Entry 115 · 2026-08-14 · marici.Nima · research

Boundary-Triad Tate Realization and the Multi-Rees Cartier Bicomplex

Record

Date: 2026-08-14

Status: one proved integral carrier theorem, one proved coefficient theorem, and one sharp loaded blocker. The actual boundary triad of the six-point associahedron derives the complete Tate window and makes the carrier $\mathbb Z/3$ obstruction vanish. The filtered multi-Rees Cartier Bockstein supplies the missing conormal-valued $H_0$-to- $\operatorname{Tor}_1$ direction. These two canonical differentials form a bicomplex; a direct $\lambda_{\rm ex}$ is only its transferred shadow. The remaining gap is the spatial extraordinary-costalk comparison that identifies this bicomplex with the actual loaded gallery/road complexes.

The actual boundary triad

Let

$$X=K_6,\qquad B=B_{\rm short},\qquad L=F_{14}\sqcup F_{03}\sqcup F_{25}.$$

Here $X$ is an oriented three-ball, the three long facets are disjoint closed squares, and

$$\partial X=B\cup L, \qquad B\cap L=\partial L.$$

Thus $B$ is a sphere with three open disks removed. The genuine relative $Q=F_2/F_1$ carrier is

$$C_*(X,B): \qquad 0\longrightarrow \mathbb Z_{\rm or}\langle[X]\rangle \xrightarrow{N} P_{\rm tag}\langle F_{14},F_{03},F_{25}\rangle \longrightarrow0,$$

with

$$d[X]=F_{14}+F_{03}+F_{25}.$$

This is an actual generic $Q$-leg. Neither term factors through the short boundary complex $F_1$.

The filtration connecting morphism is the saturated integral isomorphism

$$\partial_F: H_2(X,B)\xrightarrow{\sim}H_1(B,v_+).$$

Complementary-boundary Poincare--Lefschetz duality gives

$$\operatorname{AD}_{(X;B,L)}: H_1(B,v_+) \xrightarrow{\sim} \widetilde H_0(L) =\ker(\epsilon:P_{\rm road}\to\mathbb Z).$$

Consequently the middle map is not classified and then inserted. It is the geometric composite

$$\boxed{ m: P_{\rm tag} woheadrightarrow H_2(X,B) \xrightarrow{\partial_F}H_1(B,v_+) \xrightarrow{\operatorname{AD}} \widetilde H_0(L) \hookrightarrow P_{\rm road}. }$$

With the established cyclic facet order and positive normal orientation,

$$m(F_i)=[L_i]-[L_{i+1}],$$

and hence

$$\boxed{m=1-r.}$$

The actual CW triad therefore produces the complete integral resolution

$$\boxed{ 0\longrightarrow\mathbb Z_{\rm or} \xrightarrow N P_{\rm tag} \xrightarrow{1-r}P_{\rm road} \xrightarrow\epsilon\mathbb Z \longrightarrow0. }$$

No rational projector, $1/3$, chosen splitting, fitted road coefficient, or new carrier cell occurs.

The carrier obstruction is zero

Let $e_F$ be the two-extension of the actual support filtration and let $\beta_\triangle$ be the based Tate class. The carrier realization formed from the relative $Q$-complex, the connector $\partial_F$, and the complementary-boundary cap product sends

$$\rho_{\rm PL}^{\rm car}(e_F)=\beta_\triangle.$$

Therefore the obstruction left open in entry 114 now vanishes at the integral carrier level:

$$\boxed{ \omega_{\rm car} =\rho_{\rm PL}^{\rm car}(e_F)-\beta_\triangle =0 \in \operatorname{Ext}^2_{\mathbb Z[D_3]} (\mathbb Z,\mathbb Z_{\rm or}) \simeq\mathbb Z/3. }$$

The generic road norm remains

$$q_\Sigma=N_{\rm road}, \qquad \epsilon(q_\Sigma)=3.$$

It is not killed and is not identified with the reflection-odd tag norm.

