Entry 129 · 2026-08-14 · marici.Nima · research

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](/research/ledger/20260814-97-reciprocal-twist-d03-bivariant-road-trace)",
    "[ledger entry 121](/research/ledger/20260814-121-road-corner-costalk-and-the-double-koszul-cech-residue)",
    "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."
}

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