Absolute Support Complex, Shift-Corrected Purity, and the Marked-Correspondence Obstruction

Record

Date: 2026-08-14

Status: one scoped theorem and one sharp blocker. The absolute unlocalized original-twist/Borel–Moore support complex exists integrally, and the local positive-conductor purity shift is forced after fixed-nonzero-(\beta), characteristic-zero Koba–Nielsen completion. The literal (D03) pullback of the resulting filtration class is exactly zero, whereas the established Cousin trace is nonzero. Consequently the full chain map (G_{03}^{\rm Cousin}) remains unconstructed at one precisely typed marked ringed-support correspondence.

The absolute unlocalized complex

Let (K_6) be the labelled hexagon associahedron. A generator is a pair ([S,H]), where (S) is a noncrossing dissection and (H\subseteq S) records the normal-circle directions retained at its boundary face. Work over the tensor product of the polynomial occurrence ring (\mathbb Z[X_D]) with the universal monodromy ring, writing (u_D=q_D-1). Occurrence and monodromy variables are independent.

With (\epsilon(S,a)) the cellular incidence sign and (\operatorname{pos}_H(h)) the ordered-normal sign, define

[ \boxed{ d[S,H]

\sum_{a\ { m addable}} \epsilon(S,a)X_a[S\cup{a},H] +(-1)^{3-|S|} \sum_{h\in H} (-1)^{\operatorname{pos}_H(h)}u_h[S,H\setminus{h}]. } ]

The radial term is the original-twist costandard map (\operatorname{var}=1), with the independent occurrence coefficient (X_a). The normal term is (\operatorname{can}=u_h). The two terms anticommute, so (d^2=0) without inverting (X_D), (u_D), a Rees parameter, or an integer.

The exact census is

[ 215\ \text{generators}, \qquad (\operatorname{rk}C_0,\operatorname{rk}C_1, \operatorname{rk}C_2,\operatorname{rk}C_3) =(14,63,93,45). ]

The actual closed supports give strict (D_3)-stable subcomplexes

[ \boxed{ F_0=PC_{\rm supp}(v_+) \subset F_1=PC_{\rm supp}(B_{\rm short}) \subset F_2=PC_{\rm supp}(K_6), } ]

of total ranks (8\subset208\subset215). Thus the hypothesis left open in entry 104 is now realized. Its canonical cone roof and Yoneda class

[ e_F= [0\to F_0\to F_1\to F_2/F_0\to F_2/F_1\to0] \in\operatorname{Ext}^2(F_2/F_1,F_0) ]

are honest unlocalized objects. After adjoining the relevant (u_D^{-1}), the normal contraction (p_D\mapsto u_D^{-1}h_D) recovers the facewise nonresonant packet of entry 38.

The local purity shift is forced

The costalk (F_0) is exactly one eight-generator original three-normal packet

[ F_0=K(u_1,u_3,u_5). ]

It must not be tensored with a second Boolean carrier. For one normal, the standard/costandard packets pair perfectly by

[ K(u)\otimes K(u^\vee)\longrightarrow R[1], \qquad \beta(p,\ell^\vee)=1, \qquad \beta(\ell,p^\vee)=-q, ]

where (u^\vee=q^{-1}-1=-q^{-1}u). Hence the ordered triple gives

[ K(I_+^\vee)\simeq\mathbb D(F_0)[3], \qquad I_+^\vee=(u_1^\vee,u_3^\vee,u_5^\vee). ]

Two independent geometric placements fix the remaining shift:

  1. codimension-three Thom duality contributes ([-3]);
  2. (J_+/J_+^2) is annihilated by (J_+), so the first normal symbol is supported on (\widetilde Z_+), the terminal degree-two term of the normalization–Čech total, and contributes ([-2]).

Therefore, in the stated homological convention,

[ \boxed{ \mathcal S_{+,\mathrm{loc}}^{\mathrm{cond}} =K(I_+^\vee)[-5] \xrightarrow{\sim} \mathbb D(F_0)[-2]. } ]

The cross-geometry comparison uses

[ q_j=\exp(\beta x_j), \qquad u_j=\beta x_jv_j(x_j), \qquad v_j(0)=1, ]

at fixed nonzero (\beta) in a characteristic-zero completed coefficient ring. This is one base change of one Kummer packet; the occurrence variable (x_j) remains a coefficient. It is not a universal integral purity theorem.

The tempting double-loaded source is ruled out homologically. The full entry-99 augmented carrier has a saturated integral contraction, so its tensor product with the termwise-free reciprocal packet is acyclic. In contrast,

[ H_0(F_0)=R/(u_1,u_3,u_5)\ne0. ]

The mismatch is structural, not merely the corroborating count (64\ne8).

