Entry 251 · 2026-08-15 · marici.Nima · research

Entry143 Generic Top Defect and the dP6 Twelve-to-Six Comparison Gate

Result

The literal empty-face generator of entry 143 is a canonical graded lift in $E=F_K/F_V$ of the generic top generator in $Q=F_K/F_B$. Its promoted Čech differential has nine facet terms:

• the three long-facet terms survive in $Q$, with coefficients $X_D/u_D$ and augmentation three;
• the six short-facet terms lie in $P=F_B/F_V$ and form the defect of the graded section $Q\dashrightarrow E$.

At the cellular incidence level all nine coefficients are primitive. The six-entry short-facet defect vector has Smith factor one. Rotation and reflection preserve the long/short partition; reflection reverses the oriented top with the already fixed road-orientation character.

This refines entry 250. The target does possess a canonical interior top lift, but completing the source map now requires a concrete comparison from the twelve oriented dP6 boundary intervals to these six short-facet defect rows:

$$C_1(\partial dP_6)\longrightarrow P \quad\text{with}\quad d_E(\widetilde q_\Sigma)-\widetilde{d_Qq_\Sigma} =\Phi_\partial(d[dP_6]).$$

Entries 249, 334 give each local interval and its carrier image. They do not yet derive the global $12\to6$ mixed-variance matrix on every occurrence, normal-circle, Tor, and Čech grade. Consequently this entry does not instantiate the endpoint/Q mapping fiber.

Minimal next executable

Enumerate the twelve oriented dP6 edges and the six literal short-facet generators, expand both sides through entry143's $[S,H]$ differential, and solve

$$M_{12\to6}\,\partial[dP_6]=\delta_P(\widetilde q_\Sigma)$$

with the already fixed local Rees/KN columns, endpoint framing, reflection, and $D_3$. Its homogeneous kernel and Smith form will decide whether the interior lift is canonical, ambiguous, or obstructed. No matrix entry may be inserted merely from the carrier quotient.

Evidence

• research/voevodsky/check_entry143_generic_top_defect.rs
• research/voevodsky/check_global_k6_koszul_cech_promotion.rs
• entries 143, 249, 334, and 250.

{
  "status": "proved_scoped_target_top_defect",
  "E_top_lift_canonical": true,
  "total_facet_terms": 9,
  "Q_long_terms": 3,
  "P_short_defect_terms": 6,
  "Q_augmentation": 3,
  "P_defect_smith": [1],
  "D3_partition_covariant": true,
  "dp6_boundary_to_short_defect_rows_constructed": false,
  "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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