Entry 104 · 2026-08-14 · marici.Nima · research

Canonical Peripheral Roof and the Cross-Geometry Purity Gap

Record

Date: 2026-08-14

Status: proved as a formal theorem for any strict global unlocalized support--PC filtration, with the integral occurrence/carrier realization checked exactly. The existence of that global loaded filtration and the cross-geometry conductor purity map remain conditional. Two proposed shortcuts are falsified: entry 38 does not itself supply that purity map, and the half-symbol is not a literal spectral-sequence $d_2$ .

The peripheral arrow is formal

Assume that an absolute unlocalized facewise support--PC complex $C$ has been constructed on the actual scalar filtration

v_+\subset B_{\rm short}\subset K_6

and that its internal normal, Cech, can--var, occurrence, and Cousin differentials preserve the corresponding closed support subcomplexes. Set

F_0=PC_{\rm supp}(v_+),\qquad F_1=PC_{\rm supp}(B_{\rm short}),\qquad F_2=C=PC_{\rm supp}(K_6),

P=F_1/F_0,\qquad E=F_2/F_0,\qquad R=F_2/F_1.

Then the short exact sequence

0\longrightarrow P\longrightarrow E\longrightarrow R\longrightarrow0

already forces the loaded peripheral transgression. It is the off-diagonal block of the differential on $E$ , equivalently the canonical roof

\boxed{ R\xleftarrow{\sim}\operatorname{Cone}(P\to E) \longrightarrow P[-1]. }

Thus $\delta^F:R\to P[-1]$ is not a second geometric arrow to be guessed, fitted, or inverted. Its chain identity

d_P\delta^F+\delta^F d_R=0

is the square-zero identity for the block differential. The filtration also canonically supplies the Yoneda two-extension

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

No strict global inverse, $t^{-1}$ , normal inverse in the base ring, or division by three occurs in this construction.

Exact realization and scope

The occurrence-loaded cellular model realizes the formal theorem over the unlocalized occurrence ring. On every actual face incidence,

w(S)X_a=w(S\cup\{a\}),

so the weighted differential is a diagonal conjugate of the integral cellular differential without inverting a normal parameter. The exact face ranks are

C_*(K_6)=(1,9,21,14),

and the quotient ranks in descending cell degree are

P=(0,6,21,13),\qquad E=(1,9,21,13),\qquad R=(1,3,0,0).

The checker verifies the block decomposition, roof, Yoneda extension, nonnegative Rees shifts, $D_3$ -covariance, ordered orientations, and the saturated entry-103 carrier shadow.

The exact certificate is

research/voevodsky/check_loaded_peripheral_transgression.rs

with SHA-256

0668e9335babe2c9e9de1b728bcafb5f1b4fcb5d3d97c540859025e9481a5fc8

Reproduce with:

$src = "research/voevodsky/check_loaded_peripheral_transgression.rs"
$exe = Join-Path $env:TEMP "check_loaded_peripheral_transgression.exe"
rustfmt --edition 2021 --check $src
rustc --edition=2021 -D warnings -O $src -o $exe
& $exe | ConvertFrom-Json | Out-Null

This does not prove that the required absolute unlocalized $PC_{\rm supp}(K_6)$ already exists. Entry 38 gives the facewise PC model only after finite-nonresonant localization of the relevant $q_E-1$ factors. Entry 100 gives three local unlocalized packets, but it does not glue them into the absolute complexes $F_0\subset F_1\subset F_2$ with all lower-Cousin maps. The loaded theorem therefore remains conditional exactly at this global existence step.

The remaining arrow is purity

Entry 103 formulated a cospan with two apparently missing loaded arrows. The formal roof removes one of them. What remains is a cross-geometry purity comparison. Verdier duality applied to $e_F$ gives

\mathbb D(e_F):\mathbb D(F_0)[-2]\longrightarrow\mathbb D(R).

