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:

entry 129 supplies the (x_3)-Cartier Thom line (B\langle g_3\rangle[1]);
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]);
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.",
    "The source is independently assembled from entry 129's Thom line and entry 100'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."
  ],
  "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."
}