Origin-Level Falsification and the Derived Ward–Brauer Dictionary

Record

Date: 2026-08-13

Status: the naïve originwise strengthening of entry 54 is falsified exactly. The physical/curve realization squares commute after summing the complete five-vertex cubic-sector carrier, but not on the raw ({0,1,2}^{5}) origin basis, not after summing either endpoint of a closed edge, and not after summing both endpoints. For both one-edge and two-edge closure, every audited presentation requires transport across all five local cubic-sector coordinates.

This does not undo entry 54. It identifies the type of its missing lift: the surface dictionary must be a derived natural transformation carrying explicit Ward/V homotopies between cubic-sector origins. It cannot be the identity map on origin labels followed by ordinary augmentation.

Reproducible certificate:

research/nima/check_two_open_pair_ward_naturality.rs

The raw origin module

Each of the five Yang–Mills cubic vertices is decomposed into its three metric/handle sectors. Before summing the vertex tensor, the origin set is

[ \Omega={0,1,2}^{5}, \qquad |\Omega|=3^5=243. ]

For a spanning-tree presentation with closure edge (e), let

[ R_e(\omega)

\operatorname{Sew}^{\rm phys}e T^{\rm full}{\omega}

\operatorname{Gl}^{\rm met}e T^{\rm red}{\omega} ]

be the polynomial-valued residual while the second edge pair remains open. Entry 54 proves

[ \sum_{\omega\in\Omega}R_e(\omega)=0. ]

The new audit asks whether this equality is already supported on the two vertices incident to (e). For a set (S) of freely summed vertex-sector coordinates, fix every coordinate outside (S) and test

[ \sum_{\omega|_S}R_e(\omega)=0 ]

as an exact polynomial identity in the 48-variable Gram-free chart.

The same test is applied after both closures, using the union of the endpoints of the two closure edges.

Exact falsification

There are

[ 12\ \text{trees} \times 2\ \text{references} \times 2\ \text{first-closure choices} =48 ]

partially sewn presentations. Across their (48\times243=11{,}664) raw origins:

9,471 have a nonzero patternwise residual;
summing one closure endpoint leaves 6,903 nonzero fixed-environment defects out of 7,776 tests;
summing both closure endpoints leaves 1,200 nonzero fixed-environment defects out of 1,296 tests;
in all 48 presentations, the smallest coordinate set whose every fixed-environment residual vanishes has size five.

After both closures there are 24 tree/reference presentations. Across their (24\times243=5{,}832) raw origins:

4,491 have a nonzero patternwise residual;
the complete closure-endpoint union fails in all 144 fixed environments;
in all 24 presentations, the smallest closing coordinate set again has size five.

Thus neither of the candidate strict statements

[ R_e(\omega)=0 ]

[ \sum_{\omega|_{\partial e}}R_e(\omega)=0 \quad\text{for every fixed environment} ]

is true.

What exactly has been falsified

The audit rules out the diagonal origin map which sends a full cubic-sector word to the reduced word with the same five singleton labels, even after the ordinary endpoint-sector quotient. It also rules out any proof that uses only augmentation over the closure-incident vertex labels while treating the other vertex origins as spectators.

It does not rule out an off-diagonal map that transports one origin to a signed combination of neighboring origins. Nor does it rule out a local chain map once the local Ward/V relations are retained as one-cells. Those are now the only viable forms of origin-resolved strictification.

The required derived carrier

Let the three sectors at one cubic vertex be the vertices of a filled 2-simplex:

[ C_\bullet^{\rm cub}=C_\bullet(\Delta^2), \qquad d[i,j]=e_j-e_i, \qquad \epsilon(e_i)=1. ]

For the marked handle, the minimal formal resolution is

[ K_\bullet

\left(C_\bullet^{\rm cub}\right)^{\otimes5}. ]

Because (K_\bullet) is an augmented contractible complex, the already-proved total identity implies abstractly that each residual admits a one-chain

[ H_e\in K_1\otimes\mathcal R, \qquad dH_e=R_e. ]

But this abstract existence is not the desired theorem. A freely adjoined simplex would be as tautological as a formal cylinder. The nontrivial task is to realize:

each edge ([i,j]) by the actual local cubic Ward/V identity changing the singleton sector at a specified vertex;
each coefficient by the resolved endpoint-extension or curve carrier;
every 2-simplex and product square by the coherence between two such transports;
the resulting chain map compatibly with either partial physical trace and with surface Cuts.

