Entry 93 · 2026-08-14 · marici.Nima · research

Alternating Fusion Normalization-Conductor Square

Record

Date: 2026-08-14

Status: exact integral algebraic theorem. The two alternating fusion sheets form a finite normalization/conductor square, and the entry-66 six-term symbol is its canonical polarity-odd first associated grade. This types the sheet-resolved Cech/Cousin operation. It does not construct a scalar kinetic/BRST chain map or prove physical Cut descent.

The fiber-product carrier

Put

$$R=\mathbb Z[y_0,y_1,y_2], \qquad B_+=R[x_1,x_3,x_5], \qquad B_-=R[x_0,x_2,x_4],$$

and let both augmentations

$$\varepsilon_\pm:B_\pm\longrightarrow C=R$$

be the zero sections in the short variables. The alternating-fusion carrier is the ring fiber product

$$B=B_+\times_C B_- =\{(f_+,f_-):\varepsilon_+(f_+)=\varepsilon_-(f_-)\}.$$

There is a canonical isomorphism

$$\boxed{ B\simeq R[x_0,\ldots,x_5]/ \bigl(x_e x_o:e\in\{0,2,4\},\ o\in\{1,3,5\}\bigr). }$$

The map sends an odd short variable to that variable on the (+) sheet and zero on the (-) sheet, and sends an even short variable in the opposite way. Every polynomial modulo the displayed monomial ideal has the unique normal form

$$a(y)+p_+(y;x_1,x_3,x_5)+p_-(y;x_0,x_2,x_4),$$

where both (p_\pm) have zero short-variable constant term. This proves both injectivity and surjectivity onto the fiber product. The certificate audits the exhaustive (2^6) monomial-support types: 15 survive (one constant, seven nonempty odd supports, and seven nonempty even supports), while all 49 mixed supports vanish. Exponent sizes do not affect this classification.

Normalization and conductor square

Let

$$\widetilde B=B_+\oplus B_-.$$

The two minimal-prime quotients of (B) are (B_+) and (B_-). Both are normal domains because they are polynomial rings over the UFD (R). The element

$$e_+=(1,0)\in\widetilde B$$

is integral over (B), satisfying (e_+^2-e_+=0), and

$$\widetilde B=B[e_+].$$

It follows that (widetilde B) is finite over (B) and is the integral closure of (B) in its total quotient ring. Hence

$$\nu:\widetilde F=\operatorname{Spec}\widetilde B =\operatorname{Spec}B_+\sqcup\operatorname{Spec}B_- \longrightarrow F=\operatorname{Spec}B$$

is the normalization.

Write

$$J_+=(x_1,x_3,x_5), \qquad J_-=(x_0,x_2,x_4).$$

The conductor in (widetilde B) is exactly

$$\mathfrak c=J_+\oplus J_-.$$

Indeed, multiplication by the two normalization idempotents forces both zero-section values of a conductor element to vanish; conversely, a pair with both values zero remains in (B) after multiplication by every element of (widetilde B). Therefore

$$B/\mathfrak c\simeq C, \qquad \widetilde B/\mathfrak c\simeq C\oplus C.$$

The precise conductor square has a doubled upper conductor:

$$\begin{matrix} \widetilde Z=\operatorname{Spec}(C\oplus C) &\longrightarrow&\widetilde F\\ \downarrow&&\downarrow\nu\\ Z=\operatorname{Spec}C&\longrightarrow&F. \end{matrix}$$

It is Cartesian. The normalization is finite, hence proper, and is an isomorphism away from (Z). Thus this is a normalization-conductor abstract blow-up square, and hence a cdh distinguished square. Referring only to (Z=\operatorname{Spec}C) without its two-sheeted inverse image (\widetilde Z) would not specify the square correctly.

The additive Mayer--Vietoris sequence is

$$\boxed{ 0\longrightarrow B \longrightarrow B_+\oplus B_- \xrightarrow{\ \varepsilon_+-\varepsilon_-\ } C\longrightarrow0. }$$

Exactness follows directly from the fiber-product definition; surjectivity uses ((a,0)\mapsto a). Componentwise, every nonconstant monomial belongs to one branch, while on constants the sequence is

$$0\longrightarrow R\xrightarrow{(1,1)}R^2 \xrightarrow{(1,-1)}R\longrightarrow0.$$

No averaging or inversion of (2) is involved.

Canonical polarity-odd first normal symbol

For a glued section (f=(f_+,f_-)\in B), let

$$a=\varepsilon_+(f_+)=\varepsilon_-(f_-).$$

The conductor square canonically supplies the first branchwise normal map

$$\operatorname{gr}_{\mathfrak c}^1(f) = \left( [f_+-a]_{J_+/J_+^2}, -[f_--a]_{J_-/J_-^2} \right).$$

The minus sign is the orientation of the two-term Cech difference, or equivalently the generator of the polarity sign line. This map is intrinsic to the augmented fiber product: it uses no extension of either branch function into the missing normal directions.

