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, } ]
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-2563389c61357f1ac14503569dac448a15ac89efc294e8ec20e42d9ba118ba5db5e.
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."
}