Entry 244 · 2026-08-15 · marici.Nima · research

Strict K6-to-Entry223 Top Comparison No-Go and the Odd Pair-Homotopy Gate

Record

Date: 2026-08-15

Status: falsified for the strict integral support-preserving comparison from the literal $K_6$ top complex to the entry223 three-facet/three-pair SNC complex. The unique $D_3$-equivariant rational comparison is half-integral. Its integral obstruction is a primitive $\mathbb Z/2$ class. This is not a no-go for a derived log-excess correspondence with an additional pair/corridor homotopy generator. No graph admission is claimed.

Strict comparison problem

The literal source has cellular ranks

$$(1,9,21,14)$$

in top, facet, edge, and vertex degrees. The entry223 SNC target has ranks

$$(1,3,3)$$

in top, long-road facet, and pair-intersection degrees, with pair incidence

$$B=\begin{pmatrix} -1&0&1\\ 1&-1&0\\ 0&1&-1 \end{pmatrix}.$$

Fix the only strict top/facet comparison compatible with the named grades: the source top maps with coefficient $+1$, the three long facets map identically with their cellular signs, and the six short facets map to zero. Literal support permits an edge column only when that edge contains a unique long diagonal, and then only in the two target pair rows incident to that road. This leaves 24 integral variables.

Integral calculation

The supported chain equations have rank 21 and affine rank 3. Adding strict $C_3$-rotation covariance raises the rank to 23 and leaves affine rank 1. Adding the physical reflection equations gives a full-rank 24-variable system with a unique rational solution.

That solution has twelve nonzero coefficients, all equal to $\pm\tfrac12$. Hence it is not integral. Modulo two, the coefficient matrix has rank 23 while the augmented matrix has rank 24. An explicitly selected maximal minor has determinant $+2$. Consequently the Smith factors are

$$\underbrace{1,\ldots,1}_{23},2,$$

and the strict obstruction group is exactly $\mathbb Z/2$.

This is the earliest exact falsifier for the proposed strict support-preserving top comparison. It proves that the six short-facet contractions cannot be derived as integral strict rows under the stated literal support and $D_3$ requirements.

Why an ordinary odd column does not repair it

Solving the non-reflection equations modulo two determines a unique reflection-defect vector. It has support on twelve reflection rows: four for each long road. This is exactly the expected size of the six oriented pair packets with two corridor boundary rows each.

That numerical match does not authorize treating the pair packet as one additional degree-zero variable. Appending the twelve-row vector as an ordinary column gives a rank-25 rational system, but its unique solution sets the new scalar to zero and retains the original half-integral coefficients. A selected maximal minor has determinant $4$, and the extended system still has no integral solution.

Therefore the odd datum must enter through a genuine shifted mapping-cone differential and its homotopy equation. A same-degree stabilization—even with the correct twelve-row support—cannot absorb the obstruction.

Shifted repair

A derived correspondence can evade the no-go only by adding at least one reflection-odd pair/corridor homotopy generator whose boundary maps with an independently derived odd coefficient into the mod-two defect row. A unit coefficient is sufficient and minimal. Merely inserting halves, declaring a pair-intersection state, or choosing an AW contraction would fit the missing datum.

Entry333 supplies the required generators in the finite labelled constructible category. Expanding its primitive loaded facet boundaries on the unique twelve-row reflection orbit gives twelve degree-$-1$ mapping-cone columns. The combined exact system has 36 columns, rank 35, and augmented rank 35. Modulo two both ranks remain 35. A selected maximal minor has determinant $-1$, so every nonzero Smith factor is $1$: the strict $\mathbb Z/2$ obstruction is genuinely removed.

The repaired solution space has affine rank one. Entry223 independently derives $W_{012}\mapsto H_\Sigma$ with coefficient $+1$. Adjoining that based row gives rank 36 and augmented rank 36; an explicit maximal minor is $+1$. The finite shifted top comparison is therefore unique and integral.

