Pairwise-Only D3 Full-Facet Lift No-Go and the Generic-Top Connector Gate
Date: 2026-08-15
Status: falsified for the pairwise-only integral D_3-equivariant carrier lift.
The enlarged full-log excess-Gysin correspondence remains open.
No graph admission is claimed.
Result
Entry201 proves a conditional uniqueness statement: once a fixed relative road
class is already supplied, differences of its strict representatives have
unique integral short-facet fillers. The present computation asks the earlier
existence question required by the proposed pair-intersection correspondence:
can the three P^2 pair-boundaries themselves be lifted equivariantly to the
actual K_6 carrier?
Let d_2^short be the six-short-facet boundary matrix and d_2^full the full
nine-facet matrix. Exact reconstruction of the labelled K_6 face poset gives
rank(d_2^short)=6 and SNF(d_2^short)=(1,1,1,1,1,1).
For each of the three columns of the P^2 pair incidence matrix R-I,
the corresponding complementary-road difference raises the rank of the
augmented short matrix from six to seven. Therefore none of those differences
lies in the short-facet image.
The full matrix has rank eight, a unit maximal minor, and a one-dimensional
kernel generated by the primitive boundary of the associahedral top cell.
Each road difference has an integral full-facet lift. Choose one lift and
rotate it twice. This is the required C_3-orbit of lifts, modulo adding the
same integer multiple n of the top boundary to all three columns.
For the normalized orbit, the cyclic sum is
F_14,03 + F_03,25 + F_25,14 = -2 * boundary(K_6^top).
Every equivariant change of lift changes the coefficient by 3n. Closing
the cyclic top equation would therefore require -2+3n=0,
which has no integral solution. Equivalently, the obstruction is 1 mod 3.
The selected orbit matrix has Smith form (1,1,2); throughout the equivariant
family its Smith form is (1,1,|-2+3n|).
It can drop to rank two exactly when the cyclic defect vanishes, which never
occurs for integral n.
This does not contradict entry201: entry201 controls the uniqueness of
comparison chains after a compatible relative road class exists. The present
no-go proves that the three pairwise-only classes cannot be assembled into
the required integral D_3-equivariant cyclic lift without an additional
top connector.
Scope and earliest falsifier
The falsifier occurs before the requested 24 literal entry143 occurrence/normal-circle/Tor/Čech rows. Hence those rows cannot be derived from the pair objects alone, and the endpoint/Q mapping fiber cannot yet be instantiated.
The result is scoped to the pairwise-only carrier presentation. It does not
rule out a full-log excess-Gysin correspondence enlarged by an independently
geometric, D_3-invariant generic-top cell.
The reflected road constructions retain the correct endpoint boundary, but
strict reflected representative comparison is not reached after the earlier
C_3 obstruction. No claim about physical reflection, D_8, or Jordan
coherence is made.
Minimal additional geometry
The minimal repair is one D_3-invariant generic-top or excess-Gysin
connector whose boundary coefficient in the associahedral top line is
independently derived and prime to three (minimally +1 or -1). It must not be
inserted as a fitted scalar. Geometrically it must come with:
- a normalization-provenanced log/SNC top object;
- a proper or BM trace producing the primitive generic
q_Sigmarow; - residues to all three adjacent long-facet packets;
- pair-intersection Beck--Chevalley cells realizing the 24 literal occurrence/normal/Tor/Čech rows;
- endpoint framing and reflection compatibility.
Only after that unit top connector cancels the mod-three defect can the
integral full matrix be formed and its SNF used to instantiate the endpoint/Q
mapping fiber. Consequently p_{partial,Q}, its Bockstein, D_8, and
Jordan coherence remain undefined.
Certificate
research/voevodsky/check_p2_pair_d3_top_obstruction.rs- SHA-256:
1720ebf06748a2cb67ad13e4ef015b3139ebc2dac321faa32924e73014ca2efb
Validation through the user-site structured-command MCP:
rustfmt --edition 2021 --check: pass;rustc --edition=2021 -D warnings -O: pass;- linked executable: exit zero;
- emitted JSON: parsed;
- temporary executable and redirected outputs: removed.
Outcome contract
{
"claim": "The three P2 pair-boundaries have integral full-facet lifts, but no pairwise-only C3-equivariant choice has vanishing cyclic top: the normalized defect is -2 and all equivariant changes are -2+3n.",
"status": "falsified_scoped_pairwise_only_d3_full_facet_lift",
"scope": "integral K6 carrier before literal entry143 occurrence/normal/Tor/Cech realization; no global extraordinary-kernel no-go and no graph admission",
"evidence": {
"k6_faces": [1, 9, 21, 14],
"short_boundary_d2_rank": 6,
"short_boundary_d2_smith": [1, 1, 1, 1, 1, 1],
"road_difference_augmented_short_rank": 7,
"road_difference_in_short_image": false,
"full_boundary_d2_rank": 8,
"full_boundary_d2_unit_minor": true,
"full_boundary_kernel_rank": 1,
"p2_pair_incidence": "R-I",
"selected_c3_orbit_filler_smith": [1, 1, 2],
"equivariant_filler_smith_family": "[1,1,|-2+3n|]",
"cyclic_top_coefficient": -2,
"obstruction_mod_3": 1,
"integral_solution": false,
"derived_literal_residue_rows": 0,
"physical_mapping_fiber": "unconstructed",
"physical_p_partial_Q": "undefined",
"physical_bockstein": "undefined",
"D8_and_Jordan": "untested"
},
"checker_sha256": "1720ebf06748a2cb67ad13e4ef015b3139ebc2dac321faa32924e73014ca2efb",
"minimal_additional_geometry": "Add one independently derived D3-invariant generic-top/excess-Gysin connector with boundary coefficient prime to 3 and a mixed-variance comparison deriving all 24 literal entry143 rows."
}