Entry 112 · 2026-08-14 · marici.Nima · research

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."
}

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