Ward-Quotient One-Edge Closure and Two-Closure Coherence

Record

Date: 2026-08-13

Status: the unrestricted arbitrary-environment one-edge closure statement is false. Null on-shell momentum is not enough to make physical-projector sewing reference independent. A corrected one-edge theorem holds in the physical polarization quotient under an aligned Ward condition, and its two-edge coherence is the ordinary interchange law for two categorical traces. This does give an induction on graph cycle rank, but only conditionally on an all-graph realization hypothesis that entry 52 proves for the marked-theta cell, not for arbitrary graph environments.

The main conclusions are:

the free resolved curve-cover gluing theorem of entry 52 remains strict;
physical Gram specialization is natural for one closure only after passing to the correct Ward quotient;
two closures commute even when the product of their longitudinal projector terms is nonzero;
a proof based on universal vanishing of nested longitudinal terms is false;
premature replacement of resolved closed carriers by (D) still cannot be used in any statement that must later commute with Cuts.

Reproducible certificates:

research/nima/check_one_edge_closure_ward.rs
research/nima/check_marked_handle_symbolic_identity.rs
research/nima/check_marked_handle_x_dictionary.rs

The four levels that must remain separate

Let (E) be a set of unsewn edge pairs.

The resolved strand/curve-cover object is the free module

[ \mathsf{Cov}^{\rm res}(G\setminus E), ]

whose generators retain actual open strands and tagged closed curves. Its gluing maps

[ \operatorname{Gl}_e^{\rm res}: \mathsf{Cov}^{\rm res}(G\setminus E) \longrightarrow \mathsf{Cov}^{\rm res}(G\setminus(E\setminus{e})) ]

are topological sewing of embedded one-manifolds. For distinct open pairs,

[ \operatorname{Gl}_e^{\rm res} \operatorname{Gl}_f^{\rm res}

\operatorname{Gl}_f^{\rm res} \operatorname{Gl}_e^{\rm res}. ]

This is a statement before Gram specialization and before any closed carrier is evaluated.

Physical Gram specialization lands first in a multilinear physical-state quotient, not directly in a scalar polynomial ring. For a null momentum (p), define

[ H_p=p^\perp/\langle p\rangle. ]

A reference vector (q), with (p\cdot q\ne0), chooses the familiar tensor representative

[ \Pi_p(q)^{\mu\nu}

-\eta^{\mu\nu} +\frac{p^\mu q^\nu+q^\mu p^\nu}{p\cdot q}, ]

but the sewn state is the reference-independent trace in (H_p). The projector is a choice of representative of that trace.

The resolved closed-curve counit is still

[ \operatorname{ev}\rho [\gamma;\nu\gamma,\Delta_\gamma] =\nu_\gamma-\Delta_\gamma=D. ]

This evaluation belongs after the relevant closures. It must not be moved in front of possible later Cuts. Entries 49 and 50 already exhibit the resulting nonzero raw Cut curvature if it is moved.

The unrestricted lemma is false

Let (B_{\mu\nu}) be a two-index tensor surrounding a pair of null legs with momenta (p,-p). Physical sewing gives

[ \operatorname{Sew}_{\Pi_p(q)}B

-\eta^{\mu\nu}B_{\mu\nu} +\frac{q_\nu p_\mu B^{\mu\nu} +q_\mu B^{\mu\nu}p_\nu}{p\cdot q}. ]

Take four-dimensional rational Minkowski space with

[ \eta=\operatorname{diag}(1,-1,-1,-1), \quad p=(1,0,0,1), \quad r=(1,0,0,0), ]

and (B=r\otimes r). For the two null references

[ q_0=(1,0,0,-1), \qquad q_1=(1,1,0,0), ]

the exact sewings are

[ \operatorname{Sew}{\Pi_p(q_0)}B=0, \qquad \operatorname{Sew}{\Pi_p(q_1)}B=1. ]

Thus no reference-free resolved scalar can equal both sewings. All three momenta are null where required; what fails is the Ward condition. This is the smallest counterexample: one pair of legs and a rank-two tensor.

Even the scalar condition

[ p_\mu p_\nu B^{\mu\nu}=0 ]

does not imply Ward alignment as a matter of linear algebra. With the same (p,r), take (t=(0,1,0,0)) and (B=r\otimes t). Then (pBp=(p\cdot r)(p\cdot t)=0), but (p_\mu B^{\mu\nu}=t^\nu) is not proportional to (p^\nu). The two projector sewings are again different, now (0) and (-1). Therefore the passage from a double contraction identity to the aligned equations requires the physical Ward statement for the blob; it is not a consequence of (pBp=0) alone.

It also identifies a scope error to avoid. An arbitrary tensor built from on-shell momenta, an individual metric-pairing sector, or an individual off-shell cubic Feynman diagram is not automatically an on-shell physical blob. Ward invariance cannot be assigned to it merely because the full amplitude or completed sector sum is gauge invariant.

