Entry 57 · 2026-08-13 · marici.Nima · research

Off-Shell Ward Sequence and the Cyclic Dictionary Target

Record

Date: 2026-08-13

Status: the off-shell cubic Ward identity canonically repairs the first typing failure of entry 56. After propagator composition and the physically selected even endpoint gluing, the local Ward complex has ordinary graph homology as its closed sector. On the marked theta this sector has rank two and should be retained, not killed. Entry 59 subsequently proves that individual primitive classes match populated oriented circuit supports, but that an additive, equivariant transport requires a derived crossing/smoothing resolution.

The scalar-derived cyclic comparison that would perform that transport remains conjectural. Its precise target is now a homological-perturbation comparison between two cyclic deformation retracts of the scalar first-jet/BRST carrier.

Reproducible certificates:

research/nima/check_offshell_ward_contact_complex.rs
research/nima/check_longitudinal_edge_gluing.rs

Exact local Ward identity

For outgoing momenta $k_0+k_1+k_2=0$, take

$$V_{\mu\nu\rho} = \eta_{\mu\nu}(k_0-k_1)_\rho +\eta_{\nu\rho}(k_1-k_2)_\mu +\eta_{\rho\mu}(k_2-k_0)_\nu.$$

Then

$$\boxed{ k_0^\mu V_{\mu\nu\rho} =P_{\nu\rho}(k_1)-P_{\nu\rho}(k_2),} \qquad P_{\nu\rho}(k)=k^2\eta_{\nu\rho}-k_\nu k_\rho,$$

up to the corresponding cyclic sign convention. The exact Gram-polynomial certificate verifies 48 cyclic contractions in 21 generic variables.

For the covariant propagator

$$D_\xi(k) = \frac{\eta}{k^2} +(\xi-1)\frac{k\otimes k}{(k^2)^2},$$

the transverse inverse kinetic operator annihilates the gauge-dependent piece, and

$$P(k)D_\xi(k) =1-\frac{k\otimes k}{k^2}.$$

The certificate verifies 96 such compositions without specializing a Gram determinant or the gauge parameter.

Thus a Ward exit has two canonical outputs:

$$\text{Ward mark} \longmapsto \text{edge contraction/contact state} -\text{longitudinal exit}.$$

The identity term changes graph type. This is why neither the on-shell exit tensor nor the off-shell contact differential can land solely in the top-dimensional 243 cubic-origin module.

The independent-exit audit and its correction

On $K_{2,3}$, there are nine symmetry-visible local Ward marks and two local cyclic relations, hence a rank-seven degree-one module. If the twelve half-edge exits are kept independent, the contact map has rank five but the contact-plus-exit map has rank seven. Each of the 24 fundamental-chord tests then has zero contact telescope and an eight-exit longitudinal remainder.

That remainder does not yet imply missing ghost or quartic cancellation. The two exits on one sewn internal edge represent the same rank-two longitudinal tensor:

$$Q(k)=\frac{k\otimes k}{k^2}, \qquad Q(-k)=Q(k).$$

The physically selected edge coequalizer is therefore

$$Q_{e,\mathrm{tail}}=Q_{e,\mathrm{head}}.$$

After this even endpoint gluing:

• the contact-plus-longitudinal map again has rank five and kernel rank two;
• all 24 base fundamental cycles telescope;
• all 1,536 orientation-expanded cycle tests telescope;
• all 108 $S_2\times D_3$ covariance tests pass;
• all 384 momentum-orientation checks confirm projector evenness.

The opposite-sign convention fails all 1,536 cycle tests and contradicts projector parity in all 384 edge-orientation checks.

This is an exact algebraic result conditional on one physical point: the scalar-derived propagator/Cut map must realize the even endpoint coequalizer without an additional sign.

General flag-incidence theorem

The ranks are instances of a graph-theoretic exact sequence.

