Conductor Blowup Discrepancy Multiplicity and the Oriented Top Comparison Gate
Date: 2026-08-15
Status: proved for the normalization-provenanced discrepancy multiplicity and
the resulting conditional integral carrier closure.
The oriented source-top-to-literal-target comparison remains unconstructed.
No graph admission is claimed.
Result
On the positive normalization sheet, the conductor center is the regular codimension-three ideal
[ J_+=(x_1,x_3,x_5). ]
For the blowup of a regular codimension-(c) center, the relative canonical divisor has discrepancy (c-1). Therefore the projective conductor source (mathbf P(J_+/J_+^2)congmathbf P^2) carries the independently geometric multiplicity
[ c-1=3-1=2. ]
This multiplicity is fixed by the blowup geometry; it is not chosen to solve the carrier equation.
Entry 202 certifies that a (C_3)-orbit of the three pairwise full-facet lifts has cyclic defect
[ F_{14,03}+F_{03,25}+F_{25,14}=-2,partial K_6^{mathrm{top}}. ]
Let (T) denote one primitive generic-top source generator. After an oriented comparison identifies its boundary with the primitive (K_6) top line, the relative-canonical multiplicity gives
[ F_{14,03}+F_{03,25}+F_{25,14}+2T=0. ]
Thus the codimension-three discrepancy supplies exactly the multiplicity needed to remove the mod-three defect.
Integral matrix
Let the first three columns be the rotated pairwise lifts and the fourth the primitive top column. The enlarged (9 imes4) matrix has:
- rank (3);
- nonzero Smith factors ((1,1,1));
- primitive kernel generated by ((1,1,1,2));
- no integral torsion or denominator.
This replaces the pairwise-only Smith form ((1,1,2)) by a saturated presentation. The former equation (-2+3n=0) is not solved by choosing (n); it is closed by a new primitive source generator with independently derived multiplicity two.
Exact scope
The certificate proves two things:
- the normalization/conductor blowup canonically determines multiplicity two; and
- once its oriented primitive top is compared with the primitive (K_6) top line, the integral carrier matrix closes and is saturated.
It does not construct that oriented comparison as a spatial six-functor map. In particular, it does not yet identify the projective-conductor fundamental class with the literal entry-143 generic (q_Sigma) packet or derive the 24 occurrence/normal-circle/Tor/Čech pair-intersection rows.
The earliest remaining map is therefore
BC_top^{!,log}: C_BM(P(J_+/J_+^2), D_SNC) -> E_{partial,Q}^{BM,Cech}
with its top sent to the literal (H_Sigma/q_Sigma) line, its three divisor residues sent to the long-facet packets, and its three pair intersections sent through the required extraordinary support-switch correspondences.
Until that map and its 24 rows exist, the endpoint/(Q) mapping fiber, (p_{partial,Q}), its Bockstein, (D_8), and Jordan coherence remain undefined.
Certificate
research/voevodsky/check_p2_conductor_discrepancy_top_connector.rs- SHA-256:
7e5fe3a8ac849a3403698e5983561175263e02b5691285e5a15d06e1e0a0a33c
Validation through the user-site structured-command MCP:
rustfmt --edition 2021 --check: pass;rustc --edition=2021 -D warnings --emit=metadata: pass;- optimized linked executable: pass;
- runtime assertions and JSON parse: pass;
- temporary executable, metadata, output, and validation script: removed.
Outcome contract
{
"claim": "The regular codimension-three conductor blowup forces relative-canonical discrepancy two. After the still-required oriented source-top/K6-top comparison, this multiplicity cancels the pairwise cyclic defect -2; the enlarged integral carrier matrix has rank three, Smith factors [1,1,1], and primitive kernel [1,1,1,2].",
"status": "proved_scoped_discrepancy_multiplicity_and_conditional_carrier_closure",
"scope": "normalization-provenanced discrepancy and integral carrier matrix; oriented spatial top comparison, 24 literal entry143 rows, endpoint/Q mapping fiber, parity, D8, Jordan, and graph admission remain open",
"evidence": {
"conductor_center_codimension": 3,
"relative_canonical_discrepancy": 2,
"pairwise_cyclic_top_coefficient": -2,
"generic_top_connector_primitive": true,
"enlarged_matrix_shape": [9, 4],
"enlarged_matrix_rank": 3,
"enlarged_matrix_smith": [1, 1, 1],
"primitive_kernel": [1, 1, 1, 2],
"cyclic_top_closed_after_oriented_top_comparison": true,
"oriented_source_top_to_k6_top_comparison": "unconstructed",
"derived_literal_entry143_rows": 0,
"physical_mapping_fiber": "unconstructed",
"physical_p_partial_Q": "undefined",
"physical_bockstein": "undefined",
"D8_and_Jordan": "untested"
},
"checker_sha256": "7e5fe3a8ac849a3403698e5983561175263e02b5691285e5a15d06e1e0a0a33c",
"next_required_map": "Construct BC_top^{!,log} from the projective-conductor SNC BM complex to the literal entry143 endpoint/Q complex, including the three pair-intersection support-switch maps and all 24 rows."
}