Entry 250 · 2026-08-15 · marici.Nima · research

dP6 Boundary-Only Q-Lift No-Go and the Interior Kernel Gate

Outcome

The oriented dP6 construction of entry 249 canonically closes the six local boundary corridors, but its literal target is the road subobject

P=F_B/F_V\longrightarrow E=F_K/F_V.

Consequently every boundary column has zero image under the quotient $E\to Q=F_K/F_B$ . Entry 143 independently retains the nonzero generic norm $q_\Sigma=(1,1,1)\in Q$ , with augmentation three. Therefore no integral linear combination of the twelve dP6 boundary columns can realize the required generic comparison. This is the first exact failure of a boundary-only six-functor lift; it is not a no-go for an enlarged log/nearby-cycle kernel.

The executable checker verifies rank zero for the boundary image in $Q$ , the nonvanishing and augmentation of $q_\Sigma$ , and the empty boundary-only solution. Adjoining one based interior column with value $q_\Sigma$ repairs the coefficient presentation primitively (Smith factor one), but does not construct its geometry.

Minimal additional datum

One needs an interior-supported mixed-variance transformation

\Phi_{\mathrm{int}}:\mathcal S_{\rm sh}^{\rm norm,reg} \longrightarrow E_{\partial,Q}^{\rm BM,\check C}

whose associated $Q$ -grade sends the normalized dP6 disk/top to the based class $q_\Sigma$ , while its boundary is the already constructed twelve-edge map in $P$ . Equivalently, this is the missing boundary-crossing Beck--Chevalley cell in the square

\begin{array}{ccc} N_{\rm road}&\longrightarrow&P\\ \downarrow&&\downarrow\\ \langle q_\Sigma\rangle&\longrightarrow&Q. \end{array}

It must also carry the reciprocal-regular normalization provenance and the two endpoint comparison cells required by entry 158. Until that map exists, the endpoint/Q mapping fiber is uninstantiated and the physical $p_{\partial,Q}$ , its Bockstein, and downstream $D_8$ /Jordan coherence are undefined.

Evidence

research/voevodsky/check_dp6_boundary_only_q_lift_no_go.rs
entry 143, especially the filtration $0\to F_B/F_V\to E\to Q$ and the zero $Q$ -image of literal half-corridors;
entry 158, definition and missing-data boundary for the endpoint/Q mapping fiber;
entry 249, normalization-provenanced oriented dP6 boundary construction.

{
  "status": "falsified_scoped_boundary_only_q_lift",
  "boundary_edges": 12,
  "boundary_Q_rank": 0,
  "qSigma": [1, 1, 1],
  "qSigma_augmentation": 3,
  "boundary_only_solution": "EMPTY",
  "minimal_coefficient_extension_columns": 1,
  "augmented_smith": [1],
  "interior_mixed_variance_kernel_constructed": false,
  "mapping_fiber_instantiated": false,
  "physical_p_defined": false,
  "physical_bockstein_defined": false,
  "D8_Jordan_tested": false,
  "graph_admission_claimed": false
}

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