Entry 66 · 2026-08-13 · marici.Nima · research

Alternating Fusion Conductor Symbol and the First Cross-Normal Relation

Record

Date: 2026-08-13

Status: exact integral coefficient-symbol theorem. The two alternating scalar-scaffolded three-gluon residues canonically derive the six-point QTDS contact matrix and its marked-theta Ward-kernel lift. A scalar BRST/kinetic chain realization and physical Cut naturality remain open.

The result is the first explicit algebraic relation between two distinct normal constructions on the scalar master. It is not a relation between final amplitudes. It is a conductor symbol on the union of two complementary fusion strata.

Alternating fusion strata

Use the six short planar variables

$$x_j=X_{j+1,j+3}, \qquad j\in\mathbb Z/6,$$

and the three long variables

$$y_0=X_{14}, \qquad y_1=X_{25}, \qquad y_2=X_{36}.$$

Momentum conservation identifies $X_{26}=X_{62}=x_5$.

The scaffold pairing

$$\mathfrak f_+=(12)(34)(56)$$

has fusion-normal ideal

$$I_+=(x_0,x_2,x_4).$$

One-step cyclic rotation gives the complementary pairing

$$\mathfrak f_-=(23)(45)(61)$$

with ideal

$$I_-=(x_1,x_3,x_5).$$

Let

$$F_\pm=V(I_\pm), \qquad F=F_+\cup F_-, \qquad Z=F_+\cap F_-.$$

The normalization of the reducible fusion carrier is

$$\nu:\widetilde F=F_+\sqcup F_-\longrightarrow F,$$

and $Z$ is its conductor locus.

The two scalar-derived three-gluon residues

The documented three-gluon multi-normal residue on $F_+$ is

$$A_3^+ = y_0x_5+y_2x_1+y_1x_3 - (y_0y_1+y_0y_2+y_1y_2).$$

Cyclic rotation gives

$$A_3^- = y_1x_0+y_0x_2+y_2x_4 - (y_0y_1+y_0y_2+y_1y_2).$$

These are sections of the ordered multi-residue lines

$$L_+ = N^\vee_{13}\otimes N^\vee_{35}\otimes N^\vee_{51},$$

$$L_- = N^\vee_{24}\otimes N^\vee_{46}\otimes N^\vee_{62}.$$

One-step rotation maps the ordered fusion-normal list

$$(x_0,x_2,x_4) \longmapsto (x_1,x_3,x_5)$$

position by position. It therefore identifies the ordered residue lines with positive orientation. No arbitrary scalar trivialization is introduced.

On the conductor,

$$A_3^+|_Z = A_3^-|_Z = A_0 = -(y_0y_1+y_0y_2+y_1y_2).$$

Thus the two branch sections glue at order zero.

The intrinsic conductor symbol

Inside $F_+$, the conductor $Z$ has normal coordinates $(x_1,x_3,x_5)$. Inside $F_-$, it has normal coordinates $(x_0,x_2,x_4)$. Define the polarity-odd relative normal symbol

$$\boxed{ \sigma_{\rm alt}(A_3^+,A_3^-) = d_{Z/F_+}A_3^+ - d_{Z/F_-}A_3^-. }$$

Explicitly,

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

This symbol is intrinsic to the normalized union. It uses only derivatives in directions that exist on the corresponding branch. It does not extend $A_3^+$ into the missing even directions or $A_3^-$ into the missing odd directions.

Algebraically, if

$$J_+=(x_1,x_3,x_5)\subset\mathcal O(F_+), \qquad J_-=(x_0,x_2,x_4)\subset\mathcal O(F_-),$$

then

$$\sigma_{\rm alt} = [A_3^+-A_0]_{J_+/J_+^2} - [A_3^--A_0]_{J_-/J_-^2}.$$

Changing an ambient representative of $A_3^+$ by $I_+$, or of $A_3^-$ by $I_-$, changes neither branch function and hence neither relative normal symbol. The executable audit verifies this on all fifty-four quadratic ideal monomials at the degree relevant to $A_3$.

Shared-longitudinal symbol

Take the linear symbol in the common $y$-directions. With columns ordered as

$$(dx_0,dx_1,dx_2,dx_3,dx_4,dx_5),$$

the three coefficient rows are

$$d_0=dx_5-dx_2,$$

$$d_1=dx_3-dx_0,$$

$$d_2=dx_1-dx_4.$$

Equivalently,

$$d_y\sigma_{\rm alt} = \begin{pmatrix} 0&0&-1&0&0&1\\ -1&0&0&1&0&0\\ 0&1&0&0&-1&0 \end{pmatrix}.$$

The three long channels are the vertices opposite the three roads of the oriented channel triangle. Its Ward-star incidence is

$$\partial_\triangle(d_0,d_1,d_2) = (d_2-d_1,\ d_0-d_2,\ d_1-d_0).$$

Therefore

$$\boxed{ C_{\rm QTDS} = \partial_\triangle d_y\sigma_{\rm alt} = \begin{pmatrix} 1&1&0&-1&-1&0\\ 0&-1&-1&0&1&1\\ -1&0&1&1&0&-1 \end{pmatrix}. }$$

This is exactly the six-point QTDS contact matrix independently obtained from the scalar rank-jump presentation. Every column lies in

$$A_2=\widetilde H_0(R_3;\mathbb Z).$$

No $GL(6)$ fit or adjustable normalization occurs.