Corrected one-edge theorem

Let (R) denote all other open physical legs, and quotient tensors in those legs by their gauge ideal. Write this quotient as (Q_R). A two-index cut blob (B_{\mu\nu;R}) is Ward aligned at (e) if there are classes (N_e,N’_e\in Q_R) such that

[ p_\mu[B^{\mu\nu}] = N_e p^\nu, \qquad [B^{\mu\nu}]p_\nu = N’_e p^\mu \quad\text{in }Q_R. ]

This is weaker than literal transversality of the two-index tensor. It is the condition used in equations (39)–(41) of Carrôlo–Figueiredo for closing two legs of the same on-shell object.

Theorem: Ward-quotient one-edge closure

For a Ward-aligned blob,

[ \boxed{ \operatorname{Sew}_{\Pi_p(q)}[B]

-\eta^{\mu\nu}[B_{\mu\nu}]+N_e+N’_e } ]

in (Q_R), independently of (q).

Proof

Insert the two aligned Ward identities into the longitudinal part of the projector:

[ \frac{q_\nu p_\mu[B^{\mu\nu}]}{p\cdot q}

\frac{N_e q\cdot p}{p\cdot q} =N_e, ]

and similarly the reverse term is (N’_e). The remaining term is the metric contraction. No Ward identity for an individual contraction sector has been used. The theorem applies to the completed on-shell blob class in (Q_R).

For a separating tree sewing (B=A_\mu C_\nu), the two completed on-shell subobjects obey the ordinary Ward identities, so (N_e=N’_e=0). For a nonseparating closure of two legs of one object, these coefficients can be nonzero and are precisely the longitudinal correction that must be represented by the surface rule.

Exact factorization of the obstruction

Choose proposed aligned parts (N_ep^\nu,N’_ep^\mu) and define the Ward remainders

[ W^\nu=p_\mu B^{\mu\nu}-N_ep^\nu, \qquad W’^\mu=B^{\mu\nu}p_\nu-N’_ep^\mu. ]

Then the exact defect from the reference-free Ward formula is

[ \boxed{ \Omega_e(q;B)

\frac{q\cdot W+q\cdot W’}{p\cdot q}. } ]

Consequently the obstruction factors entirely through the Ward remainders. For a physical blob they vanish in (Q_R), even when a chosen tensor representative has terms in the gauge ideal of other still-open legs.

The curve-cover identity needs one additional, genuinely geometric realization condition. If (F_E) denotes physical Gram specialization of the resolved open cover, require

[ \boxed{ F_{E\setminus{e}} \operatorname{Gl}^{\rm res}_e(x)

-\eta:F_E(x)+N_e(F_E(x))+N’_e(F_E(x)). } \tag{ER_e} ]

Under ((ER_e)), the desired formula follows:

[ \operatorname{Sew}{\Pi(p;q)} \operatorname{sp}\widetilde\Phi{G\setminus e}

\operatorname{sp}\operatorname{ev}_\rho \widetilde\Phi_G. ]

This is a conditional all-environment theorem. The universal endpoint quadratic identity of entry 52 proves the open-strand metric part. The marked-theta symbolic checker proves the complete condition, including the longitudinal coefficients, for that graph. Those facts do not by themselves prove ((ER_e)) for every cubic graph and every partially sewn environment.

Two-edge closure coherence

Let (e=(p,-p)) and (f=(k,-k)) be two distinct open pairs of a tensor (T_{\mu\nu\alpha\beta;R}). On tensor representatives,

[ \begin{aligned} \operatorname{Sew}_e\operatorname{Sew}fT &= \Pi_p(q_p)^{\mu\nu} \Pi_k(q_k)^{\alpha\beta} T{\mu\nu\alpha\beta},\ &= \operatorname{Sew}_f\operatorname{Sew}_eT. \end{aligned} ]

This is strict interchange of contractions on disjoint index pairs. The resolved gluing maps obey the same interchange law before evaluation.

The exact additional condition needed for an edgewise proof is not vanishing of the double-longitudinal term. It is closure stability of Ward alignment: the Ward equations for (f) must hold in the quotient by all other open-leg gauge ideals before closing (e), and their coefficient maps must be natural,

[ N_f(\operatorname{Sew}_eT)

\operatorname{Sew}_eN_f(T), \qquad N’_f(\operatorname{Sew}_eT)

\operatorname{Sew}_eN’_f(T), ]

with the analogous equations after exchanging (e) and (f). These identities are automatic when the Ward equations are identities in the multi-leg physical quotient and sewing is the categorical trace there. They are not automatic for arbitrary unreduced tensor representatives.

Equivalently, the required square is

[ \begin{array}{ccc} \mathsf{Cov}^{\rm res}(G\setminus{e,f}) &\xrightarrow{\operatorname{Gl}^{\rm res}e}& \mathsf{Cov}^{\rm res}(G\setminus f) \[3pt] \big\downarrow{F{e,f}} && \big\downarrow{F_f} \[3pt] Q_{e,f} &\xrightarrow{\operatorname{Sew}_e}& Q_f, \end{array} ]

together with the corresponding (f)-square. Their common composite is the two-edge coherence condition.

