Support-Directed Can–Var Packet and Three Local Cousin Traces

Record

Date: 2026-08-14

Status: proved one-normal coefficient theorem; proved three labelled local derived road traces; falsified a strict degree-zero finite-free lift. The global augmented (D_3)-equivariant Cousin coherence remains untyped, not disproved.

Scope: the universal monodromy ring

[ R_0=\mathbb Z[q_0^{\pm1},\ldots,q_5^{\pm1}], \qquad u_j=q_j-1, ]

the plus normalization branch, and the three marked roads (F_{03},F_{25},F_{14}) of entries 97–99. Scalar occurrence variables, monodromy variables, physical normal lines, and support directions remain distinct.

This entry refines the immediate objective of entry 99. The missing object cannot be a strict map between finite Koszul stalks. Its canonical local pieces are bivariant excess correspondences followed by Koszul–Cech local duality.

The support-directed one-normal packet

Put

[ q^\vee=q^{-1}, \qquad u^\vee=q^{-1}-1=-q^{-1}u. ]

The original-twist locally finite/Borel–Moore road and its reciprocal-twist regular Verdier dual use different, paired can–var conventions.

For the costandard road object,

[ Rj_*\mathscr L[1]: \qquad \Psi=R\langle\ell\rangle \mathop{\rightleftarrows}^{\operatorname{can}=u}_{\operatorname{var}=1} \Phi=R\langle p\rangle . ]

For the reciprocal standard object,

[ j_!\mathscr L^\vee[1]: \qquad \Psi^\vee=R\langle p^\vee\rangle \mathop{\rightleftarrows}^{\operatorname{can}^\vee=1}_{ \operatorname{var}^\vee=u^\vee} \Phi^\vee=R\langle\ell^\vee\rangle . ]

In both cases the two composites are the appropriate monodromy difference. The support-directed finite complexes are

[ K(u)=[R\langle\ell\rangle\xrightarrow{u}R\langle p\rangle], \qquad K(u^\vee)=[R\langle\ell^\vee\rangle \xrightarrow{u^\vee}R\langle p^\vee\rangle]. ]

The complementary-degree pairing

[ \boxed{ \beta:K(u)\otimes K(u^\vee)\longrightarrow R[1] } ]

is fixed by

[ \beta(p,\ell^\vee)=1, \qquad \beta(\ell,p^\vee)=-q. ]

It is a chain pairing because

[ u+q u^\vee=0, ]

and it is perfect because its antidiagonal determinant is the Laurent unit (q). This is the integral one-normal source of entry 97’s bivariant pairing.

The finite complex is only the logarithmic/simple-pole stage. The honest supported Cousin object is the extended Cech complex

[ C_u^\bullet= [R\xrightarrow{\rm loc}R[u^{-1}]], \qquad H^1(C_u)=R[u^{-1}]/R. ]

There is a canonical comparison

[ \boxed{ \kappa_u: [R\xrightarrow{u}R] \longrightarrow [R\xrightarrow{\rm loc}R[u^{-1}]], \qquad \kappa_u=(1,u^{-1}). } ]

On cohomology it sends

[ \bar r\longmapsto r/u. ]

Thus entry 38’s (\ell/u) is the Koszul-to-Cech image of the integral generator. Keeping (R[u^{-1}]) as one Cousin term is not global inversion. Tensoring the whole theory with (R[u^{-1}]) instead contracts (K(u)) and erases the supported class. No single finite-free complex is both the full local-cohomology object and the literal holder of (\ell/u).