The complete cross-normal formula

Entry 64 supplies the integral suspension

$$\Gamma_3:A_2\xrightarrow{\sim}H_1(K_{2,3};\mathbb Z),$$

and entry 59 supplies the integral Ward bridge

$$\Theta:H_1(K_{2,3};\mathbb Z)\xrightarrow{\sim}\ker d_{\rm Ward}.$$

The complete coefficient relation is therefore

$$\boxed{ M_{\rm Ward} = \Theta\, \Gamma_3\, \partial_\triangle\, d_y\, \sigma_{\rm alt} \left( \mathbb J_{\mathfrak f_+}A_{\rm scalar}, \mathbb J_{\mathfrak f_-}A_{\rm scalar} \right). }$$

It gives

$$M_{\rm Ward} = \begin{pmatrix} 0&-1&-1&0&1&1\\ -1&-1&0&1&1&0\\ 0&1&1&0&-1&-1\\ 1&1&0&-1&-1&0\\ -1&-1&0&1&1&0\\ 0&1&1&0&-1&-1\\ 1&0&-1&-1&0&1 \end{pmatrix}.$$

All six columns are annihilated by the exact marked-theta Ward contact differential.

This closes the coefficient-level gap left in entry 64.

Why both fusion branches are necessary

A single residue cannot contain this relation.

The $F_+$ residue is independent of its own fusion normals

$$x_0,\ x_2,\ x_4,$$

while the $F_-$ residue is independent of

$$x_1,\ x_3,\ x_5.$$

Each branch is therefore blind to exactly three columns of $C_{\rm QTDS}$. The six-column source exists only as the polarity-odd relative symbol of the normalized two-branch carrier.

This is structurally important:

The first cross-normal relation is not a unary operator on one fusion residue. It is descent data attached to the intersection of two alternating fusion charts.

The scalar master supplies the two local sections, their cyclic line transport, their common conductor value, and their complementary relative normal directions.

What has now been proved

Proved exactly:

1. the two scaffold residues are scalar-derived and related by cyclic rotation;
2. their ordered conormal residue lines are cyclically identified with positive orientation;
3. their restrictions agree on the conductor;
4. their polarity-odd relative normal symbol is independent of ambient representatives;
5. its common-$y$ linear symbol followed by road incidence is exactly the QTDS contact matrix;
6. suspension and the Ward bridge give the exact seven-by-six Ward-kernel matrix;
7. every resulting column is Ward closed.

Still open:

1. a morphism of scalar multi-normal residue/kinetic complexes whose associated symbol is $\sigma_{\rm alt}$;
2. compatibility with gauge/BRST descent;
3. one complete-pair separating Cut square;
4. one nonseparating Cut with explicit internal-state coevaluation;
5. extension beyond the three-gluon/six-scalar local model.

Thus the result is intrinsic at coefficient-symbol level but not yet a physical chain-level natural transformation.

Consequence for the operation algebra

The relation cannot be written using only a list of unary operators such as $\operatorname{gr}_R$ and $\mathbb J_{\mathfrak f}$. It uses incidence between strata:

$$F_+\longleftarrow Z\longrightarrow F_-.$$

The new primitive operation is a bivariant normal symbol on the conductor of a reducible normal carrier,

$$\sigma_{\rm alt}: \left( \mathbb J_{\mathfrak f_+}\mathcal S, \mathbb J_{\mathfrak f_-}\mathcal S \right) \longrightarrow N^\vee_{Z/\widetilde F}\otimes\operatorname{sgn}_{\rm pol}.$$

This is exactly the kind of operation expected in a Cousin, Cech, or exit-path description of a stratified master geometry. It is further evidence that the emerging structure is a derived incidence calculus rather than a strict operator algebra on amplitudes.

Next falsifier

Construct a chain map

$$\boldsymbol{\sigma}_{\rm alt}: \operatorname{Tot} \check C \left( \{F_+,F_-\}; \mathbb J_{\mathfrak f}\mathcal S \right) \longrightarrow \mathcal W_{\rm Ward}$$

whose associated coefficient symbol is the matrix above.

It must:

1. carry the ordered residue lines rather than trivialize them silently;
2. intertwine the scalar kinetic differential with the Ward differential;
3. reproduce $\Theta\Gamma_3\partial_\triangle d_y\sigma_{\rm alt}$ on the associated grade;
4. commute with a Cut that partitions complete fusion pairs;
5. include the physical coevaluation when a Cut creates an internal gluon;
6. survive one nonseparating Cut modulo the declared hereditary total-derivative ideal.

Failure at step 2 would make the exact coefficient relation kinematic but not gauge-cohomological. Failure at steps 4--6 would make it local but not a self-factorizing physical dictionary.

Reproducible certificate

Run:

rustc --edition=2021 -D warnings -O research/nima/check_three_gluon_qtds_transgression.rs -o "$env:TEMP\\marici-three-gluon-transgression.exe"
& "$env:TEMP\\marici-three-gluon-transgression.exe"

The certificate checks the two branch formulas, conductor gluing, cyclic residue-line orientation, independence from all relevant quadratic ambient representative changes, the exact QTDS contact matrix, the complete Ward matrix, and Ward closure of all six columns.

Internal dependencies

• Entry 08: Yang--Mills as a multidegree-$(1,\ldots,1)$ normal residue.
• Entry 20: the scalar-derived six-point QTDS contact matrix.
• Entries 59 and 64: the integral Ward bridge and suspension.
• research/nima/check_three_gluon_qtds_transgression.rs: exact certificate.

View source on GitHub