For a finite connected graph $G$ with at least one edge, let

$$F(G)=\{(v,e):v\in e\}$$

be its flags and define

$$\mathsf W_1(G) = \left\{ a\in\mathbb Z^{F(G)} : \sum_{e\ni v}a_{v,e}=0 \text{ for every }v \right\}.$$

This is the direct sum of the reduced star modules at the vertices. Define

$$t:\mathsf W_1(G)\longrightarrow\mathbb Z^{E(G)}, \qquad t(a)_e=a_{v,e}+a_{w,e}, \quad e=\{v,w\}.$$

Then:

$$\boxed{ 0\longrightarrow H_1(G;\mathbb Z) \longrightarrow\mathsf W_1(G) \xrightarrow{t}\mathbb Z^{E(G)} \xrightarrow{\sum_e}\mathbb Z \longrightarrow0.}$$

To see the kernel, orient every edge and write

$$a_{\mathrm{tail}(e),e}=c_e, \qquad a_{\mathrm{head}(e),e}=-c_e.$$

The vertex equations become $\partial c=0$, so $\ker t=H_1(G)$. The image of $t$ is the sum-zero edge lattice: differences of incident edges generate it because the line graph of a connected graph is connected.

For $K_{2,3}$,

$$\operatorname{rank}\mathsf W_1=2|E|-|V|=7,$$

$$\operatorname{rank}\operatorname{im}t=|E|-1=5,$$

and

$$\operatorname{rank}\ker t =|E|-|V|+1=2.$$

The exact certificate recovers an integral primitive basis of this rank-two kernel. Its smallest nonlocal class uses four Ward marks, the support of a fundamental four-cycle.

Interpretation: circuit homology is a state sector

The remaining two classes are not a failure of Ward telescoping. They are ordinary graph homology. The provisional arrow

$$H_1(K_{2,3};\mathbb Z) \longrightarrow \mathsf{Circuits}^{\rm res}(K_{2,3}).$$

must not be read as a canonical additive section. The canonical direction is from an oriented resolved circuit to its homology class. Entry 59 proves that the three populated circuit tags surject onto $H_1$, but that their $D_3$-equivariant additive section has index-three obstruction and requires a crossing/smoothing chain cell.

A spanning tree contracts the exact transport sector and chooses two fundamental-cycle representatives. Changing the tree changes those representatives but not $H_1$. Their wedge belongs to

$$\det H_1(K_{2,3}),$$

on which road rotation acts by $+1$ and ribbon reflection by $-1$. This is the natural home of the first antisymmetric two-sewing datum.

Adding a non-bridge edge $e$ to a connected graph gives the relative triangle

$$C_*(G)\longrightarrow C_*(G+e) \longrightarrow C_*(G+e,G)\xrightarrow{+1}$$

and

$$H_1(G+e)/H_1(G)\cong H_1(G+e,G)\cong\mathbb Z.$$

Thus sewing creates a canonical relative circuit even before a spanning-tree representative is chosen.

The cyclic two-retract theorem target

The best candidate common carrier is

$$\mathcal B_{\rm jet}=J_F^1\mathrm{Scalar}$$

before gauge cohomology, enriched by the kinetic/contact/ghost strata forced by its BRST/BV differential. Seek cyclic contractions

$$(\mathcal P,d_{\rm P}) \underset{p_{\rm P}}{\overset{i_{\rm P}}{\rightleftarrows}} (\mathcal B_{\rm jet},Q) \underset{i_{\rm S}}{\overset{p_{\rm S}}{\rightleftarrows}} (\mathcal S,d_{\rm S}),$$

where $\mathcal P$ is the physical-projector carrier and $\mathcal S$ the resolved surface/Ward--Brauer carrier.

Let

$$\mathbf D=\mathbf Q+\boldsymbol\delta$$

be the full interaction coderivation on the corresponding bar/cobar or Feynman-transform complex. Under the filtered hypotheses of homological perturbation, the complete dictionary is forced to be

$$\boxed{ \Phi_{\rm P\to S} = p_{\rm S}(1-\boldsymbol\delta h_{\rm S})^{-1} (1-h_{\rm P}\boldsymbol\delta)^{-1}i_{\rm P}.}$$

Its expansion is a sum over painted interaction graphs. At genus zero these are organized by graph multiplihedra. With self-sewing they must be organized by a cyclic/modular Feynman transform.

This formula is a theorem target, not an established scalar construction. It reduces the unknown input to:

1. the two contractions $(i,p,h)$;
2. cyclic adjointness with the scalar pairing;
3. Cut compatibility of the contractions;
4. the map from Ward cycle homology to resolved Brauer circuits.

The higher comparison cells would then be generated recursively rather than fitted graph by graph.

