Local Gysin Sufficiency No-Go and the Global Mapping-Fiber Definition Gate

Record

Date: 2026-08-15

Status: proved scoped sufficiency no-go; global existence untyped.

This is a typing and provenance theorem. It corrects the claim that a new local occurrence-Gysin axiom would by itself unlock the global construction. It does not claim global nonexistence.

The local Gysin package is already proved

The tempting claim is that (\mathrm{GYS}_{\rm occ}) is a new missing axiom. That claim is false at its stated local scope. Entries 129–131 already construct, locally, the labelled principal ideal and its dual evaluation, the shifted Cartier (\operatorname{Ext}^1) class, the full Koszul–Cech lower terms, both Tor grades, the graph Bockstein, the endpoint restrictions, and scoped (D03) purity, together with its (D_3) rotations.

In particular, the proved local operation has the shifted form

[ \operatorname{Tr}i^! =\operatorname{ev}{I_i}\circ(g_i^!\otimes\mathrm{id}), \qquad g_i^!\in\operatorname{Ext}^1_R(R/I_i,I_i), ]

and its edge realization is constrained by the Bockstein-compatible purity maps (\operatorname{pur}_i). These are genuine local Cartier/purity data, not placeholders for a further local axiom. Their proved scope is essential: they are restrictions in local diagrams, not a normalization-provenanced global sheet object or a global generic component.

The local Cartier class has cohomological degree (+1). In the required global closed trace, that degree is absorbed by its coorientation shift ([-1]), so the primal mixed-variance trace has total degree zero. This degree accounting does not construct either the global source or the trace.

Established target data and source shadows

Entry 143 establishes the global target-side localization data

[ F_B/F_V \longrightarrow E:=F_K/F_V \longrightarrow Q:=F_K/F_B, ]

with its full Borel–Moore–Cech differential, fixed (Q)-generator types, and nonzero generic class (q_\Sigma). The target and its quotient are not missing objects.

There are also established source shadows:

the normalization coefficient row;
the formal branch mixed/multi-Rees totals; and
the absolute mixed identity [ dH_\Sigma=q_\Sigma-\sum_i x_i\widetilde\xi_i. ]

The coefficient shadow is the established row

[ 0\longrightarrow B\longrightarrow B_+\oplus B_- \longrightarrow C\longrightarrow0. ]

These shadows record necessary coefficients, filtrations, and local differentials. They do not yet form a normalization-provenanced global source (\mathcal S_{\rm sh}^{\rm norm,reg}). In particular, no established source object carries both the branch provenance and a branch-to-(Q) generic component retaining the nonzero (q_\Sigma).

Minimal ambient category

Let (\mathsf{Diag}_{\rm est}) be the minimal non-circular ambient category of finite (D_3)-equivariant dg diagrams over the unlocalized occurrence ring. Its objects contain only the following established data:

the normalization triangle;
the formal branch and independent multi-Rees diagrams;
the target localization triangle;
the local Cartier/purity diagrams; and
the endpoint subdiagrams and orientation lines.

Its morphisms preserve localization triangles, support and Rees filtrations, and reciprocal-regular/Borel–Moore variance. Its homotopies are (D_3)-equivariant and filtration-strict, vanish on the endpoints and based (\operatorname{gr}_Q), and commute with the Bockstein and purity maps.

No relation in (\mathsf{Diag}{\rm est}) adjoins the primal trace, its conditional (\alpha{\rm sh}^{!,\check C}) adjoint, (\tau_Q), (K_{\rm alt}), a residue, or a parity value. Those symbols may occur as future tests or names for missing data; they are not defining relations in this category.

The scoped sufficiency no-go

Theorem. Adding the proved local principal/Gysin arrows to (\mathsf{Diag}_{\rm est}) does not instantiate

[ R!\operatorname{Hom}^{\rm fr}{D_3} \left( \mathcal S{\rm sh}^{\rm norm,reg}\otimes^L E, \mathbf 1_{\chi_N} \right) ]

or its endpoint/(Q) homotopy fiber.

