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.