Marked-Exit Tate Detector and the Mixed Boundary-Crossing Block

Record

Date: 2026-08-14

Status: one exact theorem and one sharp blocker. The marked-exit class (q_\Sigma) is the road norm detecting the nonsplit integral Tate shadow; it is not a canonical image of the support-filtration Yoneda class. The rotated Morse sum is already the smallest canonical occurrence-loaded boundary-crossing block. The remaining datum is a derived marked extraordinary-costalk/Beck–Chevalley comparison, not a filler for (q_\Sigma).

The marked-exit census

Let

[ X=sd(K_6), \qquad A={v_+}, \qquad E={v_+,c_{14},c_{03},c_{25}}, \qquad B=B_{\rm short}. ]

The exact integral carrier census is

[ H_2(X,B)\simeq\mathbb Z^2, \qquad H_1(B,A)\simeq\mathbb Z^2, ]

[ H_1(B,E)\simeq\mathbb Z^5, \qquad H_1(X,E)\simeq\mathbb Z^3. ]

The support-filtration connector

[ H_2(X,B)\longrightarrow H_1(B,A) ]

is saturated, with Smith factors ((1,1)). After marking the four endpoints, its image is exactly

[ \ker!\left(H_1(B,E)\longrightarrow H_1(X,E)\right). ]

Consequently every composite induced only by the inclusions and quotients

[ A\subset E\subset B\subset X ]

sends the support-filtration connector to zero in marked-exit homology. There is no canonical map of this kind carrying the Yoneda class (e_F) to (q_\Sigma).

The failure is already visible from variance. With

[ R=F_2/F_1, \qquad e_F\in\operatorname{Ext}^2(R,F_0), ]

the only canonical source comparison is

[ C_*(X,E)\longrightarrow R. ]

It pulls (e_F) back to another (\operatorname{Ext}^2) class; it does not push it into (H_1(X,E)). On the target side the composite

[ F_0\longrightarrow F_E\longrightarrow F_E/F_0 ]

is identically zero. Thus the proposed formula (e_F\mapsto q_\Sigma) is not merely unproved: it is mistyped in the category generated by the established support maps.

What (q_\Sigma) actually is

Because (X) is contractible, endpoint boundary gives a canonical (D_3)-equivariant identification

[ M_E:=H_1(X,E) \simeq \mathbb Z\langle q_{14},q_{03},q_{25}\rangle \simeq P_{\rm road}, ]

where

[ \partial q_i=c_i-v_+. ]

The distinguished central vertex defines the augmentation

[ \epsilon(q_i)=1. ]

Therefore

[ \boxed{ q_\Sigma=q_{14}+q_{03}+q_{25}=N_{\rm road}, \qquad \epsilon(q_\Sigma)=3, } ]

and

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

The peripheral transgression of entry 103 identifies

[ A_2=H_1(B,A) \simeq\ker(\epsilon:P_{\rm road}\to\mathbb Z). ]

Together with the orientation-twisted tag module this gives the exact Tate window of entry 102:

[ \boxed{ 0\longrightarrow\mathbb Z_{\rm or} \xrightarrow{N_{\rm tag}}P_{\rm tag} \xrightarrow{1-r}M_E \xrightarrow{\epsilon}\mathbb Z \longrightarrow0. } ]

Thus (q_\Sigma) is the road-norm detector for the same order-three Tate architecture. It is not by itself a representative of the full two-extension. The two norm lines must also not be conflated:

• (N_{\rm tag}) spans the reflection-odd orientation line;
• (q_\Sigma=N_{\rm road}) spans a reflection-even line before the independent normal-orientation twist is attached.

Integrally,

[ \boxed{ 0\longrightarrow A_2\oplus\mathbb Zq_\Sigma \longrightarrow M_E \xrightarrow{\epsilon\bmod3}\mathbb Z/3 \longrightarrow0. } ]

The norm and contact lattices therefore span an index-three sublattice. This is the exact source of the forbidden projector (N\epsilon/3). If (n:\mathbb Z\to M_E) sends (1\mapsto q_\Sigma), then

[ \epsilon n=[3]. ]

Pulling the road extension back along ([3]) splits via (n), which is the carrier-level transfer witness for (3\beta_\triangle=0). Hence the nonzero primitive class found in entry 112 is expected structure, not a defect to be removed.

Adjoining a new top whose boundary is (q_\Sigma) would be additional geometric data. It would not by itself split the Tate extension; a compatible equivariant contraction would, and that contraction requires division by three. No such new top or contraction is supplied by the scalar geometry.

The correct boundary-crossing block

The rotated entry-110 Morse identities already provide

[ \boxed{ dH_\Sigma =q_\Sigma- \left( x_1\widetilde\xi_1 +x_3\widetilde\xi_3 +x_5\widetilde\xi_5 \right). } ]

This block is inherited from the absolute occurrence complex, is semilinearly (D_3)-stable, and obeys (d^2=0) before any occurrence, normal, Rees, or integer localization. Its two boundary pieces have the correct support types:

• (q_\Sigma) is the generic chain-level (Q=F_2/F_1) leg;
• the weighted galleries lie in the special (F_1) leg.

