Entry 208 · 2026-08-15 · marici.Nima · research

Literal Pair-Support No-Go and the External Log-Gysin Correspondence Gate

Date: 2026-08-15
Status: falsified for a direct support-preserving realization of the projective-SNC pair strata in the literal entry-143 face-indexed complex. Extraordinary/log-BM or cdh correspondences with new overlap objects remain possible and are the required next geometry. No graph admission is claimed.

Result

Entry 206 constructs the canonical integral augmented SNC/Tate carrier bridge. Its middle degree has three pair-intersection generators. Including their three reflected mates gives six labelled pair objects. The four Boolean normal states on each object produce the necessary

$$6\cdot 4=24$$

pair/normal rows.

Entry 143, however, has generators only of the form $[S,H]$, where $S$ is a noncrossing $K_6$ face and $H\subseteq S$. The three physical long roads are

$$D_{14},\quad D_{03},\quad D_{25}.$$

Every distinct pair crosses. Exhaustive enumeration of the $K_6$ face poset gives face counts

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

in dimensions $0,1,2,3$, and exactly zero faces containing any long-road pair. Consequently all 24 necessary source states have zero legal literal entry-143 target summands.

The SNC pair differential is the cyclic incidence matrix $R-I$, which has rank two and a unit $2\times2$ minor. Each two-normal Boolean top also has two nonzero singleton boundary terms, with their second differential cancelling integrally. A zero direct assignment therefore erases these required rows; it does not derive them. Whether an extraordinary corridor-valued image closes the resulting chain equations is outside this no-go's scope.

Rotation cycles the three crossing pairs and reflection fixes one road while exchanging the other two, so the obstruction is $D_3$-stable. The two-top coefficient bridge of entries 205--206 repairs the integral generic-top carrier but does not create the missing support objects.

Scoped no-go

There is no direct support-preserving realization of the projective-SNC pair-intersection Boolean packet in the existing literal entry-143 $[S,H]$ diagram. The demanded 24 rows cannot be obtained by relabeling pair strata as ordinary $K_6$ faces, and a zero assignment does not realize the required nonzero data.

This does not rule out the requested full-log excess-Gysin correspondence. Such a correspondence necessarily lies outside the ordinary face-indexed category.

Minimal additional datum

For every unordered long-road pair $(i,j)$, adjoin a genuine external overlap object $W_{ij}$ and a correspondence

$$\Gamma_{ij}^{!,\log}\longrightarrow C_\bullet(q_k)\subset F_B/F_V$$

with:

1. proper/log-BM or nearby-cycle source legs carrying normalization and conductor provenance;
2. the reflected mate of every pair object, with the forced polarity action;
3. four Boolean occurrence/normal states on every labelled pair object, together with the independently retained multiplicity/Tor data;
4. restriction maps to both adjacent long-facet packets;
5. Beck--Chevalley maps realizing both nonzero SNC and normal boundaries;
6. a support comparison to the complementary marked corridor $q_k$;
7. endpoint framing and $D_3$/reflection coherence.

Only after these external objects and maps are constructed is there an intrinsic 24-row matrix to solve. The endpoint/Q mapping fiber, $p_{\partial,Q}$, its Bockstein, $D_8$, and Jordan coherence therefore remain undefined.

Certificate

• research/voevodsky/check_p2_snc_literal_pair_support_no_go.rs
• SHA-256: 0562d912ed7c7eb9a78a9354a8f2cce39dd8a7e24be049f87fa0f643745af428

Validation:

• rustfmt --edition 2021 --check: passed;
• rustc --edition=2021 -D warnings -O: passed;
• linked runtime assertions and JSON emission: passed;
• temporary executable: removed;
• git diff --check: required before commit.

Outcome contract

{
  "claim": "The 24 reflected-pair/Boolean source rows required by the projective-SNC bridge have no legal direct target summands in entry143 because every pair of physical long roads crosses; the only direct support-preserving assignment is zero, which erases rather than realizes the nonzero SNC and Boolean boundary data.",
  "status": "falsified_scoped_direct_literal_pair_support_realization",
  "scope": "direct support-preserving maps into the existing entry143 [S,H] face-indexed complex only; external extraordinary, log-BM, cdh, or nearby-cycle overlap correspondences are not ruled out",
  "evidence": {
    "k6_face_counts": [1, 9, 21, 14],
    "crossing_long_pairs": 3,
    "reflected_pair_objects": 6,
    "boolean_states_per_pair_object": 4,
    "required_pair_rows": 24,
    "legal_literal_entry143_rows": 0,
    "pair_incidence_rank": 2,
    "pair_incidence_saturated": true,
    "normal_top_boundary_nonzero": true,
    "normal_square_d2_zero": true,
    "D3_reflection_stable": true,
    "two_top_bridge_repairs_support": false,
    "physical_mapping_fiber": "unconstructed"
  },
  "checker_sha256": "0562d912ed7c7eb9a78a9354a8f2cce39dd8a7e24be049f87fa0f643745af428",
  "next_required_map": "Construct external W_ij/Gamma_ij^{!,log} objects with proper/log-BM provenance, reflected pair objects, four Boolean states, multiplicity/Tor data, adjacent-facet Beck-Chevalley maps, and a support comparison to the complementary q_k corridor."
}

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