Regular Multi-Rees Diagonal and the Missing Tate Off-Diagonal
Record
Date: 2026-08-14
Status: one proved coefficient theorem and one sharp blocker. Independent, labelled Rees parameters give the smallest integral regular deformation that retains all three Cartier (\operatorname{Tor}1) lines. They do not produce the nonzero off-diagonal (1-r) extension. The remaining construction is one marked spatial extraordinary class (\lambda{\rm ex}), not another carrier cell or rational splitting.
The regular coefficient deformation
A tempting common-parameter correspondence is
[ A_t= \mathbb Z[t,x_i,q_i^{\pm1}] / (q_i-1-tx_i), \qquad i\in{1,3,5}. ]
It is an integral deformation of the unit section of (\mathbb G_m), but it is the wrong supported coefficient object. The pulled-back normal sequence
[ (tx_1,tx_3,tx_5) ]
is not regular because all three entries contain the same factor (t). Its Koszul homology contains the nonzero class represented by
[ z=x_3e_1-x_1e_3, ]
with (tz) a boundary. Equivalently, the pulled-back support contains the extra vertical component
[ V(tJ_+)=V(t)\cup V(J_+). ]
Removing that component by inverting (t) would erase the specialization data. The common-parameter model is therefore rejected.
The regular replacement is the labelled multi-Rees correspondence
[ \boxed{ A_{\rm mR}
\mathbb Z[t_i,x_i,q_i^{\pm1}] / (q_i-1-t_ix_i), \qquad i\in{1,3,5}, } ]
with (D_3) permuting the triples ((t_i,x_i,q_i)). The sequence
[ (t_1x_1,t_3x_3,t_5x_5) ]
is regular. For the reciprocal packet,
[ u_i^\vee=q_i^{-1}-1=-q_i^{-1}t_ix_i, ]
and the integral Laurent-unit change
[ \overline h_i^\vee=-q_i h_i^\vee ]
gives
[ d\overline h_i^\vee=t_ix_i p_i^\vee. ]
Thus the original and reciprocal one-normal differentials have the same regular multi-Rees diagonal. Applying the labelled (x_i)-Cartier maps retains, rather than divides out, the three conormal lines ([t_i]). This is the canonical coefficient-level selection of the three occurrence (\operatorname{Tor}_1) copies. No (x_i), (u_i), (t_i), or integer is inverted.
This theorem is deliberately coefficient-level. The original and reciprocal support variances and their endpoint pairing units remain explicit.
The unique carrier shadow
At the carrier level, let (P_{H_0}) be the abstract three-tag module and (P_{\operatorname{Tor}_1}) the abstract three-road module. Conditional on a marked global identification of the actual local Cartier (H_0) and (\operatorname{Tor}_1) lines with these two modules, integral (D_3)-equivariance gives
[ \operatorname{Hom}{\mathbb Z[D_3]} (P{H_0},P_{\operatorname{Tor}_1}) =\mathbb Z(1-r). ]
Saturation and the positive carrier orientations select the primitive coefficient (+1). Hence the only possible fully based carrier target is
[ \boxed{ 0\longrightarrow\mathbb Z_{\rm or} \xrightarrow{N_{\rm tag}}P_{H_0} \xrightarrow{1-r}P_{\operatorname{Tor}_1} \xrightarrow{\epsilon}\mathbb Z \longrightarrow0. } ]
This classifies the entry-102 Tate window; it does not derive the marked global identification from the multi-Rees diagonal. The generic class remains
[ q_\Sigma=N_{\rm road} \in P_{\operatorname{Tor}1}, \qquad \epsilon(q\Sigma)=3. ]
It is not identified with the reflection-odd tag norm.
The regular excess wedge also gives the integral evaluation/transfer pair
[ \epsilon_{\rm ex}:P_{\rm tag}\to\mathbb Z_{\rm or}, \qquad \Delta_{\rm ex}=\epsilon_{\rm ex}^{\vee}, \qquad \epsilon_{\rm ex}\Delta_{\rm ex}=3. ]
This pair is not a consecutive differential. If
[ K=\ker\epsilon_{\rm ex}, \qquad A_2^{\rm road}=\ker\epsilon, ]
then the restriction
[ (1-r)|_K:K\longrightarrow A_2^{\rm road} ]
has Smith factors ((1,3)). Therefore the split excess kernel is not the saturated peripheral lattice. Identifying them would reintroduce exactly the forbidden division by three.
The missing off-diagonal class
Each established derived Cartier base change is split:
[ R/(x_i)\otimes_R^L C \simeq[C\xrightarrow0C]. ]
Consequently their direct sum has zero (H_0)-to- (\operatorname{Tor}_1) differential. The multi-Rees equations select and orient the coefficient lines, but they do not turn that zero differential into (1-r). Inserting the carrier matrix at this point would fit the desired answer.