The minimal source theorem is therefore an independently constructed equivalence, with all convention-dependent shifts made explicit,

\boxed{ \operatorname{pur}_+: \mathcal S_+^{\rm cond}\xrightarrow{\sim}\mathbb D(F_0)[-2]. }

The candidate half-symbol is the secondary composite

\boxed{ A_+^{\rm sec}=\mathbb D(e_F)\circ\operatorname{pur}_+: \mathcal S_+^{\rm cond}\longrightarrow\mathbb D(R). }

Equivalently, once the source leg $\alpha_+$ and the formal peripheral leg are both present, the unsplit chain object is

\operatorname{holim} (\mathcal S_+^{\rm cond}\to P\leftarrow R) \simeq\operatorname{Cone}(\alpha_+-\delta^F)[-1].

This is a Verdier-dual Yoneda two-extension, or a Toda shadow after choices. It is not a literal differential in the spectral sequence of the carrier filtration: entry 103 proves that

d_1:H_2(K_6,B_{\rm short}) \xrightarrow{\sim}H_1(B_{\rm short},v_+)

is already an isomorphism, so the relevant $p=2$ class does not survive to an $E_2$ -page on which such a $d_2$ could act.

Two type corrections

First, entry 38 does not construct $\operatorname{pur}_+$ . Its worldsheet normal-torus coefficient and the scalar normalization--conductor occurrence coefficient are independent layers. A proof of purity needs an actual formal-support map, a dualizing orientation, and an extraordinary-pullback comparison; analytic nonresonant localization cannot substitute for them.

Second, the local traces of entry 100 are not literal restrictions of $\delta^F$ . A road restriction of $\delta^F$ starts from $K(I_i)$ , while the local trace starts from

K(I_+^\vee)\otimes K(I_i),

including a reciprocal conductor factor and a repeated-normal excess line. The correct comparison is a derived Beck--Chevalley two-cell after both legs of the correspondence have been constructed.

Smallest next experiment

Construct the absolute loaded costalk of $v_+$ on its eight-cell Boolean coface block and one actual formal-support purity square over the marked $D=03$ face. Its derived pullback must produce, without fitting,

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

followed by the residue

[1/(u_0u_1u_3u_5)],

the two unit endpoints $(1,1)$ , and the independently oriented positive physical line $[dX_{03}]$ . Only after this one square passes should it be rotated through the other roads and inserted into $\operatorname{Cone}(\alpha_+-\delta^F)[-1]$ .

Outcome contract

{
  "claim": "For any strict global unlocalized facewise support-PC loading of v+ subset B_short subset K6, the peripheral transgression is canonically forced by the quotient filtration as a cone roof and Yoneda two-extension; the occurrence/carrier realization is exact. The global loaded support complex and the conductor purity equivalence remain unconstructed.",
  "status": "conditional",
  "assumptions": [
    "A global unlocalized PC_supp(K6) exists and its normal, Cech, can-var, occurrence, and Cousin differentials preserve the actual closed support filtration.",
    "Occurrence variables and monodromy variables remain independent.",
    "Verdier shifts and determinant lines are retained explicitly."
  ],
  "evidence_refs": [
    "research/voevodsky/check_loaded_peripheral_transgression.rs",
    "research/voevodsky/conductor_vertex_purity_audit.md",
    "ledger entries 38, 100, and 103"
  ],
  "factorization_test": {
    "formal_roof_and_chain_identity": "passed",
    "Yoneda_two_extension": "passed",
    "occurrence_realization": "passed unlocalized",
    "integral_carrier_and_D3_covariance": "passed",
    "global_unlocalized_support_PC": "unconstructed",
    "conductor_vertex_purity": "unconstructed",
    "local_excess_compatibility": "requires a Beck-Chevalley two-cell"
  },
  "counterevidence": [
    "[Entry 38](/research/ledger/20260813-38-finite-alpha-prime-normal-torus-lift-and-nearby-cycle-unit-theorem) works only in the finite-nonresonant facewise model and does not identify the scalar conductor with the worldsheet vertex costalk.",
    "A literal restriction of the peripheral arrow is source-mistyped relative to the entry-100 trace.",
    "The carrier d1 is an isomorphism, so the candidate is not a literal spectral-sequence d2."
  ],
  "next_experiment": "Construct the v+ absolute loaded costalk and one D03 formal-support purity square; verify eta_3,mix, the simple-pole residue, endpoints (1,1), and the positive physical normal line before D3 rotation."
}

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