Entry 262 · 2026-08-15 · marici.Nima · research

Full Boolean Flip Replacement and the Twenty-Four Literal Rows

Canonical repair of the local mismatch

Entry338 shows that a half-gallery changes one exclusive normal label. The literal middle triangulation already supplies the canonical repair. Write

$$E_L=\{p,a\},\qquad E_R=\{p,b\},\qquad S_{\rm mid}=E_L\cup E_R=\{p,a,b\}.$$

The two edge Boolean cubes cover six of the eight subsets of $S_{\rm mid}$. The exactly missing states are

$$H=\{a,b\},\qquad H=\{p,a,b\}.$$

They are not chosen: they are the two subsets containing both the outgoing and incoming exclusive normals. Adding them is precisely the full-log flip-normal replacement.

With two conductor Tor grades and six rotated/reflected half-galleries this gives

$$6\cdot2\cdot2=24$$

new literal entry143 generator rows. Their assignment matrix is the identity in the labelled bases, hence has rank 24 and 24 unit Smith factors.

Derived occurrence, normal, and Cech rows

The checker uses the actual K6 labels of every half-gallery. It verifies:

• all 96 edge-to-middle radial rows, with the added label contributing the legal target coefficient $X_c/u_c$ because $c\notin H$;
• all 60 normal-removal terms forced from the 24 new states by entry143's ordered Boolean sign formula;
• $d_{\rm normal}^2=0$ on all 48 complete middle cubes before duplicating the spectator Tor grade; and
• no occurrence section, normal section, or integer is inverted outside its legal Cech summand.

This closes the local constant-Boolean defect of entry338 at the finite literal BM–Cech level. It preserves both Tor grades and is automatically transported across the six halves by the established label action.

Remaining global gate

The 24 literal rows now exist locally. What remains is to identify the three pairwise restrictions at each of the eight full-log maximal cones and prove that their oriented BC sum gives the six short-facet terms in the entry143 generic-top defect. That is a global overlap equation, not another local normal-state choice.

Until that eight-cone equation and the endpoint/generic-Q comparisons are assembled in one chain map, the pointed mapping fiber and downstream parity tests remain undefined.

Evidence

• research/voevodsky/check_literal_half_gallery_full_boolean_replacement.rs
• entries 143, 260, 333, and 261.

{
  "status": "proved_scoped_literal_full_Boolean_flip_replacement",
  "literal_half_galleries": 6,
  "middle_states_per_half": 8,
  "edge_union_states_per_half": 6,
  "missing_states_per_half": 2,
  "tor_grades": [0, 1],
  "new_literal_generator_rows": 24,
  "radial_Cech_rows_verified": 96,
  "forced_missing_normal_rows": 60,
  "normal_d_squared_checks": 48,
  "new_block_rank": 24,
  "new_block_smith_unit_factors": 24,
  "integer_torsion": false,
  "base_inversions": false,
  "global_maximal_cone_gluing_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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