The twist-aware repeated-normal excess

At the plus/(D03) intersection, order the reciprocal plus factor before the original road factor. The repeated normal is

[ D_3=K(u_3^\vee)\otimes K(u_3), ]

with

[ D_{3,2}=R \xrightarrow{(-u_3,u_3^\vee)^T} D_{3,1}=R^2 \xrightarrow{(u_3^\vee,u_3)} D_{3,0}=R. ]

Define

[ \pi_0=1, \qquad \pi_1=(1,-q_3), \qquad \pi_2=0. ]

The oriented kernel generator is

[ \boxed{ \eta_{3,\rm mix} =-q_3,\ell_3^{+,\vee}\otimes p_3^{03} -p_3^{+,\vee}\otimes\ell_3^{03}. } ]

It gives the integral exact sequence

[ \boxed{ 0\longrightarrow K(u_3^\vee)[1] \longrightarrow K(u_3^\vee)\otimes K(u_3) \xrightarrow{\pi}K(u_3^\vee) \longrightarrow0. } ]

The entry-97 twist normalization

[ p^\vee\longmapsto-q p, \qquad \ell^\vee\longmapsto\ell ]

sends the complete sequence to entry 99’s sequence with

[ \eta_3=\ell_3^+-\ell_3^{03}, ]

up to the same forced Laurent unit (-q_3) on source and image. The top coefficient and determinant orientation remain (+1). Hence twist reversal does not alter the carrier sign and requires no (u_3^{-1}) or numerical denominator.

Three local derived road traces

For the plus branch

[ I_+^\vee=(u_1^\vee,u_3^\vee,u_5^\vee) ]

and the three opposite road pairs

[ I_{03}=(u_0,u_3), \qquad I_{25}=(u_2,u_5), \qquad I_{14}=(u_4,u_1), ]

let (Q_i) be the union of the corresponding branch and road normal sequences. The actual marked paths of entry 99 distinguish the two copies of the shared normal and select a labelled excess retraction

[ \operatorname{tr}^{\rm ex}i: K(I+^\vee)\otimes K(I_i) \longrightarrow K(Q_i)[1] ]

whose shifted primitive generator maps to (1). Composing with the multi-normal Koszul–Cech comparison gives

[ \boxed{ \Theta_i^{\rm loc}: K(I_+^\vee)\otimes K(I_i) \longrightarrow C_{Q_i}[1], \qquad \eta_{i,\rm mix}\longmapsto \left[\frac{1}{\prod_{j\in Q_i}u_j}\right]. } ]

For all three roads, the exact certificate checks every Koszul degree, the actual two-flip marked path, the full road Cousin square, occurrence weights, and support twists. Each local map:

• gives the two normalized occurrence endpoint values ((+1,+1));
• kills the marked lower-Cousin interval boundary;
• uses inverses only inside the indicated Cech localization summands; and
• retains the positive physical normal line separately.

This is a local derived-correspondence theorem. Independence of the labelled retraction and compatibility among the three representatives still require a global Cousin coherence.

Strict finite-free lift no-go

There is no degree-zero (R_0)-linear chain map

[ K(I_+)\longrightarrow K(I_i) ]

whose degree-zero multiplier specializes to the carrier unit. For an unshared branch normal (u_a), the degree-one chain equation would force

[ a u_a\in I_i. ]

Modulo (I_i), (u_a) is a non-zero-divisor, so (\bar a=0). This contradicts the required identity augmentation (a(1,\ldots,1)=1). The same argument holds on (F_{03},F_{25},F_{14}).

Equivalently, the checked Hom complex has

[ \operatorname{Ext}^0=\operatorname{Ext}^1=0, \qquad \operatorname{Ext}^2\cong\operatorname{Ext}^3 \cong R_0/(I_++I_i). ]

The no-go is therefore exactly for a strict ordinary stalk map. It is positive evidence for the derived excess correspondence, not evidence that a global kernel cannot exist.

