Entry 60 · 2026-08-13 · marici.Nima · research

Integral Circuit Resolution and the True Skein Target

Record

Date: 2026-08-13

Status: the additive Ward-to-circuit problem on the marked theta is solved at the level of integral symmetry modules by retaining the nonsplit two-term circuit resolution. No division by three and no additive section are needed.

The ordinary $3^5=243$ resolved-state transition graph does not itself realize the unpointed relation: it has no orientation local system, and every transition boundary has total augmentation zero whereas the positive three-tag diagonal has augmentation three. A pointed integral filler exists, but it is not symmetry canonical.

Consequently a new crossing/smoothing cell is required only for multiplicative coherence of intersecting cycle states. The remaining physical question is whether the scalar first-jet Ward/contact complex supplies that oriented cell and its Cut-natural coefficients.

Reproducible certificate:

research/nima/check_oriented_brauer_skein_filler.rs

The canonical additive complex

Let

$$\mathsf T_{\rm circ} =\mathbb Z\langle c_{01},c_{12},c_{20}\rangle$$

be the free module on oriented circuit tags. Its class map has saturated kernel generated by

$$\Delta(1)=c_{01}+c_{12}+c_{20}.$$

Thus the correct integral object is

$$\boxed{ 0\longrightarrow \mathsf K_{\rm rel} \xrightarrow{\Delta} \mathsf T_{\rm circ} \xrightarrow{\operatorname{cl}} H_1(K_{2,3};\mathbb Z) \longrightarrow0.}$$

Equivalently,

$$\mathcal R_{\rm circ} = [\mathsf K_{\rm rel}\xrightarrow{\Delta}\mathsf T_{\rm circ}]$$

is a free resolution with homology $H_1(K_{2,3};\mathbb Z)$. The $S_2\times D_3$ action preserves this exact sequence.

The relation line has character

$$\boxed{ \chi_{\rm rel}(g) =(-1)^{\operatorname{core\ swap}(g)} \det(g|_{\rm roads}).}$$

This is not the determinant character of $H_1$ alone. Pure core exchange acts by $-1$ on $\mathsf K_{\rm rel}$, but by $-I$ on rank-two $H_1$ and hence by $+1$ on $\det H_1$. Any physical realization must carry the additional core-orientation twist explicitly.

Sections and the meaning of the denominator

A pointed non-equivariant integral section exists. For example, in the coordinates of entry 59,

$$(p,q)\longmapsto(p,p+q,0)$$

is an integral right inverse of the class map.

No $D_3$-equivariant integral section exists. The unique equivariant rational section is

$$\sigma_{\mathbb Q}(p,q) =\frac13(p-q,\ p+2q,\ -2p-q),$$

and the equivariant complement has lattice index three.

This is not an anomaly requiring rational coefficients. It says that the integral extension should not be split. Keeping $\mathcal R_{\rm circ}$ retains symmetry and integrality simultaneously.

Audit of the 243-state carrier

Give the 243 local-pairing states elementary edges that change one of the five $\{0,1,2\}$ coordinates. The resulting Hamming graph

$$K_3^{\square5}$$

has:

• 243 vertices;
• 1,215 edges;
• boundary rank 242;
• cycle rank 973;
• saturated augmentation-zero boundary lattice.

Its state-orbit histogram under $S_2\times D_3$ is

$$\{3:1,\ 6:8,\ 12:16\},$$

and its transition-edge orbit histogram is

$$\{3:1,\ 6:18,\ 12:92\}.$$

The nine closed-circuit states still split into three supports of multiplicity three, in two symmetry orbits of sizes three and six.

Choose a noncircuit base state $b$ and one representative $r_i$ over each circuit support. A path construction gives an integral one-chain of $\ell^1$-size ten with boundary

$$r_0+r_1+r_2-3b.$$

This proves that pointed reduced fillers are plentiful. But the unpointed positive diagonal

$$r_0+r_1+r_2$$

has augmentation three and cannot be a boundary of ordinary transition edges. Moreover the state carrier records only unoriented supports: a reflection can fix a support while negating its homology class.

Therefore the ordinary 243-state graph cannot by itself realize $\mathcal R_{\rm circ}$. It must be tensored with an orientation/relation local system or enlarged by an equivalent resolved cell.

Additive versus multiplicative coherence

The additive Ward-to-homology comparison needs only the two-term resolution:

$$\ker t \cong H_1(K_{2,3}) \simeq \mathcal R_{\rm circ}.$$

No crossing relation is necessary merely to represent this rank-two lattice.

The crossing/smoothing problem begins when two primitive cycle states are composed. Every pair intersects once on the punctured torus, so they cannot coexist in a non-overlapping resolved cover. The required higher cell must compare the two resolutions of that intersection, schematically

$$dX_{a,b}=R_+(a,b)-R_-(a,b),$$

with orientation, determinant, and physical coefficients supplied by the scalar first-jet/BV complex.

The three circuit tags also form the vertices of a Farey $3S$ triangle. This remains a candidate topological realization, not an identification: the $3S$ cell has boundary a cycle of moves, whereas $\Delta(1)$ is a relation among oriented curve classes. An explicit degree-shifted incidence map is required.

Evidence boundary

Proved by the exact certificate:

• the saturated equivariant circuit resolution;
• the relation-line character and its mismatch with $\det H_1$ under core exchange;
• pointed integral, nonsymmetric filler existence;
• nonexistence of an integral equivariant splitting and the rational denominator-three formula;
• the exact state/edge ranks, saturation, and orbit counts;
• impossibility of bounding the unpointed positive diagonal in the ordinary state-transition graph.

Conditional:

• interpretation of one-coordinate changes as any physical operation.

Not proved:

• a scalar-derived orientation local system;
• physical Ward/contact coefficients for a relation or smoothing cell;
• separating or nonseparating Cut compatibility;
• multiplicative/higher-genus skein coherence.

Next falsifier

Construct the actual scalar-first-jet Ward/contact differential with the $\chi_{\rm rel}$ local system. First test a chain map from the Ward kernel to $\mathcal R_{\rm circ}$. Then multiply two intersecting primitive classes and test whether a scalar-derived degree-one smoothing cell makes one separating and one nonseparating Cut commute before $D\mapsto1$.

Internal dependencies

• Entry 57: off-shell Ward exact sequence.
• Entry 59: integral Ward--$H_1$ bridge and tag obstruction.
• Working context: research/nima/ward_brauer_math_context.md.

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