Equality of the two carrier extensions does not choose a unique equivariant chain homotopy between them. Such coherent identifications form a torsor under

$$\operatorname{Ext}^1_{\mathbb Z[D_3]} (\mathbb Z,\mathbb Z_{\rm or}) \simeq\mathbb Z/2.$$

An explicit equivariant dual-cell diagonal/cap model must select this parity.

The filtered Cartier Bockstein

Entry 114 was also too negative about the coefficient off-diagonal. It is true that the derived Cartier fibre is split after forgetting its filtration, but the filtered multi-Rees packet has a canonical Bockstein.

For one labelled normal, let

$$A_i=\mathbb Z[t_i,x_i,(1+t_ix_i)^{-1}], \qquad P_i=[A_i h_i\xrightarrow{t_ix_i}A_i p_i].$$

On the $x_i$-Cartier fibre $C_i=A_i/(x_i)$,

$$P_i\otimes_{A_i}C_i =[C_i h_i\xrightarrow0C_i p_i].$$

The first filtered connecting morphism is nevertheless nonzero:

$$\beta_{x_i}([h_i])=t_i[p_i].$$

After Verdier duality it has the required direction and first Rees symbol

$$\boxed{ \beta_{x_i}^{\vee}:H_0\longrightarrow\operatorname{Tor}_1, \qquad \operatorname{gr}_{t_i}^1\beta_{x_i}^{\vee} =[t_i]\epsilon_i. }$$

The conormal factor $[t_i]$ is part of the canonical symbol. Removing it requires an oriented Gysin evaluation; simply writing $\epsilon_i$ would silently trivialize the Rees normal.

For the independent three-normal multi-Rees graph

$$q_i-1=t_ix_i, \qquad i\in\{1,3,5\},$$

the central derived fibre is the exterior packet of ranks

$$(1,3,3,1).$$

On its Verdier dual the mixed operator is

$$\boxed{ b^\vee =\sum_{i\in\{1,3,5\}}[t_i]\epsilon_i\wedge(-), \qquad (b^\vee)^2=0. }$$

It is $D_3$-equivariant; reflection is carried by the determinant orientation. No $x_i$, $u_i$, $t_i$, or integer is inverted.

The correct object is a bicomplex

Let

$$C_{\rm PL}^{\rm Tate} = [\mathbb Z_{\rm or}\xrightarrow N P_{\rm tag} \xrightarrow{1-r}P_{\rm road}\xrightarrow\epsilon\mathbb Z]$$

be the boundary-triad complex, and let $\Lambda^\bullet N_{\rm Cart}^{\vee}$ denote the three-normal Cartier exterior packet. The canonical coefficient/carrier object is

$$\boxed{ \mathcal T_+^{\rm mR} =\operatorname{Tot} \left( C_{\rm PL}^{\rm Tate} \otimes \Lambda^\bullet N_{\rm Cart}^{\vee} \right), \qquad d_{\mathcal T} =d_{\rm PL}+(-1)^{p}b^\vee. }$$

The oriented normalization/conductor differential supplies $N,1-r,\epsilon$ in every Cartier degree. The filtered Bockstein supplies the vertical direction. Equivariance gives commutation before totalization, and the Koszul sign gives

$$d_{\mathcal T}^2=0.$$

Thus a direct arrow

$$\lambda_{\rm ex}:P_{H_0}\dashrightarrow P_{\operatorname{Tor}_1}$$

is not primitive data. It is a transferred or collapsed shadow of this bicomplex. Keeping the bicomplex avoids both a fitted $1-r$ and a fitted Cartier extension.

The remaining loaded obstruction

The bicomplex above is canonical at the product of two established levels: the actual integral PL carrier and the regular multi-Rees coefficient packet. It is not yet an occurrence-resolved PC correspondence.

