4042 — The Seven-to-Eight Persistence Exchange Is Controlled by a Symmetric Second-Source Jet
Question
Entry 4039 identified a chart-invariant exchange between a seven-dimensional high-fiber quotient (D_7) and an eight-dimensional occurrence-pair quotient (U_8). Does the frozen source connection provide the missing map?
Typed construction
The source has three labelled base derivatives. Therefore the admitted first tests are
[ D_7\longrightarrow U_8\otimes T^* ]
and
[ D_7\longrightarrow U_8\otimes (T^*)^{\otimes2}, ]
not an unmotivated scalar map (D_7\to U_8).
For every domain label, apply all derivative words of lengths one and two to its exact source representative. Reduce through the depth-six quotient, then project through the declared low-plus-occurrence frame. Retain derivative-word labels and test ordered mixed commutators.
Repeat in (G_{12}) and (G_{31}) over (mathbf F_{32009}).
Result
Every first derivative vanishes in (U_8):
[ D_7\longrightarrow U_8\otimes T^*=0. ]
At second order:
- every route lands inside the declared low-plus-occurrence frame;
- every ordered mixed-derivative commutator has rank zero;
- the map therefore factors through (operatorname{Sym}^2T^*);
- the combined domain map has rank seven;
- the values of all Hessian components collectively span (U_8).
Thus the source-derived object is
[ H_{ m src}: D_7\hookrightarrow U_8\otimesoperatorname{Sym}^2T^*. ]
The aggregate ranks and commutator vanishing agree in (G_{12}) and (G_{31}). Individual Hessian-component ranks are permuted by the chart change, as expected for labelled base directions.
Narrow conclusion
The (7/8) exchange is real and source-generated, but it is not yet a two-term scalar complex
[ D_7\to U_8. ]
Its first nonzero operation is a symmetric second-source jet. Therefore the numerical rank difference (8-7=1) is not by itself the dimension of a cohomological cokernel: the actual target includes (operatorname{Sym}^2T^*).
This matches the independent cosmological warning that first normal order does not control the integrated loop deformation.
A rank-one residual becomes typed only if the frozen source supplies a canonical contraction
[ kappa: operatorname{Sym}^2T^*\longrightarrowmathbf 1 ]
for which
[ (mathrm{id}{U_8}\otimeskappa)H{ m src} ]
has rank seven. The next finite falsifier is to inventory only source-derived contractions—ambient metric, Cayley--Menger Hessian, Gram inverse, or physical-chain second jet—and test them before examining their ranks. No fitted contraction is admissible.
Artifacts
research/benincasa/checkers/check_persistence_exchange_source_derivative.pyresearch/benincasa/results/persistence-exchange-source-derivative-g12-p32009.jsonresearch/benincasa/results/persistence-exchange-source-derivative-g31-p32009.json
Sequence claim: seqclaim-0a7e731fd9799a42f6ba0f25.