Entry 131 · 2026-08-14 · marici.Nima · research

D03 Cartier Edge Purity and the Scoped PC Promotion

Record

Date: 2026-08-14

Status: proved in the definitionally scoped absolute/unlocalized D03 road-face model. The coefficient/Cousin source left unnamed in entry 130 has an independent construction, finite Cartier purity identifies it with the actual edge costalk, and the Bockstein-compatible normalized identification is unique.

Scope: the $x_3$ normal of the entry-105 absolute support complex, with the entry-100 reciprocal-standard/original-Borel--Moore pairing and the entry-129 occurrence Cartier map frozen. This does not construct the scalar specialization differential, $G_{03}^{\rm Cousin}$, a nonzero generic $Q$-leg, or an unrestricted enlargement of the road-face PC complex.

The actual target packet

Factor out the common physical line and all normal packets unrelated to $x_3$. The closed-star quotient of entry 105 is

$$P_3= \left[ A\langle g_3,h_3\rangle \xrightarrow{(x_3,u_3)} A\langle p_3\rangle \right].$$

Here $g_3$ is radial/occurrence and $h_3$ is the original normal-circle generator. No occurrence or monodromy variable is inverted. Its complete primitive syzygy is forced:

$$z_3=u_3g_3-x_3h_3.$$

Let $B=A/(x_3)$. Since $x_3$ is Cartier and $P_3$ is perfect,

$$\boxed{ i_{x_3}^{!}P_3 =R\!\operatorname{Hom}_A(B,P_3) \simeq \left(B\otimes_A P_3\right) \otimes\operatorname{or}(x_3)[-1]. }$$

Thus, up to the displayed coorientation shift, the target costalk is

$$E_{3,\rm tar}= \left[ B\langle g_3,h_3\rangle \xrightarrow{(0,u_3)} B\langle p_3\rangle \right].$$

It contains one free radial line and one $B/(u_3)$ normal line. There is no integer torsion. In particular, rank one is not automatic.

Independent construction of the source

The same packet is obtained without copying the target:

1. entry 129 supplies the $x_3$-Cartier Thom line $B\langle g_3\rangle[1]$;
2. entry 100 supplies the original/Borel--Moore can--var packet (K_B(u_3)=[B\langle h_3\rangle\xrightarrow{u_3} B\langle p_3\rangle]);
3. their support-typed direct sum is

$$E_{3,\rm src} = B\langle g_3\rangle[1]\oplus K_B(u_3) = \left[ B\langle g_3,h_3\rangle \xrightarrow{(0,u_3)} B\langle p_3\rangle \right].$$

The reciprocal-regular factor is not discarded. The strict filtered quotient/excess decomposition of $K(u_3^\vee)\otimes P_3$, followed by entry 97's perfect reciprocal/original pairing, curries the bivariant packet to $E_{3,\rm src}$. The Laurent unit in

$$u_3^\vee=-q_3^{-1}u_3$$

is absorbed by that pairing; it does not identify the two support variances.

This is the missing definition of entry 130's $\mathcal C_{x_3\text{-edge}}^{\rm coeff/Cousin}$. It is derived from the occurrence Thom and can--var data, not from the desired endpoint fractions.

The graph Bockstein selects one map

On the derived graph

$$u_3=t_3x_3,$$

the first Cartier Bockstein is forced:

$$\beta_{x_3}(g_3)=p_3, \qquad \beta_{x_3}(h_3)=t_3p_3.$$

Its primitive kernel is

$$t_3g_3-h_3=z_3/x_3.$$

Before Bockstein compatibility, a filtration- and support-preserving degree-zero chain endomorphism of the edge packet has

$$f_1= \begin{pmatrix} a&b\\ 0&e \end{pmatrix}, \qquad f_0(p_3)=ep_3.$$

The ambient homotopy classes contain one free scalar and two $B/(u_3)$-ambiguities. Hence rank one is not inherited from interval contractibility.

The equation

$$\beta_{x_3}f=f\beta_{x_3}$$

forces

$$a=e, \qquad b=0.$$

Therefore the admissible map module is one torsion-free scalar line. Positive Cartier coorientation, endpoint normalization, and $[dX_{03}]=+1$ fix its generator to $e=1$.

The scoped purity theorem

The independently constructed source and the actual target are canonically identified:

$$\boxed{ \operatorname{pur}_{x_3,\partial}^{\rm PC}: E_{3,\rm src}\otimes\operatorname{or}(x_3)[-1] \xrightarrow{\sim} i_{x_3}^{!}P_3. }$$

This is finite Cartier purity, and it is the unique positively normalized Bockstein-compatible map in the frozen filtration/support type.