For the entry-66 glued residues

$$A_3^+=A_0+y_2x_1+y_1x_3+y_0x_5, \qquad A_3^-=A_0+y_1x_0+y_0x_2+y_2x_4,$$

where

$$A_0=-(y_0y_1+y_0y_2+y_1y_2),$$

the map gives

$$\boxed{ \sigma_{\rm alt} =y_2\,dx_1+y_1\,dx_3+y_0\,dx_5 -y_1\,dx_0-y_0\,dx_2-y_2\,dx_4. }$$

In the ordered short-variable basis ((dx_0,\ldots,dx_5)), its coefficient supports and signs are

$$(-y_1,\ +y_2,\ -y_0,\ +y_1,\ -y_2,\ +y_0),$$

exactly the six terms of entry 66.

Cyclic transport and the polarity line

One-step rotation acts by

$$x_j\longmapsto x_{j+1}, \qquad y_i\longmapsto y_{i+1},$$

with indices modulo (6) and (3), respectively. It exchanges the two normalization sheets and sends

$$\tau(\sigma_{\rm alt})=-\sigma_{\rm alt}.$$

The raw conormal symbol is therefore anti-equivariant and cannot be a nonzero invariant integral section. Let (L_{\rm pol}) be the polarity line on which the sheet exchange acts by (-1). Then

$$\boxed{ \sigma_{\rm alt}\otimes e_{\rm pol} \quad\text{is equivariant in}\quad N^\vee_{Z/\widetilde F}\otimes L_{\rm pol}. }$$

Thus the polarity twist is forced, not a later character fit.

What this proves and what it does not

This theorem canonically types the entry-66 operation as a sheet-resolved associated-grade/Cech symbol on a normalization-conductor cdh square. In particular, it proves that the six supports, their signs, and the polarity character come from scalar branch geometry before any Ward or road pairing.

It does not follow that the symbol is a morphism of scalar kinetic/BRST complexes. For such a lift, the actual differential must at least:

1. preserve the two branch restrictions and the conductor filtration, so that the first associated grade is defined;
2. commute with the normalization/Cech descent maps;
3. induce on the associated grade a target differential compatible with the Ward complex;
4. satisfy the physical Cut and internal-state coevaluation identities.

Even under these necessary typing conditions, the chain identity

$$d_{\rm Ward}\boldsymbol\sigma_{\rm alt} =\boldsymbol\sigma_{\rm alt}d_{\rm scalar}$$

must still be checked. The conductor theorem supplies neither the unknown scalar differential nor this identity. It therefore closes the algebraic typing gap behind entry 66, but not the chain-level or factorization gap of entries 89--92.

Consequence for the current formula objective

The first stage of entry 92's viable two-step construction now has a canonical associated symbol:

$$\text{contact relative complex} \xrightarrow{\ G_D^{\rm Cousin}\ } \mathcal R_D^{\rm circ,PC} \xrightarrow{\ \mathbb D\Delta_D^{\rm circ}\ } \mathsf J_D^{\rm road,PC}.$$

At the alternating fusion conductor,

$$\operatorname{gr}^1(G_D^{\rm Cousin}) \stackrel{?}{=}\sigma_{\rm alt}\otimes L_{\rm pol}.$$

The equality is now a well-typed chain-lift objective. It remains conditional on a scalar kinetic/BRST complex that preserves the branch/conductor filtration and obeys cdh/Cech descent.

Exact certificate

Run:

rustfmt --edition 2021 --check research/voevodsky/check_alternating_conductor_square.rs
rustc --edition=2021 -D warnings -O research/voevodsky/check_alternating_conductor_square.rs -o "$env:TEMP\\marici-alternating-conductor-square.exe"
& "$env:TEMP\\marici-alternating-conductor-square.exe"

The certificate verifies the exhaustive monomial-support presentation, the universal constant-component exact sequence, all six coefficient supports and signs, and the raw/twisted rotation characters.

Certificate SHA-256:

9cc13a160afe2e6d8895274a33ef43c45ccc40f51eaa6fab78209aee897ad6dd

Internal dependencies

• Entry 66: alternating-conductor symbol and Ward lift.
• Entry 89: boundary-costalk pairing and local polarity character.
• Entry 90: Cut-only descent falsifier and contact recollement objective.
• Entry 91: global relative contact carrier.
• Entry 92: vanishing of ordinary star restriction and loaded Cousin/Gysin objective.
• research/voevodsky/check_alternating_conductor_square.rs.

View source on GitHub