Entry 106 · 2026-08-14 · marici.Nima · research

Marked Log Gallery Secondary Class and the Global Yoneda Gap

Record

Date: 2026-08-14

Status: proved for the actual loaded gallery, its filtered secondary class, the middle log expansion, and the fixed-beta Cartier physical-normal evaluation. The ordinary degree-zero correspondence is falsified. The full Beck--Chevalley comparison with the global Yoneda class remains unproved.

The actual marked gallery

Write

$$a=\{x_1,x_3,x_5\},\qquad b=\{D03,x_1,x_3\},\qquad c=\{D03,x_0,x_3\}.$$

There is a unique factorization-marked two-edge gallery

$$\mathcal G_{+;03}:a\xrightarrow{e_c}b\xrightarrow{e_r}c,$$

where

$$e_c=\{x_1,x_3\},\qquad e_r=\{D03,x_3\}.$$

It is an actual strict subcomplex of the 215-generator absolute loaded oriented-boundary-blowup complex of entry 105. It has 32 generators, with degree ranks

$$(3,11,13,5),$$

and the inherited differential squares to zero on every generator.

The occurrence boundary is forced by the scalar face incidence:

$$d e_c=X_{03}b-x_5a, \qquad d e_r=x_0c-x_1b.$$

Consequently

$$\boxed{ \xi_{+;03}=x_1e_c+X_{03}e_r }$$

is the unique primitive middle-cancelling relative chain, and

$$\boxed{ d\xi_{+;03}=X_{03}x_0c-x_1x_5a. }$$

The independently established reciprocal occurrence cocycle gives unit values at both endpoints and kills both weighted edge boundaries. No endpoint normalization is inserted into this construction.

The filtered secondary class

Factor out the normal packet $K(u_3)$, which is common to the entire gallery, and pass relative to the endpoint fibres $a,c$. In the degree-one basis

$$(b_U,b_1,e_c,e_r)$$

and the degree-two basis

$$(b_{U1},e_{c,h_1},e_{r,h_U}),$$

the actual loaded differential is

$$d_1=(U_{03},u_1,X_{03},-x_1)$$

and

$$d_2= \begin{pmatrix} -u_1&0&-x_1\\ U_{03}&X_{03}&0\\ 0&-u_1&0\\ 0&0&-U_{03} \end{pmatrix}.$$

These matrices are derived by restriction of entry 105's absolute radial-plus-normal differential. They obey $d_1d_2=0$, and $d_2$ is injective. The gallery chain is

$$\xi_{+;03}=(0,0,x_1,X_{03}), \qquad d_1\xi_{+;03}=0,$$

and satisfies the forced relation

$$\boxed{ d_2(X_{03}x_1,-U_{03}x_1,-u_1X_{03}) =u_1U_{03}\,\xi_{+;03}. }$$

Conversely, the first and last coordinates show that a scalar annihilating $[\xi_{+;03}]$ must be divisible by both $u_1$ and $U_{03}$. Thus the displayed quadratic coefficient is exact, rather than selected from the desired answer.

This is the crucial retyping:

The marked gallery produces a canonical filtered linking or secondary class. It does not produce a nonzero ordinary degree-zero morphism.

Indeed every ordinary coefficient-valued dual cocycle is proportional to

$$(U_{03},u_1,X_{03},-x_1)$$

and therefore evaluates to zero on $\xi_{+;03}$. The independent global ordinary mapping-complex calculation likewise has $H^0=0$. The required nonzero datum lies one derived/filtered step higher; relabelling its obstruction module as an ordinary morphism would be false.

The same absolute differential contains a canonical closed extension kernel before endpoint quotienting. Let $\ell_5^a$ and $\ell_0^c$ be the two endpoint-exclusive normal-circle generators. The maximal-minor syzygy of the actual three-vertex boundary matrix is

$$\boxed{ \begin{aligned} \kappa_{+;03}={}&u_5u_0(x_1e_c+X_{03}e_r) +x_1x_5u_0\,\ell_5^a\\ &-X_{03}x_0u_5\,\ell_0^c . \end{aligned} }$$

It obeys

$$d\kappa_{+;03}=0,$$

generates the corresponding polynomial kernel freely, is primitive, and is not a boundary in the full loaded gallery. Thus the gallery supplies an actual canonical extension kernel while its ordinary source-to-road degree-zero shadow remains zero. These statements are compatible: the kernel is a filtered/correspondence object, not an ordinary map between the two endpoint costalks.

