4034 — The Depth-Six Persistent Image Is Transverse to the Rank-Twenty-Six Low Sector
Question
The source-reachable quotient has the finite persistence profile
[ 53\longrightarrow 33\longrightarrow 27 ]
from (K)-depth three through five. Does the rank-(27) image persist at depth six, and is it merely the old rank-(26) low sector plus one occurrence class?
The second possibility was numerically tempting because the earlier degree-two Boolean adapter had dimension
[ 36=26+10. ]
Frozen test
For each residue chart (G_{12}) and (G_{31}), over (mathbf F_{32009}):
- construct the depth-three source-reachable quotient;
- map it directly to the depth-six exact quotient;
- verify that every depth-three relation reduces to zero at depth six;
- compare the persistent image (I) with the internally reduced rank-(26) low sector (L);
- test all ten labelled square-free degree-two occurrence generators.
No equality was inferred from dimensions.
Result
Both charts give
[ dim Q_3=53,\qquad dimoperatorname{im}(Q_3\to Q_6)=27,\qquad dimker(Q_3\to Q_6)=26. ]
All depth-three relations descend.
The comparison with the low sector is also identical in both charts:
[ dim I=27,\qquad dim L=26, ]
but
[ dim(I\cap L)=19,\qquad dim(I+L)=34. ]
Hence (L) is not contained in (I). Equivalently, the persistent image contains eight directions outside the low sector and omits seven low directions.
Each of the ten square-free occurrence generators adds one direction to (L), but no individual generator completes (I) over (L).
Narrow conclusion
The equality
[ 26=36-10=53-27 ]
is a dimension coincidence, not an identification of typed subspaces.
The depth-six persistence result strengthens the candidate stability of the rank-(27) image, but the image is a mixed quotient of low, occurrence, (K)-pole, and fiber data. It is not the rank-(26) low sector plus one distinguished occurrence line.
The next finite test should compute the canonical principal-angle analogue over the finite field: export bases for the common rank-(19) intersection and the complementary (7+8) directions, then identify their source-label and filtration provenance. No separate (K) and occurrence grading may be imposed after quotient.
Artifacts
research/benincasa/checkers/check_consecutive_kdepth_source_quotient_transition.pyresearch/benincasa/checkers/check_persistent_image_low_occurrence_comparison.pyresearch/benincasa/results/kdepth3-to6-source-quotient-transition-g12-p32009.jsonresearch/benincasa/results/kdepth3-to6-source-quotient-transition-g31-p32009.jsonresearch/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-05879b1bd4d57547b504a4e0.