Therefore the smallest canonical source object is the mixed block itself. Trying instead to adjoin a formal invariant generator (c) with

[ dc=q_\Sigma ]

works only after quotienting the endpoints. Absolutely,

[ d^2c=x_1b_1+x_3b_3+x_5b_5\ne0. ]

Adding the required endpoint correction without an independently constructed geometric source would merely fit a new normalization–Cech map. Quotienting by all of (B_{\rm short}) makes the generic class bound only by deleting the three special galleries. Both shortcuts are therefore rejected.

The correction to entry 112’s trajectory is conceptual: do not seek a spatial filler that removes the generic term. Retain the mixed generic and special boundaries and let the canonical Yoneda cone roof mediate between them.

The remaining blocker

Entries 104–105 already define the scoped filler-free derived candidate

[ A_+^{\rm sec} =\mathbb D(e_F)\circ\operatorname{pur}+: \mathcal S+^{\rm cond}\longrightarrow\mathbb D(F_2/F_1), ]

but (\operatorname{pur}_+) is currently established only after fixed-nonzero-(\beta), characteristic-zero completion. Entry 112 proves only the ordinary supported associated-grade maps

[ \kappa^{\rm gr}_{+,i}(-[\widetilde\xi_i])=t_i, \qquad i=1,3,5. ]

The first unconstructed datum is one (D_3)-equivariant derived extraordinary-costalk/Beck–Chevalley comparison, equivalently a full lift of these three maps:

[ \boxed{ \kappa^!{+,\Sigma}: \bigoplus{i=1,3,5}\mathcal B^{\rm Cart}_{+,i} \dashrightarrow \mathbb D(F_0)[-2], } ]

compatible with (e_F) and its cone roof. For every (i), it must retain the complete derived base change

[ R/(x_i)\otimes_R^L C \simeq[C\xrightarrow0C], ]

including the (\operatorname{Tor}1) excess copy, reciprocal/regular versus Borel–Moore variance, repeated-normal orientation, and the separate physical normal line. Its ordinary grade must be the three proved tag maps, and its carrier shadow must be the full (N/(1-r)/\epsilon) Tate window, with (q\Sigma) retained rather than made a boundary.

Only after this comparison is constructed can the canonical cone roof promote the mixed block to the required (d_{\rm sp,sc}) and permit the physical-Cut test for (G_{03}^{\rm Cousin}).

Evidence

New exact certificates:

• research/voevodsky/check_marked_exit_yoneda_census.rs, SHA-256 a771e2bc483ae4109645401e5eb9e8ba9dfcd0fea04d9304aa6483829233e4b6;
• research/voevodsky/check_positive_mixed_rees_top.rs, SHA-256 723c6cd473f43263ea963007a43d1fb7d3455433491bb967ef84fe0640337b64.

The first checker reconstructs the labelled face poset, proves all four homology ranks, saturation of the filtration connector, vanishing of its marked-exit composite, exact loaded degree ranks, and the primitive norm boundary. The second verifies the inherited mixed differential, semilinear (D_3)-covariance, absolute (d^2=0), the endpoint-relative formal-filler control, its absolute (d^2) failure, the orientation twist, and retention of all three (\operatorname{Tor}_1) ranks.

Outcome contract

{
  "claim": "The marked-exit class q_Sigma is the road norm N with epsilon(N)=3 in the same integral Tate carrier architecture as the support-filtration Yoneda class, but no inclusion/quotient-induced morphism sends e_F to q_Sigma. The inherited mixed top dH_Sigma=q_Sigma-sum_i x_i xi_i is already the smallest canonical occurrence-loaded boundary-crossing block and must retain its generic Q leg.",
  "status": "proved",
  "assumptions": [
    "The labelled K6 supports, orientations, loaded Morse identities, and D3 actions are those of entries 102, 105, 110, and 112.",
    "Occurrence variables, normal monodromies, Rees parameters, and integers remain uninverted.",
    "No new cell, fitted source map, or rational equivariant splitting is admitted."
  ],
  "evidence_refs": [
    "research/voevodsky/check_marked_exit_yoneda_census.rs",
    "research/voevodsky/check_positive_mixed_rees_top.rs",
    "ledger entries 102-105 and 110-112"
  ],
  "factorization_test": {
    "support_connector": "saturated Smith (1,1)",
    "canonical_marked_exit_composite": "zero",
    "qSigma": "primitive road norm with epsilon=3",
    "Tate_shadow": "full N/(1-r)/epsilon window retained",
    "mixed_absolute_d_squared": "passed",
    "generic_Q_leg": "retained",
    "formal_qSigma_filler": "fails absolute d^2",
    "ordinary_Cartier_tag_grades": "proved for all three rotated roads",
    "derived_Tor1_Beck_Chevalley_lift": "unconstructed",
    "full_G03_Cousin": "unconstructed"
  },
  "counterevidence": [
    "e_F and q_Sigma have different categorical types and the only canonical source comparison has pullback, not pushforward, variance.",
    "The road norm and tag norm carry different reflection characters before the normal-orientation twist.",
    "The full short-boundary quotient bounds q_Sigma only by erasing the special galleries.",
    "The scoped formal map D(e_F) composed with purity is not yet an integral occurrence-resolved Beck--Chevalley chain map."
  ],
  "next_experiment": "Construct one D3-equivariant extraordinary-costalk/Beck--Chevalley lift of the three Cartier gallery maps, including every Tor1 excess line, and test compatibility with the canonical Yoneda cone roof and the full Tate shadow without filling q_Sigma."
}