurrence-pair directions}. ]
The net rank increase from the common core is one, but that one is the Euler characteristic of a (7\to8) exchange, not a distinguished scalar occurrence generator.
This is the first explicit source-labelled evidence that the correct object is a mixed (K)-occurrence-fiber filtered complex. The next falsifier is to derive a source differential from the seven discarded fiber jets to the eight surviving occurrence pairs. Its rank must be seven with one-dimensional cokernel. A fitted (7\times8) matrix is inadmissible.
Artifacts
research/benincasa/checkers/check_persistent_image_low_occurrence_comparison.pyresearch/benincasa/results/persistent-image-low-occurrence-comparison-g12-p32009-k3-to6.jsonresearch/benincasa/results/persistent-image-low-occurrence-comparison-g31-p32009-k3-to6.json
Sequence claim: seqclaim-df5e11f94d7d9128d591cf53.