Geometrically, a normalization-provenanced realization of these generators must still come from an actual multiplicity-sensitive log/excess overlap object $\Gamma_{ij}^{!,\log}$, with a support-typed map to the complementary literal corridor $C_\bullet(q_k)\subset F_B/F_V$. Its boundary must derive the odd row while retaining occurrence ideals, normal-circle states, Tor grades, Čech signs, both adjacent long-facet restrictions, endpoint framing, reflection, and $D_3$.

Downstream status

The finite shifted top comparison does not yet instantiate the endpoint/$Q$ mapping fiber. Until its normalization-provenanced six-functor realization and endpoint cells exist, $p_{\partial,Q}$, its Bockstein, and the $D_8$/Jordan coherence tests remain undefined.

Executable evidence

Checker: research/voevodsky/check_k6_entry223_strict_top_comparison_no_go.mjs

SHA-256: 56533e18eacf13e5e4f43a2fc558c5bf53d3d76cb46f9055ccd11f484cedc7d2

The user-site structured-command MCP executed Node with exit code zero. The checker asserts the complete census, rational ranks, mod-two rank jump, half-integral unique solution, determinant-two maximal minor, and Smith factors before emitting its JSON result. Delegation-worker validation was attempted first at low cognition, but its admitted root is limited to the Narada site and correctly refused the Marici repository cwd.

Outcome contract

{
  "claim": "No strict integral D3-equivariant support-preserving comparison exists from the literal K6 top complex to the entry223 three-facet/three-pair SNC complex.",
  "status": "falsified_scoped_strict_integral_K6_to_entry223_top_comparison",
  "scope": "strict literal support rows with fixed top and long-facet images; derived log-excess overlap generators excluded",
  "matrix": {
    "variables": 24,
    "supported_rank": 21,
    "supported_affine_rank": 3,
    "rotation_equivariant_rank": 23,
    "rotation_equivariant_affine_rank": 1,
    "D3_equivariant_rank": 24,
    "D3_equivariant_affine_rank": 0,
    "unique_solution_nonzero_coefficients": 12,
    "unique_solution_denominators": [2],
    "integral_solution": false,
    "mod2_coefficient_rank": 23,
    "mod2_augmented_rank": 24,
    "selected_maximal_minor_determinant": 2,
    "smith": [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2],
    "obstruction_group": "Z/2"
  },
  "naive_odd_column_control": {
    "unique_mod2_reflection_correction_rows": 12,
    "ordinary_extended_rank": 25,
    "ordinary_new_scalar": 0,
    "ordinary_integral_solution": false,
    "selected_minor_determinant": 4,
    "shifted_mapping_cone_required": true
  },
  "shifted_pair_facet_repair": {
    "columns": 12,
    "rank": 35,
    "augmented_rank": 35,
    "affine_rank": 1,
    "mod2_rank": 35,
    "mod2_augmented_rank": 35,
    "selected_unimodular_minor": -1,
    "smith_nonzero_all_ones": true,
    "integral_obstruction": false
  },
  "external_top_normalization": {
    "free_classes_before_normalization": 1,
    "coefficient": 1,
    "rank": 36,
    "augmented_rank": 36,
    "selected_unimodular_minor": 1,
    "integral_unique": true
  },
  "general_derived_correspondence_no_go": false,
  "derived_pair_facet_repair_constructed": true,
  "finite_W012_qSigma_top_normalization_constructed": true,
  "remaining_geometric_datum": "normalization-provenanced six-functor realization into the literal entry143 BM-Cech complex",
  "unconstructed": [
    "normalization-source six-functor realization of the finite Gamma_ij rows",
    "endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_k6_entry223_strict_top_comparison_no_go.mjs",
  "checker_sha256": "56533e18eacf13e5e4f43a2fc558c5bf53d3d76cb46f9055ccd11f484cedc7d2"
}

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