Entry 114 · 2026-08-14 · marici.Nima · research

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-256 3820fe6ce63cae922aba86151867d787ca781a48d3c112832918f84ee880ccab.

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."
}

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