Whole-Gallery Cartier–Tag Gysin and the Primitive Three-Top Obstruction

Record

Date: 2026-08-14

Status: proved for the supported associated-grade (D03) whole-gallery map and for the failure of the three rotated tops to glue in the existing absolute/barycentric scalar carrier. No full PC extraordinary-costalk lift, positive-sheet totalization, or factorization-natural half-object is claimed.

Claim

Let (B_{+,03}^{\rm Cart}) be entry 110’s endpoint-and-generic-relative Cartier complex. After the natural lcm-line cap, its expanded gallery is

[ C_2=R\langle H\rangle \xrightarrow{-x_3(1,1,1)} C_1=R\langle e_c,h_E,e_r\rangle \xrightarrow{ \left(\begin{smallmatrix}1&-1&0\0&1&-1\end{smallmatrix}\right)} C_0=R\langle b_1,b_D\rangle . ]

Hence

[ H_1(B_{+,03}^{\rm Cart}) =R/(x_3)\langle[n]\rangle, \qquad n=e_c+h_E+e_r, ]

and the positive Cartier normal applied to the negative Morse thimble gives entry 110’s class (-[\widetilde\xi]=-[n]).

The marked (D03) road meets the gallery on (e_r). Its positive costalk orientation is (c\to b_D), opposite to the gallery orientation (b_D\to c). Therefore the strict representative (-e_r^\vee) satisfies

[ (-e_r^\vee)(-[n])=1. ]

The three negative edge cochains are cohomologous modulo the two internal vertex coboundaries. This is a single whole-gallery Borel–Moore integration, not a segmentwise assignment of three source edges to three tags.

With (C=R/(x_1,x_3,x_5)), the canonical quotient (R/(x_3)\to C), entry 93’s conormal map ([x_3]\mapsto dx_3), and entry 94’s label (dx_3\mapsto t_3=d_1) give the intrinsic supported associated-grade map

[ \boxed{ \kappa^{\rm gr}{+,03}: \beta{x_3}^{\rm Cart}(B_{+,03}) \longrightarrow C,t_3, \qquad \kappa^{\rm gr}_{+,03}(-[\widetilde\xi])=+t_3 . } ]

No occurrence variable and no integer is inverted. A free target is impossible:

[ \operatorname{Hom}_R(R/(x_3),R)=0, \qquad (-e_r^\vee)dH=x_3\ne0. ]

The next hoped-for step does not exist inside the current source carrier. Rotating the three Morse identities gives

[ d(H_{14}+H_{03}+H_{25}) =q_\Sigma- (x_1\widetilde\xi_1+x_3\widetilde\xi_3+x_5\widetilde\xi_5), ]

where

[ \partial q_\Sigma =c_{14}+c_{03}+c_{25}-3v_+. ]

For

[ E={v_+,c_{14},c_{03},c_{25}}, ]

the class is

[ [q_\Sigma]=(1,1,1) \in H_1(sd(K_6),E;\mathbb Z) \simeq\ker!\left(H_0(E)\to H_0(K_6)\right) \simeq\mathbb Z^3. ]

It is primitive, nonzero, and (D_3)-invariant. Thus no integral polynomial higher chain in the existing endpoint-relative carrier cancels the generic terms while retaining the three special galleries. Quotienting by the full short boundary (B_{\rm short}) makes (q_\Sigma) bound, but kills all three special galleries at the same time.

The target norm map itself is unobstructed:

[ N:Rf_+\longrightarrow R\langle t_1,t_3,t_5\rangle, \qquad N(f_+)=(1,1,1). ]

If a common source top existed, its map to (f_+) would be unique and integral. The obstruction is therefore on the scalar source side, not a need to divide by three.

Evidence

Exact certificates:

  • research/voevodsky/check_d03_whole_gallery_tag_gysin.rs
  • research/voevodsky/check_three_rotated_gallery_top_gluing.rs

SHA-256:

971df4192b644a408193551b8bb02cc6c0036c93c4e2c7037824b97e450a2e20
b1fd94f004e97226c55bfc3b943c398773434eda9dee44adf1f03b3673b907d5

The first certificate checks the actual blown-up gallery supports, lcm-line naturality, relative homology, road/costalk orientation, internal-coboundary independence, supported base change, conormal label, and free-target no-go.

