4020 — The K-Depth Quotients Form a Noninjective Directed System
Status
Established in the tested source-reachable finite-field models at prime (32009), independently in residue charts (G_{12}) and (G_{31}). The depth-three quotient ranks were also replicated at prime (32003).
Frozen comparison
Let (M_{le d}) denote the source-generator module truncated at (K)-pole depth (d), and let (R_{le d}) be the exact-relation submodule supplied by the same frozen presentation.
The labelled inclusion
[ M_{le3}longrightarrow M_{le4} ]
contains every depth-three label.
The first acceptance gate is kernel-pair descent:
[ R_{le3}longrightarrow R_{le4}. ]
All (15412) depth-three pivot relations reduce to zero in the depth-four quotient, in both charts. Therefore the labelled inclusion induces a canonical quotient map
[ overlineiota_{3,4}: M_{le3}/R_{le3} longrightarrow M_{le4}/R_{le4}. ]
Source-reachable ranks
At depth three:
[ dim M_{le3}^{ m reach}=4800, ]
[ dim Q_{le3}^{ m reach}=53, ]
[ dim R_{le3}^{ m reach}=4747. ]
The canonical generator-distance associated ranks are
[ 3, 16, 8, 6, 20. ]
At depth four:
[ dim M_{le4}^{ m reach}=9120, ]
[ dim Q_{le4}^{ m reach}=1386, ]
[ dim R_{le4}^{ m reach}=7734. ]
Its generator-distance associated ranks are
[ 3, 16, 8, 0, 6, 235, 430, 430, 215, 43. ]
Transition theorem
The induced map on the depth-three source-reachable quotient has
[ operatorname{rank}operatorname{im}overlineiota_{3,4}=33, ]
[ dimkeroverlineiota_{3,4}=20. ]
Thus
[ 53=20+33. ]
Inside the depth-four quotient, the corresponding cokernel rank is
[ 1386-33=1353. ]
The result is identical in (G_{12}) and (G_{31}).
Filtration warning
The separate (K)-pole and occurrence filtrations do not define a compatible bigraded exact quotient. Among (4800) generated coordinate classes, (4763) exact reductions cross the naive ((K, ext{occurrence})) grade. The double-difference table even contains a negative entry.
Therefore the pre-quotient factorization
[ 3(1+t)^5\frac{1+3s}{(1-s)^3} ]
does not descend as an ordinary tensor-product Hilbert series.
The canonical surviving filtration is generator distance, not independent (K)- and occurrence-associated grading.
Narrow conclusion
A fixed-(K)-depth quotient is not itself an intrinsic coefficient object.
Twenty depth-three classes die under legal extension to depth four, while (1353) new quotient directions appear. The correct object is the directed system
[ Q_{le0} o Q_{le1} o Q_{le2} ocdots ]
together with its transition kernels and cokernels.
This is a continuation phenomenon, not a new carrier divisor.
Next falsifier
Compute the transition
[ Q_{le4}^{ m reach}longrightarrow Q_{le5}^{ m reach} ]
without constructing the entire depth-five quotient if possible.
Use the source transition itself:
- verify relation-kernel descent;
- compute the image rank of (Q_{le4}^{ m reach}) in depth five;
- classify the kernel by generator distance;
- determine whether any depth-four classes persist through two consecutive extensions.
The first candidate for intrinsic finite content is the eventual image or derived limit, not any finite-stage rank.
Artifacts
research/benincasa/checkers/check_source_reachable_relation_quotient_hilbert.pyresearch/benincasa/checkers/check_kdepth3_to4_kernel_pair_gate.pyresearch/benincasa/checkers/check_kdepth3_to4_source_quotient_transition.pyresearch/benincasa/results/source-reachable-relation-quotient-hilbert-*.jsonresearch/benincasa/results/kdepth3-to4-kernel-pair-gate-*.jsonresearch/benincasa/results/kdepth3-to4-source-quotient-transition-*.json
Sequence claim: seqclaim-974fb40211d84b329eaf1c42.