Evidence

Exact certificates:

• research/voevodsky/check_one_normal_can_var_cousin.rs
• research/voevodsky/check_unlocalized_plus_recollement_obstruction.rs

SHA-256:

f2342969a0623742846fa538ef32e307d5efbf479b232389c307e9800144f401
e0cfc26031c78ae2c9050ac96cbc672a0de59f11b547e683d6d93aa56d57d448

Reproduce with:

$sources = @(
  "research/voevodsky/check_one_normal_can_var_cousin.rs",
  "research/voevodsky/check_unlocalized_plus_recollement_obstruction.rs"
)
foreach ($src in $sources) {
  $exe = Join-Path $env:TEMP ((Split-Path $src -LeafBase) + ".exe")
  rustfmt --edition 2021 --check $src
  rustc --edition=2021 -D warnings -O $src -o $exe
  & $exe | ConvertFrom-Json | Out-Null
}

Inherited inputs are entries 38 and 93–99.

Boundary

This entry does not prove a global map

[ A_+^{\rm Cous,PC}: \mathcal D_+^{\rm Cous,reg,\vee} \longrightarrow \mathbb D\operatorname{PC}(K_6,B_{\rm short})\otimes\chi_N. ]

In particular:

1. the three local Cech targets are not yet glued through the lower (q)-vertices and augmentation;
2. Cech localization terms are not finite free over (R_0);
3. the labelled local excess retractions have not been proved independent of representative;
4. no (D_3)-equivariant homotopy coherence has been constructed; and
5. absence of that coherence in the present audit is not a nonexistence theorem.

Calling the localization arrow itself can is also invalid:

[ \operatorname{Hom}_{R_0}(R_0[u^{-1}],R_0)=0, ]

so no reverse var can make both composites (u). Can–var belongs to the finite perfect stage; the Cech comparison realizes its supported Cousin class.

Consequence and next formula

The global objective should be formulated as a bivariant kernel on the marked flag correspondence, not as a strict map of finite stalk complexes. The first unresolved coherence occurs at the lower source vertex (q_2), shared by the (F_{03}) and (F_{25}) incidence terms. After passing both local traces to their common Cech refinement, construct or falsify

[ \boxed{ \rho_{q_2}^{03}\Theta_{03}^{\rm loc} -\rho_{q_2}^{25}\Theta_{25}^{\rm loc} =d_{\rm Cous}H_{q_2}+H_{q_2}d. } ]

The occurrence pullbacks, reciprocal/Borel–Moore twist, and all localization summands must be fixed before solving for (H_{q_2}). If this homotopy exists, rotate it to the other two lower vertices and test the remaining top coherence. If it does not, its nonzero class is the first intrinsic obstruction to the global half-object lift.

Outcome contract

{
  "claim": "The support-directed one-normal can-var packet and its Koszul-to-Cech realization canonically produce the twist-aware repeated-normal excess line and three labelled local derived road traces; no strict degree-zero finite-free Koszul lift can carry the unit road coefficient.",
  "status": "proved",
  "assumptions": [
    "The original BM and reciprocal regular support directions use their paired, not identical, can-var conventions.",
    "C_Q is the support Cech complex, with inverses only in its localization summands.",
    "The local trace retains the marked occurrence path, twist reversal, and physical normal line."
  ],
  "evidence_refs": [
    "research/voevodsky/check_one_normal_can_var_cousin.rs",
    "research/voevodsky/check_unlocalized_plus_recollement_obstruction.rs",
    "ledger entries 38 and 93-99"
  ],
  "factorization_test": {
    "one_normal_can_var": "passed",
    "twist_pairing": "passed and perfect",
    "mixed_repeated_normal": "passed integrally",
    "three_local_Cech_traces": "passed",
    "strict_finite_free_lift": "falsified on all three roads",
    "global_D3_Cousin_coherence": "untyped, not disproved"
  },
  "counterevidence": [
    "The local-cohomology realization is not a bounded finite-free R0 complex.",
    "The three labelled representatives are not yet connected through lower-vertex homotopies.",
    "Globally inverting u would erase the supported excess class."
  ],
  "next_experiment": "Compare the F03 and F25 labelled Cech residues at their shared q2 vertex and solve the first support- and twist-compatible chain homotopy."
}