4026 — The First K-Depth Persistence Profile Is Fifty-Three to Thirty-Three to Twenty-Seven
Status
Established at prime (32009) in residue charts (G_{12}) and (G_{31}).
Consecutive continuation
Entry 4020 constructed the canonical quotient map
[ Q_{le3}^{ m reach}longrightarrow Q_{le4}^{ m reach} ]
and found
[ 53longrightarrow33 ]
with kernel rank (20).
The next source-derived map
[ Q_{le4}^{ m reach}longrightarrow Q_{le5}^{ m reach} ]
also passes kernel-pair descent. It has
[ 1386longrightarrow1366 ]
with kernel rank (20).
Thus a rank-20 boundary-loss mechanism recurs at two consecutive extensions.
Two-step persistence
The composite source map
[ Q_{le3}^{ m reach}longrightarrow Q_{le5}^{ m reach} ]
passes exact-relation descent and has image rank
[ 27. ]
Equivalently,
[ 53=26+27, ]
where (26) depth-three classes vanish after two legal extensions.
Since the one-step image has rank (33), exactly six of those 33 classes die at the second extension:
[ 33=6+27. ]
The complete persistence profile presently known is therefore
[ 53supset33supset27. ]
Both charts give the same ranks.
Narrow conclusion
The recurring rank-20 kernel of consecutive maps does not imply that each finite source stage loses an independent rank-20 block from every earlier stage. Persistence must be computed by composites.
The rank-27 two-step image is the first candidate for stable finite coefficient content, but it is not yet an eventual-image theorem.
No new carrier divisor appears. The phenomenon belongs to continuation and exact-relation transport.
Next falsifier
Compute one of the following equivalent finite gates:
- the image of (Q_{le3}^{ m reach}) in (Q_{le6}^{ m reach});
- the image of the known rank-27 subspace under (Q_{le5} o Q_{le6}).
The second is preferable if a basis for the rank-27 image can be exported, because it avoids reducing all 4800 depth-three generators again.
Possible outcomes:
- rank (27): first evidence of stabilization;
- smaller positive rank: persistence filtration continues;
- rank (0): all finite-depth classes are boundary artifacts.
Do not identify rank (27) with a physical coefficient system before this gate.
Artifacts
research/benincasa/checkers/check_consecutive_kdepth_source_quotient_transition.pyresearch/benincasa/results/kdepth4-to5-source-quotient-transition-g12-p32009.jsonresearch/benincasa/results/kdepth3-to5-source-quotient-transition-g12-p32009.jsonresearch/benincasa/results/kdepth3-to5-source-quotient-transition-g31-p32009.json
Sequence claim: seqclaim-1f42fd415e1bc3e8d95803cc.