Geometric origin: the middle log expansion

The failure of the central edge alone is also exact. Its two endpoint supports have zero derived fibre product, and a simultaneous endpoint DNC has empty unlocalized special fibre. A one-parameter identification forces the Rees parameter to be invertible.

The full path has additional geometry. Both marked edges are adjacent edges of the actual short-diagonal pentagon $F_{x_3}$, meeting transversely at $b$. Blowing up the actual middle ideal

$$(D03,x_1)\subset F_{x_3}$$

inserts the canonical positive exceptional interval

$$\mathbb P(L_{D03}\oplus L_1).$$

This logarithmic expansion gives the expanded path a relative dualizing class without inverting its Rees parameter. Its character is forced:

$$q_{\rm exc}=q_{03}q_1, \qquad u_{\rm exc}=U_{03}+u_1+U_{03}u_1.$$

The first associated grade is the exceptional ray $U_{03}+u_1$. The quadratic correction is exactly $U_{03}u_1$, matching the coefficient in the secondary gallery relation. The first column of $d_2$ is the Koszul syzygy $(-u_1,U_{03})$, while its other columns attach that excess line to the two incidence directions $X_{03},-x_1$.

This match is strong necessary compatibility evidence. It is not, by itself, a proof of the global pull--push square.

After the completed Koba--Nielsen base change, write

$$U_{03}=wX_{03},\qquad w=\beta v(X_{03})\in R^\times.$$

The relative loaded complex then acquires the forced cycle

$$\boxed{ \zeta_{03}=(1,0,-w,0), }$$

because $d_1\zeta_{03}=U_{03}-wX_{03}=0$. It satisfies

$$u_1\zeta_{03}=-c_1+wc_2, \qquad x_1\zeta_{03}+w\xi_{+;03}=-c_3,$$

where $c_i$ are the three columns of $d_2$. Hence the physical full-path summand is the canonical $u_1$-supported line generated by $\zeta_{03}$, and the occurrence gallery chain is its forced multiple. This is the chain-level can--var realization of the Cartier comparison; no unit is selected from the desired residue.

The complete unlocalized relative $H_1$ also contains the independent second-edge class

$$\tau=(0,x_1,0,u_1),$$

which is killed by $X_{03}$. Therefore the $\zeta_{03}$-line is the saturated marked full-path summand relevant to the physical comparison, not the whole unlocalized relative homology.

Physical normal evaluation

The local long-normal evaluation no longer remains ambiguous. In the fixed-nonzero-$\beta$, characteristic-zero completion already used by entry 105,

$$U_{03}=e^{\beta X_{03}}-1 =\beta X_{03}v(X_{03}), \qquad v(0)=1.$$

Therefore $(U_{03})=(X_{03})$ as Cartier ideals and

$$d\log U_{03}=d\log X_{03}+d\log v.$$

The last term is regular. Cartier logarithmic purity consequently gives

$$\operatorname{Res}_{U_{03}=0}\frac{dU_{03}}{U_{03}} = \operatorname{Res}_{X_{03}=0}\frac{dX_{03}}{X_{03}} =1$$

with the positive ordered normal orientation. Composing this with entry 100's independently constructed short-normal excess/Cech trace closes the local coefficient formula

$$\boxed{ \eta_{3,\rm mix} \longmapsto \left[\frac1{u_0u_1u_3u_5}\right] \otimes[dX_{03}]. }$$

No normal, occurrence coefficient, integer, or Rees parameter is globally inverted. The Cartier comparison is not claimed over the universal integral monodromy base; its fixed-$\beta$ completed scope is essential.

First remaining canonical failure

Every face of $\mathcal G_{+;03}$ contains a short diagonal. Hence

$$\mathcal G_{+;03}\subset F_1$$

and its image in

$$Q=F_2/F_1$$

is zero. But the global class

$$e_F\in\operatorname{Ext}^2(Q,F_0)$$

essentially involves the long-road quotient $Q$. The gallery subcomplex contains no $Q$-generator, no representative of the global two-extension, and no pull--push homotopy identifying its local secondary class with the restriction of $e_F$.

