Cox Principal-Line Trace and the Extraordinary Cousin Boundary

Record

Date: 2026-08-14

Status: proved integral occurrence-level theorem; falsified the bare coherent-Cousin interpretation; loaded PC promotion and the scalar-source lift remain open.

Scope: the occurrence variables of the fixed (D=03) road square. Monodromy/Kummer packets, reciprocal-standard versus original-Borel–Moore variance, the repeated-normal excess map, and the physical line ([dX_{03}]) are not reconstructed here.

Setup

Put

[ R=\mathbb Z[x_0,x_1,x_3,x_4], \qquad U=\operatorname{Spec}R\setminus \bigl(V(x_0,x_1)\cup V(x_3,x_4)\bigr). ]

The fixed weighted road square of entry 97 is

[ R\langle F\rangle\xrightarrow{d_2} R\langle a,b,c,d\rangle\xrightarrow{d_1} R\langle v_{00},v_{10},v_{01},v_{11}\rangle, ]

with

[ d_2F=x_3a-x_4b-x_0c+x_1d ]

and

[ \begin{aligned} d_1a&=-x_0v_{00}+x_1v_{10},& d_1b&=-x_0v_{01}+x_1v_{11},\ d_1c&=-x_3v_{00}+x_4v_{01},& d_1d&=-x_3v_{10}+x_4v_{11}. \end{aligned} ]

Define the principal-lcm generators

[ \bar a=x_3a, \quad \bar b=x_4b, \quad \bar c=x_0c, \quad \bar d=x_1d, ]

[ \bar v_{00}=x_0x_3v_{00}, \quad \bar v_{10}=x_1x_3v_{10}, \quad \bar v_{01}=x_0x_4v_{01}, \quad \bar v_{11}=x_1x_4v_{11}. ]

In these generators the differential is ordinary oriented square incidence:

[ F\longmapsto \bar a-\bar b-\bar c+\bar d, ]

[ \begin{aligned} \bar a&\longmapsto-\bar v_{00}+\bar v_{10},& \bar b&\longmapsto-\bar v_{01}+\bar v_{11},\ \bar c&\longmapsto-\bar v_{00}+\bar v_{01},& \bar d&\longmapsto-\bar v_{10}+\bar v_{11}. \end{aligned} ]

Let

[ M=x_0x_1x_3x_4. ]

The weighted augmentation

[ (v_{00},v_{10},v_{01},v_{11}) \longmapsto (x_1x_4,x_0x_4,x_1x_3,x_0x_3) ]

sends every (\bar v_{ij}) to (M).

Theorem: the formal road trace is one principal-line functional

The principal-lcm subcomplex is an integral cellular resolution of the principal Cartier ideal

[ (M)=\mathcal O_U(-D), \qquad D=D_0+D_1+D_3+D_4. ]

Its normalized differential ranks are (1) and (3); every nonzero Smith factor is a unit. Consequently it is saturated and has no integral torsion. Duality gives the single coefficient line

[ \mathcal O_U(D)=\operatorname{Hom}_{\mathcal O_U} (\mathcal O_U(-D),\mathcal O_U), ]

not four independently chosen Laurent coefficients.

The distinguished functional (M\mapsto1) has, in the original road bases, the four representatives

[ \boxed{ \Theta_{03}^{\rm occ,formal}

\left( \frac1{x_0x_3}, \frac1{x_1x_3}, \frac1{x_0x_4}, \frac1{x_1x_4} \right). } ]

Thus the entry-97 occurrence coefficients are the four local expressions of one normalized dual principal-line functional. Once its dense-torus value is fixed to one, Laurent injectivity makes the solution unique.

The generic normalization is nevertheless independent information. The (v_{10}) residue alone permits

[ f\in1+(x_1,x_3), ]

and even all four corner residues permit

[ f\in1+(x_0x_1,x_3x_4). ]

This is the generic-unit obstruction of the D03 ledger entry Four-Corner Cellular Nerve and the Generic-Unit Obstruction in its smallest coefficient form.

The occurrence-level (x_3) Gysin

The first extraordinary step can be constructed without fitting a fraction. On the (x_3) edge let

[ S_3=(R/(x_3))|_U. ]

For (i=0,1), the endpoint Cartier divisor is cut out by the non-zero-divisor (x_i), and

[ R!\operatorname{Hom}_{S_3}(S_3/(x_i),S_3) \simeq[S_3\xrightarrow{x_i}S_3]. ]

Hence

[ \operatorname{Ext}^0_{S_3}(S_3/(x_i),S_3)=0, \qquad \operatorname{Ext}^1_{S_3}(S_3/(x_i),S_3)=S_3/(x_i), ]

with one primitive orientation generator and no integer torsion.

The canonical Koszul–Cech comparison for the ordered normals ((x_i,x_3)) is

[ 1\longmapsto1, \qquad e_i\longmapsto(1/x_i,0), \qquad e_3\longmapsto(0,1/x_3), \qquad e_i\wedge e_3\longmapsto1/(x_ix_3). ]

Equivalently, the degree-one extraordinary map from the (x_3) Cech object to the product Cech object is

[ g_i^0(r)=(r/x_i,0), \qquad g_i^1(t)=t/x_i. ]

The chain equation is exact. With cellular incidence ((-v_{00},+v_{10})) and the retained endpoint orientation lines, the displayed corner coefficients are

[ +\left[\frac1{x_0x_3}\right], \qquad +\left[\frac1{x_1x_3}\right]. ]

This proves the occurrence Koszul–Cech Gysin on one road edge. It does not yet identify that map with the loaded PC Gysin.