Proof. The displayed derived Hom requires an actual global source as its first input. The normalization coefficient row and the formal branch/multi-Rees totals supply shadows of such an input, but no object (\mathcal S_{\rm sh}^{\rm norm,reg}) with normalization provenance is an object of the established data. Therefore the tensor factor, and hence the derived Hom, is not instantiated.

Even if one temporarily names a formal source, the required closed degree-zero primal mixed-variance trace is absent. The local arrows prove the Cartier restrictions but do not supply a trace/correspondence with the required based generic (Q) comparison retaining (q_\Sigma\ne0). The two endpoint connector cells are absent as well, so no endpoint/(Q) pointing data are available from which a homotopy fiber could be formed.

Finally, every existing local endpoint-and-generic-relative gallery has killed its (q_i). Such a gallery cannot determine a nonzero global (q_\Sigma) component in (Q): the generic class that would have to be tracked has already been quotiented away. Thus local principal/Gysin data are necessary compatibility data, but are not sufficient source, generic, or endpoint data. This proves non-instantiation in the stated ambient category; it does not prove that a global construction cannot later exist. (\square)

Earliest missing global construction

The earliest missing global construction has two ordered parts:

a normalization-provenanced source (\mathcal S_{\rm sh}^{\rm norm,reg}); and
a closed degree-zero primal mixed-variance trace [ \operatorname{Tr}^{\rm biv}{\rm sh,\partial,Q}: \mathcal S{\rm sh}^{\rm norm,reg} \mathbin\otimes^L \mathcal E_{\partial,Q}^{\rm BM,\check C} \longrightarrow \mathbf1_{\chi_N}, ] together with two endpoint connector cells and a based nonzero (Q) comparison.

The endpoint connector cells have the required restriction form

[ h_+,h_-: \left.\operatorname{Tr}^{\rm biv}{\rm sh,\partial,Q}\right|{ \widetilde Z_\pm} \Longrightarrow \operatorname{ev}{v\pm}. ]

The local (\operatorname{Ext}^1) degree (+1) of the Cartier operation is absorbed by its coorientation ([-1]) in this global degree-zero trace. Its local restrictions must recover the proved (g_i^!) and (\operatorname{pur}i), including the (D03) scope and its (D_3) rotations. The two endpoint connector cells compare the endpoint restrictions of the trace with the established endpoint evaluations. The based generic (Q) comparison must retain the fixed nonzero (q\Sigma) class. These are required compatibility conditions of the primal trace/correspondence, not a freely assigned map.

Only after an independently proved supported closed-duality theorem for the nonperfect extended-Cech target may one introduce (\alpha_{\rm sh}^{!,\check C}) as the adjoint of this primal trace. Until then, (\alpha_{\rm sh}^{!,\check C}) is conditional shorthand, not the first constructed arrow or a presently typable morphism of triangles.

Conditional definition of the mapping fiber

Only after the source, the primal trace restriction diagram, both endpoint connector cells, and the based nonzero (Q) comparison exist may their induced ((\tau_+,\tau_-,\tau_Q)) point the definition

[ \mathcal M_{\partial,Q}= \operatorname{hofib}{(\tau+,\tau_-,\tau_Q)} \left[ R!\operatorname{Hom}^{\rm fr}{D_3} \left( \mathcal S{\rm sh}^{\rm norm,reg}\otimes^L \mathcal E_{\partial,Q}^{\rm BM,\check C}, \mathbf 1_{\chi_N} \right) \longrightarrow B_+\oplus B_-\oplus B_Q \right]. ]

At present this mapping fiber is not instantiated. Parity remains undefined now.