The first unconstructed datum is therefore
[ \boxed{ \lambda_{\rm ex}: P_{\rm Cart,H_0}\dashrightarrow P_{\rm Cart,\operatorname{Tor}1}, \qquad \operatorname{gr}{\rm car}\lambda_{\rm ex}=1-r, } ]
realized by a marked spatial multi-Rees extraordinary pull–push. It must carry every ([t_i]) line, reproduce the three entry-100 excess generators, and be compatible with the actual support filtration and its Yoneda cone roof.
At the integral carrier level the obstruction is now a single class
[ \boxed{ \omega= ho(e_F)-\beta_\triangle \in \operatorname{Ext}^2_{\mathbb Z[D_3]} (\mathbb Z,\mathbb Z_{\rm or}) \simeq\mathbb Z/3. } ]
The identity (\epsilon(q_\Sigma)=3) proves only that three times any such obstruction vanishes; it does not prove (\omega=0). If (\omega=0), the coherent lifts form a torsor under
[ \operatorname{Ext}^1_{\mathbb Z[D_3]} (\mathbb Z,\mathbb Z_{\rm or}) \simeq\mathbb Z/2. ]
The positive normal and whole-gallery orientations are the candidate geometric datum that should select this parity coherence. They cannot be used until (\lambda_{\rm ex}) itself has been constructed.
Evidence
New exact certificate:
research/voevodsky/check_positive_cartier_tate_costalk.rs, SHA-2563820fe6ce63cae922aba86151867d787ca781a48d3c112832918f84ee880ccab.
The checker proves the conditional carrier Hom classification, exact abstract Tate window, dihedral signs, common-parameter torsion, regular multi-Rees diagonal, reciprocal normalization, retention of the Rees conormal lines, the ((1,3)) excess-lattice Smith obstruction, and failure of a strict extension diagram morphism. Entries 100, 102, 105, 112, and 113 supply the inherited local traces, Tate class, absolute filtration, whole-gallery maps, and mixed generic/special block.
Reproduce with rustfmt --check, rustc --edition 2021 -D warnings -O, and
execution of the certificate. Its JSON result and git diff --check pass.
Boundary and consequence
- The multi-Rees correspondence is not an identity-base substitution between independent occurrence and monodromy variables.
- It proves a regular coefficient diagonal, not the spatial Beck–Chevalley transformation.
- The inherited (1-r) carrier is classified uniquely but is not thereby lifted to the split local costalks.
- Ordinary restriction of (e_F) remains zero, while the desired local tuple is nonzero.
- No full (G_{03}^{\rm Cousin}), negative-sheet assembly, physical-Cut theorem, or CHY identification follows yet.
The next discriminating experiment is singular: construct (\lambda_{\rm ex}) on one marked multi-Rees support correspondence, compute (\omega), and, only if it vanishes, use the positive orientation to test the remaining parity coherence. Rotation then supplies the other two roads.
Outcome contract
{
"claim": "Independent D3-permuted Rees parameters give a regular integral coefficient diagonal q_i-1=t_i x_i that retains every Cartier Tor1 and conormal line; a common parameter is nonregular. The unique possible carrier extension is the full N_tag/(1-r)/epsilon Tate window, but the multi-Rees coefficient geometry does not construct its off-diagonal 1-r Beck-Chevalley class.",
"status": "inconclusive",
"assumptions": [
"The ordinary Cartier tag maps and orientations are those of entries 100 and 112.",
"Occurrence, monodromy, Rees, and integer parameters remain uninverted.",
"The coefficient diagonal is not promoted to a spatial correspondence without an explicit extraordinary pull-push."
],
"evidence_refs": [
"research/voevodsky/check_positive_cartier_tate_costalk.rs",
"ledger entries 100, 102, 105, 112, and 113"
],
"factorization_test": {
"common_parameter": "falsified by nonregular t-torsion",
"multi_Rees_diagonal": "proved regular and D3-equivariant",
"Tor1_and_conormal_lines": "retained",
"carrier_extension": "unique candidate N_tag/(1-r)/epsilon window after the unconstructed marked global identification",
"excess_kernel_to_peripheral": "Smith (1,3), not an isomorphism",
"lambda_ex": "unconstructed",
"omega_in_Z_mod_3": "uncomputed",
"parity_torsor": "unselected",
"full_G03_Cousin": "unconstructed"
},
"counterevidence": [
"The local derived Cartier packets are split and contain no intrinsic 1-r differential.",
"The excess augmentation and its dual norm are an evaluation/transfer pair, not consecutive Tate differentials.",
"The literal support restriction of e_F is zero."
],
"next_experiment": "Construct the marked spatial multi-Rees extraordinary class lambda_ex with carrier 1-r, then compute rho(e_F)-beta_triangle and the residual parity coherence while retaining q_Sigma."
}