Sharp blocker: coherent restriction is the wrong variance

The tempting simplification

[ \text{generic regular Cox section} \xrightarrow{\text{ordinary Cousin boundary}} \text{corner simple pole} ]

is false. In (A[x^{-1}]/A), the regular unit represents

[ [1]=0. ]

The same holds for a section regular in the invertible sheaf (\mathcal O_U(D)). The nonzero class ([1/x_i]) is an extraordinary Cartier fundamental class, not an ordinary restriction of the generic line.

The occurrence calculation therefore does not supply any of the following:

• the reciprocal-standard/original-Borel–Moore comparison;
• the repeated-normal excess trace and its (q)-units;
• the physical orientation ([dX_{03}]=+1);
• physical-Cut/Beck–Chevalley naturality;
• a nonzero scalar-specialization (Q)-leg; or
• the source differential (d_{\rm sp,sc}) and chain map (G_{03}^{\rm Cousin}).

The first unconstructed target arrow is the promotion

[ \boxed{ g_{3}^{!,\rm occ} \longrightarrow g_{3}^{!,\rm PC}, } ]

including both (v_{00}) and (v_{10}), all lower Cech terms, the independently established normal-excess packet, and ([dX_{03}]). Only after this one-edge map is typed should the (x_4) edge or the complete four-edge coherence be attempted.

The first missing source arrow remains the independently normalized map into the generic road term with a genuinely nonzero (Q)-leg. The target coefficient theorem cannot create it.

Cross-sector status

This is a (D03)-local occurrence theorem, not Marici-core machinery. Under the cross-sector promotion rule, a cosmological partial-energy analogue must construct its own extraordinary boundary class from independently fixed source geometry. Matching rational fractions or incidence patterns is insufficient.

Evidence

Executable certificate:

research/voevodsky/check_d03_toric_cox_cousin_trace.rs

SHA-256:

852652cbe3f8d20076c526e3adb493857e7859d6f33ad9cc2daede03750bfce4

Reproduction:

rustfmt --edition 2021 --check research\voevodsky\check_d03_toric_cox_cousin_trace.rs
rustc --edition 2021 -D warnings -O research\voevodsky\check_d03_toric_cox_cousin_trace.rs -o "$env:TEMP\check_d03_toric_cox_cousin_trace.exe"
& "$env:TEMP\check_d03_toric_cox_cousin_trace.exe"

The checker verifies the weighted and normalized chain identities, the principal ideal augmentation, Smith saturation, uniqueness of the formal trace, residue-only ambiguity ideals, the endpoint (\operatorname{Ext}^1) groups, the full Koszul–Cech lower terms and chain equations, the two (x_3)-edge corner coefficients, the coherent-residue zero control, and the torsion-free global Cox cohomology profile.

It must be read together with ledger entries 97 and 121 and the D03 ledger entry Four-Corner Cellular Nerve and the Generic-Unit Obstruction. Entry 97 supplies the complete road trace, entry 121 supplies the independently loaded (v_{10}) corner comparison, and the four-corner entry proves why supported corners alone cannot recover the generic unit.

Next experiment

Construct the loaded PC realization of the already-fixed occurrence map (g_3^!). Test both endpoint squares before any all-edge assembly:

[ \operatorname{Res}{v{00}}^{\rm PC}g_3^! \stackrel?= \Theta_{03,v_{00}}^{\rm corner}, \qquad \operatorname{Res}{v{10}}^{\rm PC}g_3^! \stackrel?= \Theta_{03,v_{10}}^{\rm corner}. ]

The second target is entry 121. Require the same reciprocal/BM variance, ordered normal orientation, excess retraction, and positive physical line at both endpoints. Reject the promotion if either square needs a support change, a chosen splitting, a fitted sign, or deletion of lower Cech terms.

Outcome contract

{
  "claim": "The D03 principal-lcm road complex resolves one Cartier line, and the x3 edge carries a canonical occurrence-level Koszul-Cech Gysin with primitive v00 and v10 endpoint classes.",
  "status": "proved",
  "assumptions": [
    "The scope is the occurrence coefficient system on the fixed entry-97 road square.",
    "Endpoint and product-normal orientation lines are retained rather than scalarized.",
    "Previously established normal-excess and physical-normal data are frozen external factors, not outputs of this checker."
  ],
  "evidence_refs": [
    "research/voevodsky/check_d03_toric_cox_cousin_trace.rs sha256:852652cbe3f8d20076c526e3adb493857e7859d6f33ad9cc2daede03750bfce4",
    "ledger entry 97",
    "ledger entry 121",
    "D03 ledger entry Four-Corner Cellular Nerve and the Generic-Unit Obstruction"
  ],
  "factorization_test": {
    "principal_line_resolution": "passed integrally with unit Smith factors",
    "x3_Koszul_Cech_chain_map": "passed at both endpoints with all lower terms",
    "product_Cartier_provenance": "passed; one unit gives the diagonal endpoint pair and edge incidence supplies the relative signs",
    "loaded_PC_promotion": "open",
    "nonzero_source_Q_leg": "open"
  },
  "counterevidence": [
    "A regular section of O or O(D) has zero ordinary coherent Cousin boundary.",
    "The occurrence theorem does not construct reciprocal/BM variance, the repeated-normal PC comparison, physical-Cut naturality, or G03."
  ],
  "next_experiment": "Promote the same x3 occurrence Gysin to the loaded PC category and test both v00 and v10 endpoint squares before rotating to any other edge."
}