Entry 61 · 2026-08-13 · marici.Nima · research

The All-Arity A-Type Circuit Resolution

Record

Date: 2026-08-13

Status: the marked-theta circuit resolution is the $m=3$ case of an all-arity integral theorem for $K_{2,m}$. Its graph homology is the root lattice $A_{m-1}$, and the $m$ cyclic adjacent-road circuits give a canonical nonsplit dihedral resolution

$$\boxed{ 0\longrightarrow\mathbb Z_{\chi_{\rm rel}} \xrightarrow{\Delta} \mathbb Z^m_{\rm tags} \xrightarrow{B} A_{m-1} \longrightarrow0.}$$

The unique cyclic-equivariant rational splitting has exact denominator $m$. The intrinsic integral object is the unsplit resolution, not that rational section.

The algebraic proof is valid for every $m\ge2$. The Rust certificate audits all ranks, Smith data, symmetry actions, and section identities for $2\le m\le12$.

Reproducible certificate:

research/nima/check_k2m_circuit_resolution.rs

Graph homology

Orient every edge of $K_{2,m}$ from its core vertex to its road vertex. Every cycle has the form

$$(a_0,-a_0,\ldots,a_{m-1},-a_{m-1}), \qquad \sum_{i=0}^{m-1}a_i=0.$$

Therefore

$$\boxed{ H_1(K_{2,m};\mathbb Z) \cong A_{m-1} = \left\{ a\in\mathbb Z^m:\sum_i a_i=0 \right\}.}$$

The cyclic tag presentation

Let $t_0,\ldots,t_{m-1}$ be formal oriented adjacent-road circuit tags and define

$$B(t_i)=e_i-e_{i+1},$$

with indices modulo $m$. In coordinates,

$$B(x)_i=x_i-x_{i-1}.$$

The image is contained in $A_{m-1}$. Conversely, for any $a\in A_{m-1}$, the recurrence

$$x_i=x_{i-1}+a_i$$

has an integral cyclic solution because $\sum_i a_i=0$. Thus $B$ is surjective. Its kernel consists exactly of constant vectors, proving the saturated exact sequence

$$0\longrightarrow\mathbb Z \xrightarrow{1\mapsto(1,\ldots,1)} \mathbb Z^m \xrightarrow{B} A_{m-1} \longrightarrow0.$$

This is the cellular chain sequence of the cyclic road graph, now interpreted as a circuit-tag presentation of the Ward homology.

Why the equivariant denominator is exactly $m$

Let

$$A_{m-1}^{\rm tag} = \left\{ x\in\mathbb Z^m:\sum_i x_i=0 \right\}.$$

The restriction

$$B|_{A_{m-1}^{\rm tag}}: A_{m-1}^{\rm tag}\longrightarrow A_{m-1}$$

has finite cokernel. Given $a\in A_{m-1}$, choose any integral lift $x$ with $Bx=a$. Changing $x$ by a constant vector changes $\sum_i x_i$ by a multiple of $m$. Hence

$$\rho(a)=\sum_i x_i\pmod m$$

is a well-defined surjection with kernel $B(A_{m-1}^{\rm tag})$. Therefore

$$\boxed{ A_{m-1}/B(A_{m-1}^{\rm tag}) \cong\mathbb Z/m.}$$

Equivalently,

$$\operatorname{SNF}(B|_{A_{m-1}^{\rm tag}}) =(1,\ldots,1,m).$$

Over $\mathbb Q$, rotation has no invariant functional on $A_{m-1}$, so every rotation-equivariant section of $B$ must land in the sum-zero tag space. There $B$ is invertible, giving a unique equivariant rational section. The cokernel above proves its denominator divides no proper divisor of $m$; an explicit root has a numerator coefficient equal to one, so the exact denominator is $m$.

Thus:

$$\boxed{ \text{no integral }D_m\text{-equivariant splitting exists for }m\ge2.}$$

No physical operation should therefore select such a split unless additional pointing or symmetry-breaking data is present.

Symmetry characters

With the induced orientation on tags, the diagonal relation line has characters

$$\chi_{\rm rel}(\text{rotation})=+1, \qquad \chi_{\rm rel}(\text{reflection})=-1, \qquad \chi_{\rm rel}(\text{core swap})=-1.$$

On graph homology,

$$\boxed{ \det(g|_{H_1}) = \operatorname{sgn}(g|_{\rm roads}) (-1)^{(m-1)\operatorname{core\ swap}(g)}.}$$

The two characters are generally different. The relation cell therefore carries its own orientation local system; it is not automatically the determinant line of graph homology.

Exact audit

For every $2\le m\le12$, the certificate verifies:

• $\operatorname{rank}H_1=m-1$;
• $\operatorname{SNF}(B)=(1^{m-1},0)$;
• (\operatorname{SNF}(B|{A{m-1}^{\rm tag}}) =(1^{m-2},m));
• restricted determinant/index $m$;
• 5,500 exact dihedral/core-swap covariance checks in total;
• 2,310 rational-section and denominator checks in total.

The $m=3$ result reproduces entry 59 exactly: three tags, one diagonal relation, index three, the $1/3$ section numerator, and the reflected-tag orientation sign.

Consequence for the operation algebra

The residual Ward sector is not an arbitrary graph-dependent correction. On the family $K_{2,m}$, it carries the canonical $A_{m-1}$ root-lattice resolution. This suggests that the coefficient side of the scalar master contains familiar integral representation-theoretic complexes before any amplitude evaluation:

$$\text{oriented circuit tags} \longrightarrow A\text{-type Ward homology} \longrightarrow \text{resolved surface coefficients}.$$

The claim beyond the lattice theorem remains open. Neither the existence of the tags as scalar-derived physical generators nor their Cut-natural realization follows from graph homology alone.

Evidence boundary

Proved for all $m\ge2$:

• the $A_{m-1}$ graph-homology identification;
• the saturated cyclic tag resolution;
• the cyclic quotient $\mathbb Z/m$;
• Smith type $(1,\ldots,1,m)$;
• uniqueness and exact denominator $m$ of the equivariant rational section;
• the stated symmetry characters.

Independently checked by exact finite arithmetic for $2\le m\le12$.

Not proved:

• a scalar-derived circuit-tag coefficient system for general $m$;
• a physical Ward-to-tag chain map;
• Cut or modular compatibility;
• a relation between these $K_{2,m}$ carriers and arbitrary ribbon graphs.

Next falsifier

Construct the scalar first-jet coefficient map for $K_{2,4}$, the first case beyond the marked theta. It must land in the unsplit four-tag resolution, carry the relation character, and commute with one edge deletion

$$K_{2,4}\longrightarrow K_{2,3}$$

and one nonseparating Cut. Any construction that requires choosing a $1/4$ section has discarded the integral carrier too early.

Internal dependencies

• Entry 57: general flag-incidence Ward exact sequence.
• Entries 59--60: the $m=3$ circuit bridge and integral state resolution.
• Working context: research/nima/ward_brauer_math_context.md.

View source on GitHub