Only such a realized augmentation upgrades the polynomial equality to an intrinsic carrier map.

The non-tautological local generator is a moving Ward mark

There is a canonical candidate for the one-cells. For the ordinary cubic gluon vertex with outgoing momenta (p+q+r=0),

[ V_{\mu\nu\rho}

\eta_{\mu\nu}(p-q)\rho +\eta{\nu\rho}(q-r)\mu +\eta{\rho\mu}(r-p)_\nu, ]

direct contraction gives the off-shell Ward identity

[ \boxed{ p^\mu V_{\mu\nu\rho}

P_{\nu\rho}(r)-P_{\nu\rho}(q), \qquad P_{\nu\rho}(k)=k^2\eta_{\nu\rho}-k_\nu k_\rho .} ]

On the massless three-point locus this becomes

[ p^\mu V_{\mu\nu\rho} =q_\nu q_\rho-r_\nu r_\rho. ]

A longitudinal mark entering one half-edge is therefore the signed boundary of the two ways it can leave through the other half-edges. The mark propagates locally from vertex to vertex until it reaches a transverse external state or closes around a circuit. This explains why the residual has global support in the vertex-sector coordinates without requiring a nonlocal physical interaction.

It also replaces the formal simplex by a testable carrier. Let (\mathsf W_\bullet(G)) be generated by cubic-sector words together with a marked oriented half-edge, with differential given by the displayed Ward identity. The actual task is to construct a comparison

[ \mathsf W_\bullet(G) \longrightarrow \left(C_\bullet(\Delta^2)\right)^{\otimes V(G)} ]

whose image of a moving mark is the endpoint-extension transport. If this comparison exists, the all-five-coordinate support measured above is the expected footprint of a local mark traversing the connected marked-theta network. If it does not, the simplex filler remains formal bookkeeping.

Revised Ward–Brauer target

The source of the dictionary is not merely

[ \bigotimes_{e\in E}\langle M_e,L_e^+,L_e^-\rangle. ]

It must also retain the cubic-sector resolution:

[ \boxed{ \mathcal K_E

\left(C_\bullet(\Delta^2)\right)^{\otimes V(G)} \otimes \bigotimes_{e\in E} \langle M_e,L_e^+,L_e^-\rangle .} ]

The formal source must therefore be physically realized by the marked Ward complex. The desired surface dictionary is a derived transformation

[ \Phi_E: \mathsf W_\bullet(G) \otimes \bigotimes_{e\in E}\langle M_e,L_e^+,L_e^-\rangle \longrightarrow \mathsf{Cov}^{\rm res}(G;E) ]

such that augmentation gives the entry-54 trace-strict representative and each edge trace commutes with (\Phi_E) up to the specified one-chain (H_e). For two closures, the difference between the two composites is a cycle which must be filled by a specified two-chain. This is the first place where genuine higher coherence, rather than final equality, is unavoidable.

Relation to Nima’s talk

The transcript of Scattering Amplitudes and Dualities at Infinity emphasizes that the self-factorizing carrier precedes the function and that a separate dictionary transports carrier factorization into amplitude factorization. The present falsification is a concrete instance of that distinction:

after valuation/augmentation, the physical and surface polynomials agree;
on the raw carrier origins, the diagonal dictionary fails;
the missing information is precisely the transport between origins.

Thus the correct new claim is not that the surface formula is termwise equal to physical-projector sewing. It is that the two may be related by a derived, sewing-natural dictionary on a resolved carrier.

Next executable test

Construct an integral one-chain (H_e) for each of the 48 partial presentations using only nearest-neighbor changes of one cubic singleton. Then impose three non-tautological restrictions:

every edge coefficient must be generated by the corresponding local three-gluon Ward/V relation;
the construction must be covariant under the order-three road rotation and independent of the null-reference choice up to an admitted boundary;
for the two closure orders, the induced one-cycle must be the boundary of a curve-resolved two-chain.

Failure of the first restriction means the formal simplex resolution is only homological bookkeeping. Failure of the third means entry 54 has no coherent origin-level lift despite its exact polynomial naturality.

Internal dependencies

Entry 51: literal endpoint-extension origins and homotopy-sensitive curve labels.
Entry 52: final symbolic physical/graphical identity.
Entry 53: corrected Ward-quotient closure theorem.
Entry 54: exact two-open-pair partial realization and closure stability.
Source annotations: research/sources/nima/talks/scattering-amplitudes-and-dualities-at-infinity/annotations.md.