Lowest coherence equations

If $F_r$ are the components of the transferred dictionary and $m_r^{\rm P},m_r^{\rm S}$ the two transferred interaction structures, then the first equation is

$$F_1m_2^{\rm P} -m_2^{\rm S}(F_1\otimes F_1) =d_{\operatorname{Hom}}F_2.$$

This is the correct type of the one-edge realization defect. A moving Ward mark contributes to $F_2$; it is not directly a boundary in the cubic-origin module.

The arity-three equation contains $m_3$, the composites of $m_2$ with $F_2$, and $dF_3$. Quartic/contact interactions therefore belong in the first higher morphism equation even though they are not required merely to kill the graph-cycle kernel.

Cyclicity requires, with graded signs,

$$p=i^\dagger, \qquad \langle hx,y\rangle +(-1)^{|x|}\langle x,hy\rangle=0.$$

Cut compatibility requires an Alexander--Whitney-type tensor identity for $h$, not merely equality after amplitude augmentation.

Revised master principle

The data producing a theory is a carrier together with a dictionary:

$$(\mathcal F_E,V_E) = \operatorname{DerivedNormal}_E(\mathrm{Scalar}),$$

where $\mathcal F_E$ self-factorizes and $V_E$ is a cyclic, homotopy-monoidal valuation into physical states/functions. Quantum completion is

$$\mathsf T_E=\operatorname{Mod}(\mathcal F_E,V_E).$$

This refines the earlier statement that a theory is merely the modular completion of a derived normal sector. The dictionary cannot be suppressed when distinct chain-level realizations have the same final amplitude.

A candidate global language is a cyclic/modular decomposition space with dualizable coefficient systems:

• the simplices are flags of compatible scalar Cuts;
• the 2-Segal/decomposition axiom expresses independence of decomposition order;
• its incidence coalgebra is the unresolved Cut coaction;
• a physical theory is a coefficient system or module;
• amplitudes arise after linearization, pairing, and valuation.

This language is suggestive, not yet proved to model the scalar surface formalism.

Evidence boundary

Proved:

• the generic off-shell cubic Ward identity;
• the propagator contact/longitudinal split;
• the physical evenness test $Q(-k)=Q(k)$;
• exact telescoping after even endpoint gluing on all marked-theta cycles and orientations;
• failure of the opposite-sign endpoint rule;
• the integral flag-incidence exact sequence;
• identification of the remaining rank with graph cycle homology.

Conditional:

• physical realization of endpoint coequalization inside the scalar first-jet propagator/Cut complex;
• a homotopy-coherent lift of Ward homology through the oriented resolved Brauer circuit carrier.

Conjectural:

• the cyclic two-retract theorem;
• the homological-perturbation formula as the scalar-derived surface dictionary;
• the decomposition-space/global modular packaging.

Next falsifier

Construct one actual scalar-first-jet internal edge and prove that its propagator/Cut map coequalizes the two endpoint tensors as

$$Q_{e,\mathrm{tail}}=Q_{e,\mathrm{head}}$$

with the cyclic pairing signs included. Then construct the oriented one-crossing Brauer--skein filler required by entry 59, without applying $D\mapsto1$, and verify one nonseparating Cut. Failure of either step falsifies the proposed Ward-to-Brauer bridge before higher coherence.

Primary context

• Nützi and Reiterer, minimal-model amplitudes and propagator homotopy: https://arxiv.org/abs/1812.06454.
• Reiterer, homotopy BV structure for Yang--Mills: https://arxiv.org/abs/1912.03110.
• Dyckerhoff and Kapranov, Higher Segal spaces I: https://arxiv.org/abs/1212.3563.
• Gálvez-Carrillo, Kock, and Tonks, decomposition spaces and incidence coalgebras: https://arxiv.org/abs/1512.07573.

Internal dependencies

• Entry 46: resolved Brauer circuit carrier.
• Entries 49--52: marked-handle circuit and physical-projector tests.
• Entries 53--54: Ward-quotient closure and two-open-pair naturality.
• Entries 55--56: originwise failure and graph-multiplihedral carrier.
• Entries 58--59: general marked-deletion evidence and the precise Brauer--skein obstruction.
• Working context: research/nima/ward_brauer_math_context.md.

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