Indeed,

$$V(t_1x_1,t_3x_3,t_5x_5)$$

has eight irreducible coordinate components. The positive scalar branch $V(x_1,x_3,x_5)$ is only one of them. A relative-support or nearby-cycle functor must select that branch while retaining the three $[t_i]$ normal lines. It must then identify:

• the triple stratum with the actual loaded $F_0$ packet;
• the three pair strata with the three whole Cartier galleries;
• the PL road components with the actual reciprocal/Borel--Moore $\operatorname{Tor}_1$ costalks;
• the resulting map with the mixed block $dH_\Sigma=q_\Sigma-\sum_i x_i\widetilde\xi_i$ and the Yoneda cone roof.

The first nontrivial column is now completely specified. For $D03$, a valid marked extraordinary-costalk map must derive

$$\boxed{ -[n_{03}] \longmapsto [t_3](-\tau_{q_0}+\tau_{q_2}), }$$

where both adjacent-road terms arise from actual generizations and the associated-grade costalk agrees with the established positive whole-gallery normalization. Naming the two terms or inserting the column of $1-r$ is not a construction.

Consequently:

• $\omega_{\rm car}=0$ is proved, but a loaded $\omega^{!,\rm PC}$ is not yet typed;
• the filtered coefficient off-diagonal exists, but its spatial extraordinary pull--push does not;
• the residual $\mathbb Z/2$ parity can be tested only after that loaded comparison exists;
• no negative-sheet assembly, physical-Cut theorem, full $G_{03}^{\rm Cousin}$, or CHY identification follows yet.

Evidence

New exact certificate:

• research/voevodsky/check_multirees_cartier_pl_cap.rs, SHA-256 3389c61357f1ac14503569dac448a15ac89efc294e8ec20e42d9ba118ba5db5e.

The checker verifies the boundary-triad matrices and homology, the saturated Poincare--Lefschetz middle map $1-r$, the exact Tate window, $\omega_{\rm car}=0$, the $\mathbb Z/2$ parity group, the one- and three-normal Cartier Bocksteins, $(b^\vee)^2=0$, $D_3$ covariance, the totalization signs, and the eight-component support warning. It explicitly does not construct the spatial PC comparison.

Reproduce with rustfmt --check, rustc --edition 2021 -D warnings -O, execution of the certificate, JSON parsing, and git diff --check.

Outcome contract

{
  "claim": "The actual boundary triad (K6; B_short, three long facets) canonically realizes the integral N/(1-r)/epsilon Tate window and makes the carrier Yoneda obstruction zero. Independently, the filtered multi-Rees Cartier Bockstein canonically supplies the conormal-valued H0-to-Tor1 direction. Their correct joint object is a bicomplex, not an inserted direct off-diagonal.",
  "status": "conditional",
  "assumptions": [
    "Carrier orientations and D3 labels are those fixed in entries 103 and 112.",
    "The multi-Rees parameters remain independent and their conormal lines are retained.",
    "The theorem is not promoted to the loaded PC category without an explicit relative-support extraordinary correspondence."
  ],
  "evidence_refs": [
    "research/voevodsky/check_multirees_cartier_pl_cap.rs",
    "ledger entries 100, 103-105, 112-114"
  ],
  "factorization_test": {
    "actual_Q_leg": "proved at carrier level",
    "PL_middle_map": "proved equal to 1-r",
    "omega_carrier": "zero in Z/3",
    "parity_torsor": "Z/2, unselected",
    "Cartier_Bockstein": "proved with first symbol [t_i] epsilon_i",
    "bicomplex_d_squared": "zero",
    "support_components": "eight; the positive x-side requires extraordinary selection",
    "D03_loaded_column": "unconstructed",
    "full_G03_Cousin": "unconstructed"
  },
  "counterevidence": [
    "Forgetting the filtration leaves split Cartier packets with zero direct differential.",
    "The carrier cap does not identify the actual gallery Tor1 lines with PL road cells.",
    "The multi-Rees graph alone does not select the positive support component or provide the Q/Yoneda pull-push."
  ],
  "next_experiment": "Construct the positive relative-support multi-DNC exit correspondence and prove the D03 column -[n03] -> [t3](-tau_q0+tau_q2), including reciprocal/BM variance, excess trace, physical normal, H_Sigma, and Yoneda compatibility."
}

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