Endpoint transitivity is the standard adjunction

$$R\!\operatorname{Hom}_B \left(B/(x_i),R\!\operatorname{Hom}_A(B,P_3)\right) \simeq R\!\operatorname{Hom}_A\left(A/(x_3,x_i),P_3\right), \qquad i=0,1.$$

After the complete occurrence and normal Koszul--Cech comparisons, every lower term is retained and the two endpoint restrictions are

$$+\left[ \frac1{x_0x_3u_0u_1u_3u_5} \right]\otimes[dX_{03}], \qquad +\left[ \frac1{x_1x_3u_0u_1u_3u_5} \right]\otimes[dX_{03}].$$

Thus entry 130's $\Gamma_0,\Gamma_1$ are actual morphisms in the definitionally scoped road-face PC model once its source is retyped as $E_{3,\rm src}$. They are two restrictions of one edge purity arrow, not two fitted maps.

Sharp remaining blocker

The local target-side PC promotion is closed only for the established road-face complex. It does not produce the scalar-side specialization correspondence. The first remaining formula objective is

$$\boxed{ d_{\rm circ}^{\rm PC}G_{03}^{\rm Cousin} =G_{03}^{\rm Cousin}d_{\rm sp,sc}, \qquad \operatorname{gr}(G_{03}^{\rm Cousin})=K_{\rm alt}, \qquad \operatorname{Res}_{x_3}G_{03}^{\rm Cousin} =\operatorname{pur}_{x_3,\partial}^{\rm PC}, \qquad q_Q(G_{03}^{\rm Cousin})\ne0. }$$

The final condition is decisive. Any construction supported wholly in $F_1$ has zero ordinary $Q=F_2/F_1$ leg and cannot solve this problem. Nothing in the Cartier theorem manufactures that generic component.

Admissible ablations

• Without the graph Bockstein, two $B/(u_3)$ ambiguities remain.
• Deleting the radial/Tor-zero line makes unit normalization require a forbidden $t_3^{-1}$.
• Deleting the normal/Tor-one line loses $\eta_{3,\mathrm{mix}}$ and the primitive excess trace.
• Dropping lower Cech terms breaks the endpoint chain equations.
• Globally inverting $x_3$ or $u_3$ kills the supported costalk.
• Defining the source from the desired endpoint fractions is circular; the Thom-plus-can--var construction is required.

Evidence

The proof is the explicit packet, finite Cartier duality, and two-by-two Bockstein calculation above. Its independently established inputs are:

• research/voevodsky/check_absolute_unlocalized_support_pc.rs (entry 105);
• research/voevodsky/check_one_normal_can_var_cousin.rs (entry 100);
• research/voevodsky/check_d03_bivariant_pc_hom.rs (entry 97);
• research/voevodsky/check_d03_toric_cox_cousin_trace.rs (entry 129);
• research/voevodsky/check_d03_x3_loaded_pc_endpoint_boundary.rs (entry 130).

No new checker is required: the new assertion is standard Cartier duality for the already enumerated entry-105 packet plus the displayed admissibility calculation.

Outcome contract

{
  "claim": "In the definitionally scoped absolute D03 road-face model, the independently assembled x3 Thom-plus-BM source is canonically the finite Cartier costalk of P3, and its unique positively normalized Bockstein-compatible purity map has the two entry-130 endpoint restrictions.",
  "status": "proved_scoped_pc_edge_purity__global_scalar_specialization_open",
  "assumptions": [
    "P3 is the actual closed-star packet extracted from [entry 105](/research/ledger/20260814-105-absolute-support-complex-shift-corrected-purity-and-the-marked-correspondence-obstruction).",
    "The source is independently assembled from [entry 129](/research/ledger/20260814-129-cox-principal-line-trace-and-the-extraordinary-cousin-boundary)'s Thom line and [entry 100](/research/ledger/20260814-100-support-directed-can-var-packet-and-three-local-cousin-traces)'s original/BM packet.",
    "The graph Bockstein u3=t3*x3 is part of the admissible filtered structure.",
    "Reciprocal-standard and original/BM variances remain distinct until currying through [entry 97](/research/ledger/20260814-97-reciprocal-twist-d03-bivariant-road-trace)."
  ],
  "evidence_refs": [
    "ledger entries 97, 100, 105, 129, and 130",
    "the explicit Cartier and two-by-two Bockstein calculation in this entry"
  ],
  "factorization_test": {
    "ambient_costalk": "one free line plus one B/(u3) line; no integer torsion",
    "unconstrained_endomorphisms": "larger than rank one",
    "Bockstein_compatible_maps": "one torsion-free scalar line",
    "positive_normalization": "unique unit",
    "v00_endpoint": "entry-130 positive residue",
    "v10_endpoint": "entry-130 positive residue",
    "generic_Q_leg": "not constructed"
  },
  "counterevidence": [
    "The theorem is scoped to the established road-face PC model.",
    "A correspondence supported wholly in F1 has zero ordinary Q leg.",
    "No scalar total-specialization differential or G03 chain map is constructed."
  ],
  "next_experiment": "Construct d_sp,sc and G03^Cousin with nonzero generic Q leg, and require its x3 residue to be the canonical purity map proved here."
}

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