Mandatory counterevidence and ablations

The ordinary zero-section test gives (0=\pm1) on the conductor (entry 156).
Relabeling a principal line alone leaves the nonzero operation in (\operatorname{Ext}^1), rather than producing an ordinary degree-zero global trace (entry 157).
Every local endpoint-and-generic-relative gallery killed (q_i), so its relative quotient cannot determine the nonzero global (q_\Sigma) leg.
A quotient that makes (q_\Sigma) bound does so by deleting the special galleries that would have carried the required comparison.
Freely adjoining the primal trace, its conditional (\alpha_{\rm sh}^{!,\check C}) adjoint, or (\tau_Q) would encode the desired answer instead of testing whether the normalization geometry supplies it.

Consequence and evidence

The correct status vocabulary is: proved scoped sufficiency no-go; global existence untyped. This is a provenance boundary, not a claim of global nonexistence. The next construction must produce the source and primal trace, with the nonzero generic comparison and endpoint cells, together.

Evidence:

entry 113: mixed boundary-crossing block and the nonzero (q_\Sigma) leg;
entries 129–131: local principal-line, Koszul–Cech, Tor, Bockstein, endpoint, and scoped Cartier-purity package;
entry 143: fixed endpoint/(Q) target and full BM–Cech realization;
entry 154: global source and endpoint-cell typing gate;
entries 156–157: zero-section and relabeling ablations.

Outcome contract

{
  "claim": "proved scoped sufficiency no-go; global existence untyped",
  "status": "proved",
  "assumptions": [
    "The occurrence ring remains unlocalized.",
    "Only established normalization, branch, target, local Cartier/purity, endpoint, and orientation data occur in the ambient category.",
    "Morphisms and homotopies preserve the stated localization, filtration, variance, Bockstein, purity, endpoint, and based gr_Q conditions.",
    "No primal trace, conditional alpha_sh adjoint, tau_Q, K_alt, residue, or parity relation is freely adjoined."
  ],
  "evidence_refs": [
    "src/ledger/20260814-113 Marked-Exit Tate Detector and the Mixed Boundary-Crossing Block.md",
    "src/ledger/20260814-129 Cox Principal-Line Trace and the Extraordinary Cousin Boundary.md",
    "src/ledger/20260814-130 Simultaneous D03 Endpoint Cousin Map and the PC Purity Boundary.md",
    "src/ledger/20260814-131 D03 Cartier Edge Purity and the Scoped PC Promotion.md",
    "src/ledger/20260815-143 Two-Endpoint Road Carrier and the Loaded Conductor Cospan Blocker.md",
    "src/ledger/20260815-154 Primal Bivariant Trace Retyping and the Double Object Gate.md",
    "src/ledger/20260815-156 Zero-Section Trace No-Go and the Principal-Dual-Line Gate.md",
    "src/ledger/20260815-157 Principal-Line Relabeling No-Go and the Ext-One Globalization Gate.md"
  ],
  "factorization_test": {
    "local_Gysin": "proved",
    "global_source": "unconstructed",
    "primal_trace": "unconstructed",
    "supported_duality": "unconstructed",
    "alpha_adjoint": "conditional shorthand",
    "based_Q_comparison": "unconstructed",
    "endpoint_connector_cells": "unconstructed",
    "mapping_fiber": "not_instantiated",
    "parity": "undefined"
  },
  "counterevidence": [
    "The ordinary zero-section specialization gives 0=+/-1 on the conductor.",
    "Principal-line relabeling leaves the nonzero operation in Ext1.",
    "Every local endpoint-and-generic-relative gallery killed q_i.",
    "A quotient making q_Sigma bound deletes the required special galleries.",
    "Freely adjoining the primal trace, alpha_sh adjoint, or tau_Q encodes the desired answer."
  ],
  "next_experiment": "Construct the normalization-provenanced source and closed degree-zero primal trace, including the endpoint connector cells and based nonzero Q comparison; then independently prove supported closed duality and test whether an adjoint alpha_sh exists before instantiating the mapping fiber."
}