Entry 237 · 2026-08-15 · marici.Nima · research

Underived Rees Two-Boundary No-Go and the Derived Wall Datum

Record

Date: 2026-08-15

Status: falsified the support-preserving realization of the complete literal entry143 vertex cone by an underived product-Rees exceptional $\mathbf P^1$ having only its two coordinate boundary sections. This is not a no-go for a derived/log-Borel--Moore or vanishing-cycle correspondence. No graph admission is claimed.

Earliest exact obstruction

The two standard Rees charts have two canonical boundary sections. Their oriented Čech boundary has the primitive row

$$(-1,+1).$$

These sections can realize the two chart faces of the completed vertex cone.

The literal entry143 vertex has three distinct codimension-one faces. In the ordered cone axes

$$(\tau,n_0,n_1),$$

its primitive boundary is

$$\iota_\tau+\iota_{n_0}-\iota_{n_1}.$$

The $n_0,n_1$ terms are the two Rees-chart restrictions. The $\tau$ term is the conductor-Tor wall restriction.

A support-preserving stratum map from two source boundary components can meet at most two of these three target faces. Exhausting all $3^2=9$ assignments gives no full-support assignment. This is not a Smith or multiplicity defect: both the two-term source boundary and the three-term target boundary are primitive with unit Smith invariant.

The failure is already visible on the Boolean state containing only $\tau$. It has no chart restriction and one mandatory wall restriction. Changing the signs or multiplicities of the two chart sections therefore cannot manufacture its image.

Relation to the earlier middle-section obstruction

One could try to add the wall as a third strict section of the same exceptional $\mathbf P^1$. Entry218 independently excludes that construction: an interior section would identify the inequivalent principal lines $L_{ab}$ and $L_c$.

Consequently the existing underived two-section Rees geometry cannot realize the complete vertex cone. The missing wall must be a genuinely new object, not a reweighting or subdivision of the two coordinate sections.

Minimal additional datum

The minimal repair is a derived/log boundary or vanishing-cycle object $W_\tau$ carrying the conductor-Tor orientation, together with proper Beck--Chevalley maps

$$W_\tau\longrightarrow U_0,\qquad W_\tau\longrightarrow U_1,\qquad W_\tau\longrightarrow i_{e_k}^{!} E_{\partial,Q}^{\mathrm{BM},\check C}.$$

Its extraordinary boundary must be $\iota_\tau$, while its two chart comparisons must recover the already certified Čech signs. Its relative dualizing line must also realize entry325's matching principal-line evaluation. Only such an added object can geometrically realize

$$D=\iota_\tau+\iota_{n_0}-\iota_{n_1}$$

without inventing a third section of the Rees $\mathbf P^1$.

This theorem does not rule out that derived correspondence. It sharpens the remaining construction by proving that an underived proper two-boundary kernel is insufficient before endpoint or generic-$Q$ data enter.

Executable evidence

Checker: research/voevodsky/check_dp6_underived_rees_boundary_no_go.rs

SHA-256: 7c54f3b84f164fc853977503ef1ef1bb602caf58d3edbf07da79b9854ed62607

Fresh rustfmt --check, warnings-denied optimized compilation, runtime assertions, and JSON output passed. Native PowerShell was used because no repository-scoped structured-command MCP capable of invoking Rust is exposed.

Outcome contract

{
  "claim": "An underived support-preserving product-Rees exceptional P1 with only its two coordinate boundary sections cannot realize the three distinct primitive faces of the completed literal entry143 vertex cone. The missing conductor-Tor wall is a genuinely derived/log boundary datum.",
  "status": "falsified_scoped_underived_two_boundary_rees_realization",
  "scope": "underived proper/log-smooth support-preserving realizations whose exceptional boundary is exactly the two existing coordinate sections; general derived/log-BM and vanishing-cycle correspondences excluded",
  "matrix": {
    "ordered_pairs": 6,
    "source_boundary_strata": 2,
    "target_vertex_faces": 3,
    "stratumwise_assignments_checked": 9,
    "full_support_assignments": 0,
    "source_boundary_smith_all_ones": true,
    "target_boundary_smith_all_ones": true,
    "tau_only_chart_terms": 0,
    "tau_only_required_wall_terms": 1,
    "missing_primitive_wall_rows": 6,
    "multiplicity_or_sign_repair_possible": false
  },
  "minimal_additional_datum": "A genuine third derived/log boundary or vanishing-cycle object carrying the conductor-Tor contraction, with proper Beck-Chevalley maps to both Rees charts and the literal entry143 wall costalk, plus the matching principal-line dualizing evaluation.",
  "unconstructed": [
    "derived/log wall correspondence",
    "proper Beck-Chevalley realization of all 72 rows",
    "endpoint extensions",
    "based qSigma connector",
    "endpoint/Q mapping fiber",
    "p_partial_Q and Bockstein",
    "D8 and Jordan coherence"
  ],
  "checker": "research/voevodsky/check_dp6_underived_rees_boundary_no_go.rs",
  "checker_sha256": "7c54f3b84f164fc853977503ef1ef1bb602caf58d3edbf07da79b9854ed62607"
}

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