The second reconstructs the full (K_6) face poset and barycentric differential. At unit occurrence coefficients over (\mathbf F_{101}),

[ \operatorname{rk}\operatorname{im}d_2=126, \qquad \operatorname{rk}\langle\operatorname{im}d_2,q_\Sigma\rangle=127. ]

Any unlocalized polynomial filler would specialize to a filler here, so the rank jump is a valid no-go. Relative to (B_{\rm short}), both ranks are (64), while the special galleries vanish.

Reproduce with rustfmt --check, rustc --edition 2021 -D warnings -O, and execution of each certificate. Both JSON outputs and git diff --check pass.

Boundary

  • The local theorem is an ordinary supported associated-grade statement. The derived base change contains an excess copy: [ R/(x_3)\otimes_R^L C\simeq[C\xrightarrow0 C]. ] Its (\operatorname{Tor}_1) line, reciprocal twist, normal shifts, and full PC extraordinary costalk have not been identified.
  • Principal occurrence ideals are retained as labelled dualizable lines and evaluated before base change. Replacing them by their image ideals in (C) would wrongly kill the ((x_1)) and ((X_{03}x_1)) factors.
  • The global calculation falsifies only gluing in the current absolute/barycentric carrier and its two natural relative quotients. It is not a no-go for a new normalization–conductor, multi-Rees, or bivariant specialization correspondence.
  • The coefficient (-3v_+) does not license averaging by (1/3). The class ((1,1,1)) is primitive, and the target norm attachment is integral.
  • No negative-sheet assembly, full (K_{\rm alt}), physical-cut Beck–Chevalley theorem, CHY comparison, or higher-multiplicity conclusion follows.

Consequence

The first stage proposed in entry 111 is complete: the whole (D03) Cartier gallery has a canonical map to its single conductor tag. The failure is now one level higher and has an exact address.

The smallest missing arrow is a positive-sheet normalization–conductor comparison

[ \boxed{ \alpha_+: \mathcal S_+^{\rm cond} \longrightarrow R\Gamma_{v_+}^{F}(\mathcal P_{\rm abs}) } ]

or an equivalent bivariant/multi-Rees kernel whose differential supplies a common top (H_+) with

[ dH_+=- (x_1\widetilde\xi_1+x_3\widetilde\xi_3+x_5\widetilde\xi_5), ]

while retaining the three special galleries and the excess (\operatorname{Tor}1) data. Its associated-grade image must be (df+=t_1+t_3+t_5). Constructing this arrow, rather than another local sign or a rational projector, is the next discriminating experiment.

Outcome contract

{
  "claim": "The whole D03 Cartier gallery has a canonical supported associated-grade map sending the intrinsic Bockstein class -xi_tilde to the positive tag t3. The three rotated source tops do not glue in the current absolute/barycentric carrier: their generic sum is the primitive D3-invariant class (1,1,1) in H1(sd(K6),E;Z), and the only existing quotient that bounds it also kills all three special galleries.",
  "status": "proved",
  "assumptions": [
    "The absolute occurrence complex, expanded gallery, Cartier class, and orientations are those of entries 93, 94, and 105-111.",
    "The local map is scoped to the supported ordinary associated grade C*t3, not the full derived PC costalk.",
    "Occurrence variables, normal monodromies, and integers remain uninverted."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_whole_gallery_tag_gysin.rs",
    "research/voevodsky/check_three_rotated_gallery_top_gluing.rs",
    "ledger entries 93, 94, and 105-111"
  ],
  "factorization_test": {
    "whole_gallery_relative_homology": "passed integrally",
    "D03_costalk_orientation_and_sign": "passed",
    "supported_conormal_tag_map": "passed",
    "free_target_map": "falsified",
    "rotated_generic_sum_endpoint_relative": "nonzero primitive class; rank 126 to 127",
    "short_boundary_quotient": "generic class bounded but all special galleries killed",
    "full_PC_extraordinary_costalk": "unconstructed",
    "physical_Cut_Beck_Chevalley": "unconstructed"
  },
  "counterevidence": [
    "Derived base change has a nonzero Tor1 excess copy.",
    "No current relative quotient both kills the generic sum and retains the three special galleries.",
    "The target norm map is integral; division by three would solve the wrong problem."
  ],
  "next_experiment": "Construct alpha_+ or an equivalent bivariant multi-Rees kernel that retains a nonzero generic/source leg, cancels q_Sigma, and maps the three local Cartier galleries to the integral conductor norm boundary."
}