Product-Branch Rees Clutching and the Literal Realization Gate
Record
Date: 2026-08-15
Status: proved in the universal product-branch Rees coefficient geometry. The literal entry143 six-functor realization and adjacent-facet Beck--Chevalley maps remain unconstructed. No graph admission is claimed.
Construction
For two ordered adjacent branch sections (a,b) and the complementary long-road section (c), use the Rees blowup [ Y=operatorname{Bl}_{(ab,c)}operatorname{Spec}mathbf Z[a,b,c]. ] With homogeneous coordinates ([A:B]), its equation is [ cA-abB=0. ] On (A e0), writing (t=B/A) gives the polynomial chart (c=abt). On (B e0), writing (s=A/B) gives the normal conifold chart [ cs=ab. ]
The tautological Rees line carries (ab) and (c) as its two labelled sections. It therefore interpolates the product branch and complementary road without identifying their multidegrees and without inverting any base section. This bypasses, but does not contradict, entry 215's direct line-isomorphism no-go.
After (b) is made a unit, the construction is the ordinary blowup of ((a,c)); after (a) is made a unit, it is the blowup of ((b,c)). Thus the two adjacent branch packets are recovered with multiplicities ((1,1)). Over (ab=c=0), the exceptional fibre is canonically (mathbf P^1).
The center ((ab,c)) is a regular sequence. Its Koszul row has (d_1=[ab,c]), (d_2=[-c,ab]^T), so (d_1d_2=0), and the derived self-intersection has exterior ranks ((1,2,1)). Rotation permutes the three labelled coordinates. Reversing the ordered branch pair fixes the product and Rees equation while reversing the separate log orientation.
Exact scope boundary
This establishes a canonical multiplicity-sensitive clutching carrier and its two branch restrictions. It does not yet identify the exceptional (mathbf P^1), its KN/log-BM cap, or its derived Tor packet with the literal entry143 ([S,H]) stalks and corestrictions. In particular, the adjacent-facet Beck--Chevalley squares, endpoint framing, generic (q_Sigma) connector, endpoint/Q mapping fiber, physical parity, Bockstein, and (D_8)/Jordan tests remain open.
The next exact step is to construct the oriented log-excess cap. At the characteristic-lattice level this means comparing the product map [ mathbf Zlongrightarrowmathbf Z^2,\qquad 1longmapsto(1,1) ] with its primitive anti-diagonal quotient, and proving that its exterior contraction realizes the four legal Boolean/Tor states and the two adjacent facet restrictions. That calculation must then be promoted to the literal entry143 support diagram.
Certificate
Executable:
research/voevodsky/check_dp6_product_branch_rees_clutching.rs
SHA-256:
8d35cd75398302c8fb912ba2ea39ef01373f9e83e3f1895684b82d91ebf7c28a
Rustfmt, warnings-denied optimized compilation, runtime assertions, and JSON parsing passed. Native PowerShell was used only because structured-command MCP was unavailable.
Outcome contract
{
"claim": "The blowup of the product-branch ideal (ab,c) canonically interpolates the two adjacent multiplicity-one Rees packets and the complementary road through its tautological line, without a direct multigraded line isomorphism or base localization.",
"status": "proved_scoped_product_branch_rees_clutching_geometry",
"scope": "universal algebraic/log coefficient geometry of Bl_(ab,c), excluding literal entry143 six-functor realization",
"factorization": {
"center_ideal": "(a*b,c)",
"rees_equation": "c*A-a*b*B=0",
"A_chart": "Z[a,b,t] with c=a*b*t",
"B_chart": "Z[a,b,c,s]/(c*s-a*b)",
"B_chart_normal": true,
"branch_restrictions": [
"Bl_(a,c) after b is a unit",
"Bl_(b,c) after a is a unit"
],
"branch_multiplicities": [1, 1],
"triple_center_fibre": "P1",
"center_regular_sequence": true,
"derived_self_intersection_tor_ranks": [1, 2, 1],
"tautological_rees_interpolation": true,
"direct_line_isomorphism": false,
"base_inversions": false,
"D3_covariant": true,
"reflection_log_orientation_odd": true,
"literal_entry143_realization_constructed": false,
"adjacent_facet_BC_as_six_functor_map_constructed": false
},
"minimal_remaining_datum": "The KN/log-BM realization of the Rees P1 and a support-typed comparison identifying its two branch restrictions, Tor/excess grades, and Cech rows with the literal adjacent entry143 facet packets.",
"evidence_refs": [
"research/voevodsky/check_dp6_product_branch_rees_clutching.rs",
"src/ledger/20260815-214 Finite Ordered Log-Link Gysin Matrix and the Spatial Realization Gate.md",
"src/ledger/20260815-215 Toric Star-Subdivision Line No-Go and the Clutching Datum.md"
],
"checker_sha256": "8d35cd75398302c8fb912ba2ea39ef01373f9e83e3f1895684b82d91ebf7c28a",
"next_experiment": "Construct the primitive characteristic-lattice excess contraction and derive its literal four-state entry143 branch/corestriction rows."
}