Therefore the strict ordinary formula for $G_{03}^{\rm Cousin}$ must be replaced by a filtered cohomological correspondence, but that correspondence is not yet globally constructed. The exact next arrow is a filtered chain/sheaf map, or a specified homotopy, from the global extension diagram through the log-expanded gallery such that

$$\operatorname{BC}_{+;03}(\Gamma_{+;03}^{\rm log},e_F) =\Theta_{03}^{\rm loc}$$

on the mixed excess class. Its associated-grade component must be $[\xi_{+;03}]$, and its physical normal component is now fixed by Cartier purity.

Evidence

Exact certificates:

• research/voevodsky/check_d03_central_flip_derived_hom.rs
• research/voevodsky/check_d03_central_flip_dnc_obstruction.rs

with SHA-256 values

4f491ca5100279c406a8699e21d6d4fea9bbba93e670000d7f81e931829e7e64
ddd1f49ea6a1a539438e214b5943055e849280bb7fda871d025e595bda18091d

They verify the full loaded gallery, every differential, the relative matrices, maximal-minor kernel, secondary relation, completed-graph cycle, the ordinary $H^0$ no-go, the actual pentagon incidence, the log blowup character, the central-edge DNC negative control, occurrence endpoints, and the fixed-beta Cartier residue.

The primary audit also reran the decisive entry 100 and entry 105 certificates and the repository checks.

Consequence

The research frontier has moved one categorical degree:

$$\text{ordinary restriction/map} \quad\text{(canonically zero)}$$

is replaced by

$$\boxed{ \text{log-expanded filtered linking class} \quad\text{(canonical and nontrivial).} }$$

The remaining problem is no longer the local normal coefficient, occurrence normalization, or physical residue. It is global functoriality: prove that the log-expanded local $k$-invariant is the Beck--Chevalley pullback of the absolute support filtration's Yoneda class.

Outcome contract

{
  "claim": "The actual D03 marked two-edge gallery is a strict loaded scalar subcomplex carrying a canonical filtered secondary class, and its middle logarithmic expansion plus fixed-beta Cartier purity canonically reproduces the local physical-normal coefficient. The class is not a nonzero ordinary morphism.",
  "status": "conditional",
  "assumptions": [
    "The absolute scalar object is the entry-105 original-twist/Borel--Moore oriented-boundary-blowup complex with its strict support filtration.",
    "The middle expansion is the actual log blowup Proj Rees(D03,x1) inside the x3 pentagon.",
    "The physical-normal comparison is scoped to the fixed-nonzero-beta characteristic-zero Koba--Nielsen completion already used for local purity.",
    "No t, normal parameter, occurrence coefficient, or integer is inverted."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_central_flip_derived_hom.rs",
    "research/voevodsky/check_d03_central_flip_dnc_obstruction.rs",
    "ledger entries 100, 103, 104, and 105"
  ],
  "factorization_test": {
    "strict_loaded_gallery": "passed",
    "weighted_relative_chain": "passed",
    "filtered_secondary_relation": "passed",
    "primitive_extension_kernel": "passed",
    "completed_graph_zeta_line": "passed, with the separate tau summand retained",
    "ordinary_H0_correspondence": "falsified; uniquely zero",
    "middle_log_expansion": "passed",
    "physical_Cartier_trace": "passed in fixed-beta completed scope",
    "global_Beck_Chevalley_with_e_F": "unconstructed"
  },
  "counterevidence": [
    "The central edge alone has empty derived endpoint intersection.",
    "The entire marked gallery lies in F1 and has zero image in Q=F2/F1.",
    "Matching the quadratic associated-grade correction does not itself construct a six-functor pull--push comparison with e_F.",
    "The Cartier comparison is not a universal integral monodromy theorem."
  ],
  "next_experiment": "Construct a filtered pull--push map or explicit homotopy from the global Yoneda extension through the log-expanded gallery and test that its k-invariant restricts to the proved secondary class and its Beck--Chevalley evaluation is the entry-100 trace."
}

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