4013 — The Source Jet Generates a Boolean–K-Fiber Module with Rational Hilbert Series
Status
Established for the frozen source generator graph through (K)-depth four in residue charts (G_{12}) and (G_{31}) at prime (32009). The depth-three orbit was also constructed before this prediction.
Source generators
A basis label has the form
[ (k;ell_0,ell_1,ell_2,ell_3,ell_4;(a,b)), ]
where:
- (k) is the (K)-pole grade;
- each (ell_iin{1,2}) is a labelled pole level for the (q)-pole and four marked poles;
- ((a,b)) is fibre bidegree.
The source connection has two typed generator families.
The (K)-generator acts by
[ kmapsto k+1 ]
and shifts fibre degree by a monomial appearing in an external derivative of (K).
The five occurrence generators act by
[ ell_imapstoell_i+1 ]
and shift fibre degree by a monomial appearing in the corresponding external derivative of (q_i).
These generators were extracted from the frozen connection formula, not inferred from failed residuals.
Orbit census
Starting from the three source numerator monomials, the depth-three reachable graph has rank
[ 4800 ]
in both charts. Its generator-distance profile is
[ 3, 33, 165, 519, 1065, 1383, 1083, 465, 84. ]
The Hilbert census factorizes between (K)-pole and occurrence degree. Through depth three:
[ H_{le3}(s,t)
3(1+t)^5 left( 1+6s+15s^2+28s^3
ight). ]
The occurrence coefficients are
[ 1,5,10,10,5,1, ]
the Boolean rank profile on five labelled pole occurrences.
The (K)-fibre coefficients are
[ 1,6,15,28, ]
which equal
[ (k+1)(2k+1) ]
for (k=0,1,2,3).
This predicted the depth-four coefficient
[ 45 ]
and total rank
[ 3cdot 2^5(1+6+15+28+45)=9120. ]
The independently rerun depth-four graph has rank (9120) in both charts and the predicted (K)-marginal (4320=3cdot2^5cdot45).
Rational series
The resulting source-generator Hilbert law is
[ H(s,t)
3(1+t)^5 sum_{kge0}(k+1)(2k+1)s^k
3(1+t)^5\frac{1+3s}{(1-s)^3}. ]
This is a statement about the labelled source-generator module before exact-relation quotienting.
Narrow conclusion
Finite marked-pole closure failed because it retained the Boolean occurrence factor while truncating the intrinsically unbounded (K)-fibre factor.
The correct next object is therefore a source-derived module or bicomplex with:
- a finite Boolean occurrence algebra on five labels;
- an unbounded filtered (K)-fibre module with cubic-denominator Hilbert growth;
- chart transitions exchanging the two fibre-degree directions.
No new carrier divisor is indicated.
Next falsifier
Construct the exact-relation submodule inside this factored generator module. Test whether its quotient preserves the Hilbert factorization or introduces genuine coupling between the Boolean and (K)-fibre factors.
The first finite test is at (K)-depths three and four:
- compute the relation-submodule Hilbert table;
- compute the quotient Hilbert table;
- test rank-one factorization in ((k, ext{occurrence degree}));
- transport both tables between (G_{12}) and (G_{31}).
Artifacts
research/benincasa/checkers/check_source_generator_trigraded_orbit.pyresearch/benincasa/checkers/factor_source_generator_trigraded_orbit.pyresearch/benincasa/results/source-generator-trigraded-orbit-*.jsonresearch/benincasa/results/source-generator-trigraded-orbit-*-factorization.json
Sequence claim: seqclaim-7d88ac33f582769632b41685.