Expanding each projector as (M+L), both orders contain

[ M_eM_f, \qquad L_eM_f, \qquad M_eL_f, \qquad L_eL_f. ]

The (L_eL_f) term can be nonzero. The new small checker has an exact example with (L_eL_f=1) and a commuting final square. More importantly, the marked-theta certificate has four spanning-tree presentations with

[ [M,L_1,L_2,L_1L_2]=[-2056,8,8,-8]. ]

Thus nested longitudinal vanishing is neither the coherence law nor a valid universal induction hypothesis.

What happens to cycle-rank induction

Choose a spanning tree of a connected graph of cycle rank (L), leaving (L) edge pairs to close. The corrected one-edge theorem yields an induction on (L) if all of the following hold:

the tree-level resolved curve map realizes the physical tree blob;
every partially sewn blob is a class in the multi-leg physical quotient;
Ward alignment is stable under every remaining closure;
((ER_e)) holds for every edge in every such environment;
resolved carriers are retained until all required Cut operations are past.

Under these hypotheses, one-edge naturality and the two interchange laws make the result independent of closure order, and induction proves the fully sewn identity.

This is a valid conditional theorem, not yet an unconditional all-graph theorem in the present ledger. The missing step is exactly hypothesis 4 in an arbitrary multi-leg environment. Testing more final closed graphs cannot replace it; the decisive next certificate must retain at least two open state pairs and verify the quotient-valued Ward/naturality equations before either closure.

A representative-level two-closure warning

The new checker also gives a minimal reason not to scalarize intermediate representatives. Take

[ T=(r\otimes r)\otimes(k\otimes k). ]

The first factor has the reference-dependent one-edge values (0) and (1) above, while the second factor is pure gauge and is annihilated by (Pi_k). Hence both final double closures are zero and the physical square commutes, but one intermediate representative depends on the first reference. The correct intermediate object is its class in the remaining physical quotient, where the pure-gauge factor is already zero.

This is the two-edge analogue of the resolved-circuit warning: equality after final evaluation does not justify forgetting the structure required by the next operation.

Necessary clarification to entry 52

Entry 52 says, immediately after the endpoint identity, that multiplication over strands gives the metric-sector tensor contraction. Read literally as the full cubic all-metric network, this conflicts with its later theorem and with its checker.

The checker has two distinct handle functions:

full_handle       -> naive_metric_polynomial
reduced_handle    -> graphical_polynomial

and verifies that the two polynomials are unequal. The curve-cover endpoint identity realizes the gauge-reduced graphical network. It does not equal the naive full-handle all-metric network; the physical projectors supply the longitudinal correction that relates the full cubic tensor calculation to the reduced graphical result. Entry 52 should be read with this qualification. No change to entry 52 is made here.

Evidence and scope

Proved here:

the unqualified rank-two arbitrary-tensor lemma has an exact null-kinematic counterexample;
the Ward-quotient one-edge formula;
exact factorization of its defect through Ward remainders;
strict two-projector interchange;
nonvanishing nested longitudinal terms are compatible with coherence;
the precise hypotheses under which cycle-rank induction is valid.

Evidence, not an all-graph proof:

entry 52’s 24 symbolic marked-theta presentations realize the complete physical/curve identity;
its exact denominator cancellation shows reference independence on that cell;
the source paper supplies the graphical left-turn rule for the one-loop Ward coefficients in arbitrary on-shell attachments.

Not proved:

that every origin-resolved cubic graph cell lands in the required multi-leg physical quotient before all loop closures;
((ER_e)) for every edge and arbitrary partially sewn environment;
an all-graph induction from the marked-theta result alone;
any Cut-commuting theorem after premature (D)-evaluation.

Next executable test

Use the smallest tree leading singularity with four distinguished forward legs (p,-p,k,-k) and at least one generic physical external attachment. Retain the rank-four tensor and the origin-resolved curve cover. Verify:

the Ward equations for the (p)-pair modulo the gauge ideal of the (k)-pair, and conversely;
the two coefficient naturality equations for (N,N’);
both one-edge realization squares before the second closure;
the complete (M_eM_f,L_eM_f,M_eL_f,L_eL_f) decomposition;
equality of the two closure orders without evaluating any carrier needed by a later Cut.

The smallest failure among these equations is the true obstruction to the all-graph induction. A nonzero (L_eL_f) by itself is not a failure.

Primary source

Carrôlo and Figueiredo, How gluon leading singularities discover curves on surfaces, especially equations (39)–(41), the exclusively-left-turning rule, the arbitrary on-shell attachment statement at one loop, and the higher-loop projector discussion: https://arxiv.org/html/2512.17019.

Internal dependencies

Entry 46: resolved closed-circuit carrier.
Entry 49: premature-evaluation Cut defect.
Entry 51: exact nonzero nested marked-theta correction.
Entry 52: symbolic marked-theta physical realization and strict resolved Cut/gluing naturality.