The exact (D03) blocker

The absolute complex makes the next comparison decidable. In the literal (D03) road packet, the off-diagonal boundary lands in a strict (D03)-supported subcomplex (G_{03}\subset F_1) which is disjoint from (F_0). The first connecting morphism therefore lifts through (G_{03}), and its Yoneda product with (0\to F_0\to F_1\to F_1/F_0\to0) vanishes:

[ \boxed{ \operatorname{pb}^{\mathrm{lit}}_{03}(e_F)=0. } ]

But entry 100’s independently constructed local Cousin trace obeys

[ \boxed{ \Theta_{03}^{\mathrm{loc}}(\eta_{3,\mathrm{mix}})

\left[\frac{1}{u_0u_1u_3u_5}\right]\ne0. } ]

Thus the local trace is not a literal road restriction of the filtration class. This falsifies that shortcut, not a derived factorization correspondence.

The first missing arrow is now exact. The marked road endpoint ({D03,x_1,x_3}) and the central conductor vertex (v_+={x_1,x_3,x_5}) are related by removing (D03); they are not joined by a face inclusion in the absolute complex. One must construct a marked, ringed support correspondence across this central flip. Neither the support filtration nor the one-normal perfect pairing supplies it automatically.

Consequence for the formula objective

The absolute scalar differential and the local source placement are now fixed. The remaining formula is not a new differential but a correspondence component

[ \boxed{ \Gamma_{+;03}^{\mathrm{mark}}: \mathcal S_{+,\mathrm{loc}}^{\mathrm{cond}} \dashrightarrow \mathbb D(F_2/F_1) } ]

whose composition with the canonical Yoneda class must induce

[ \eta_{3,\mathrm{mix}} \longmapsto [1/(u_0u_1u_3u_5)], ]

the two endpoint values ((1,1)), and the separate positive physical normal line ([dX_{03}]). It must arise from the marked normalization–conductor and central-flip geometry, not be defined by these required values.

Only after this one correspondence and its Beck–Chevalley two-cell exist can one assemble the full

[ G_{03}^{\rm Cousin}: (\mathcal S_F^{\rm sp},d_{\rm sp,sc}) \longrightarrow (\mathcal R_{03}^{\rm circ,PC},d_{\rm circ}^{\rm PC}) ]

and test factorization naturality. The north-star objective remains open, but its first unconstructed datum is now singular and geometric.

Evidence

Exact certificates:

  • research/voevodsky/check_absolute_unlocalized_support_pc.rs, SHA-256 55234ea577d528838bf91d7a641947ee79c51f24fa48c0915221a8e168cb9d2d;
  • research/voevodsky/check_d03_formal_support_purity.rs, SHA-256 f3a585c5a2091f2b500b9c76f878be83e4e48f676f648acde64ab828ed2be0d1.

The entry-104 certificate remains unchanged at 0668e9335babe2c9e9de1b728bcafb5f1b4fcb5d3d97c540859025e9481a5fc8.

Reproduce with:

$sources = @(
  "research/voevodsky/check_absolute_unlocalized_support_pc.rs",
  "research/voevodsky/check_d03_formal_support_purity.rs"
)
foreach ($src in $sources) {
  rustfmt --edition 2021 --check $src
  $exe = Join-Path $env:TEMP ((Split-Path $src -LeafBase) + ".exe")
  rustc --edition=2021 -D warnings -O $src -o $exe
  & $exe | ConvertFrom-Json | Out-Null
}

Outcome contract

{
  "claim": "The n=6 scalar target has a canonical 215-generator integral unlocalized original-twist/Borel--Moore support-PC complex with strict F0 subset F1 subset F2. At fixed nonzero beta in characteristic-zero Koba--Nielsen completion, its positive-conductor source has the forced placement K(I_plus^vee)[-5] equivalent to D(F0)[-2]. The literal D03 pullback of the filtration Yoneda class is zero and therefore cannot equal the nonzero entry-100 Cousin trace.",
  "status": "proved",
  "assumptions": [
    "The positive real chamber fixes the radial basepoint on every oriented normal circle.",
    "Occurrence coefficients and universal monodromy variables are independent.",
    "The cross-geometry purity comparison is restricted to fixed nonzero beta in a characteristic-zero completed coefficient ring.",
    "Complexes use the homological shift convention stated above."
  ],
  "evidence_refs": [
    "research/voevodsky/check_absolute_unlocalized_support_pc.rs",
    "research/voevodsky/check_d03_formal_support_purity.rs",
    "ledger entries 93, 99, 100, and 104"
  ],
  "factorization_test": {
    "absolute_unlocalized_d_squared": "passed on 215 generators",
    "strict_support_filtration": "passed with ranks 8 subset 208 subset 215",
    "D3_covariance": "passed",
    "local_purity_shift": "passed as [-3] Thom plus [-2] terminal conductor placement",
    "double_loading": "falsified by homology",
    "literal_D03_Yoneda_pullback": "zero",
    "entry100_local_trace": "nonzero",
    "full_marked_Beck_Chevalley_map": "unconstructed"
  },
  "counterevidence": [
    "The local purity statement is not a universal integral cross-geometry theorem.",
    "The literal road pullback cannot produce the required Cousin trace.",
    "No marked ringed support correspondence across the central flip has yet been constructed."
  ],
  "next_experiment": "Construct one marked D03 ringed support correspondence across {D03,x1,x3} to v_plus={x1,x3,x5}; test its canonical Beck--Chevalley composite against eta_mix, the four-normal residue, endpoints (1,1), and the separate [dX03] line."
}