Trading Global Associativity for Recoverable Local Boundaries in Full-Color Nonplanar QCD Amplitudes
Executive summary
The most promising version of the program is not to replace QCD color with a fundamentally nonassociative algebra at the outset. It is to keep ordinary color exactly as it is, but to enlarge amplitude representations so that every local color-flow patch carries an explicit bracketing datum and neighboring patches are related by associator maps rather than by an implicit assumption of one global associative composition law. In planar settings, a single cyclic order makes locality, factorization channels, and bracketings mutually compatible; this is why the associahedron and its canonical form organize tree-level amplitudes so efficiently. In full-color nonplanar QCD, that compatibility is lost: loop amplitudes require multiple trace/flow sectors, and recent full-color calculations show that nonleading-color coefficients and rational prefactors proliferate even when the function space is under control. The obstruction is therefore best understood as a mismatch between local boundary data and global associativity, not as a failure of locality itself. citeturn6view0turn23view0turn5view3turn5view5turn25view1turn5view0
The mathematically sharp way to implement this is a semistrict, chart-based model. Each chart is a bracketed color-flow graph or double-line patch carrying primitive-amplitude data; overlaps between charts are governed by associator matrices assembled from Jacobi, Kleiss–Kuijf, BCJ, shuffle, and trace/color-flow identities. The natural frameworks are, in order of priority, a colored operad / associahedral complex for local compositions, a monoidal category with explicit associators for gluing patches, and an (A_\infty/C_\infty/L_\infty) layer for controlled failures of strict associativity and for mapping those failures into kinematic numerator defects. A semistrict higher-categorical layer is useful for coherence auditing, while quasi-Hopf or braided structures should be added only if branch-cut or crossing data force genuine braiding. Octonions and division algebras are best treated as a diagnostic overlay—a source of latent (SU(3)) selection rules and nonassociative labels—rather than as the primary computational algebra. citeturn5view3turn5view4turn6view0turn5view2turn32view0turn23view6turn23view7turn23view8
The best launch targets are exactly the ones you specified, but with a staged helicity strategy. First, tree-level six-gluon full-color is the cleanest place to prototype the bracketing machinery because it has rich cubic factorization structure but no loop integration. Second, one-loop four-gluon full-color is the minimal integrated test with nontrivial trace mixing and universal infrared poles. Third, one-loop five-gluon full-color is where the method becomes genuinely informative: one can test subleading-color coefficients, rational finite sectors such as all-plus, and IR-sensitive sectors such as MHV or single-minus. Success should be judged not only by exact reconstruction of known amplitudes, but by whether the bracketed model yields compression, predictivity, or a clean localization of color-kinematics defects that ordinary bases obscure. citeturn22view0turn10search0turn23view1turn6view2turn31view0turn11search1
Geometry of the obstruction
Planar is easy because a fixed cyclic order turns all admissible factorization channels into noncrossing diagonals of one (n)-gon, and the compatible bracketings of that ordered product form a single associahedron. In the ABHY picture, the tree amplitude is the canonical form of this positive geometry, and its boundaries are precisely the physical factorization channels. That is the deep reason planar locality feels “boundary visible”: one global ordering already does most of the bookkeeping for you. citeturn6view0turn6view1
Full-color nonplanar QCD breaks that compatibility in three different ways at once. The trace basis keeps explicit external boundary orderings but splits the amplitude into several trace sectors; the color-flow basis keeps explicit color lines but not an intrinsic record of the local parenthesization by which those lines were assembled; and color-dressed cubic representations keep Jacobi relations manifest but allow generalized gauge choices that can hide the very local boundaries one would like to track. Loop-level full-color constructions accordingly do not reduce to a single ordered object; instead they retain color on unitarity cuts and reconstruct the amplitude from many locally compatible sectors. Recent leading-colour six-gluon and full-color five-parton results show the same pattern from another angle: even when the functional space is known, the rational prefactors and subleading-color structure proliferate. citeturn23view0turn23view1turn5view5turn26view0turn25view1turn5view0turn34search0
That diagnosis explains why existing solutions “fail” for your program. They are not wrong; they simply fuse together two logically distinct tasks. One task is to keep local boundary information visible. The other is to impose a single globally associative or globally ordered composition law. In planar problems those tasks coincide, so the conflation is harmless. In full-color nonplanar QCD they do not. Your proposal is therefore to decouple them: keep local boundaries chart by chart, and replace one global associative product with recoverable reassociation data on overlaps. citeturn6view0turn23view0turn23view1turn25view1
A useful mental picture is that planar amplitudes live on one associahedron, while full-color amplitudes require a network of local associahedra, each attached to a color-flow chart and glued to its neighbors by explicit associators. The recent (\Delta)-algebra work is suggestive here because it provides algebraic moves between nonplanar on-shell diagrams—precisely the sort of local move set one wants for a bracketing complex beyond the planar world. citeturn5view2turn6view0graph LR A["(((12)3)4)"] B["((12)(34))"] C["((1(23))4)"] D["(1((23)4))"] E["(1(2(34)))"] A --- B A --- C C --- D D --- E B --- E
The pentagon above is the elementary bracketing complex for four inputs. In the planar case, one such complex suffices for a fixed ordering. In the full-color case, the program is to attach many such complexes to local flow patches and solve the gluing problem between them. citeturn6view0turn32view0
Frameworks for semistrict boundary recovery
The core object should be a local chart
[
U_\alpha=(\Gamma_\alpha,\;T_\alpha,\;\mathcal B_\alpha),
]
where (\Gamma_\alpha) is a color-flow or double-line graph, (T_\alpha) is a rooted binary forest recording how local color subblocks are bracketed, and (\mathcal B_\alpha) is the vector of primitive-amplitude coefficients compatible with that chart. The amplitude is then a global section of a sheaf-like or prestack-like assignment (U_\alpha\mapsto \mathcal F(U_\alpha)), with transition maps
[
\Phi_{\alpha\beta}(s,\epsilon):\mathcal F(U_\alpha)\to \mathcal F(U_\beta)
]
generated by elementary reassociation, Jacobi, shuffle, and trace/flow moves. This is the simplest precise meaning of “trade global associativity for recoverable local boundaries.” The frameworks below differ mainly in how they encode the (\Phi_{\alpha\beta}) and their coherence. citeturn5view3turn6view0turn23view0
The comparison below is a synthesis of the amplitude papers, higher-structure papers, and tool documentation most relevant to implementation. The essential source-level facts are that tree amplitudes can be encoded via homotopy transfer of (L_\infty) and (C_\infty) structures; weak (2)-groups are monoidal categories with associators and unitors; nonplanar on-shell diagram moves can be organized algebraically; and Catlab/homotopy.io already provide computational infrastructure for monoidal and higher-categorical bookkeeping. citeturn5view3turn32view0turn5view2turn22view9turn23view5 Framework What it encodes naturally Best role in this program Main advantage Main risk Monoidal category with explicit associator Local gluing of color-flow patches and reassociation between neighboring bracketings Primary bookkeeping layer Minimal overhead; direct control of local-to-local transitions Too weak if higher coherence or gauge homotopies matter Colored operad plus associahedra Trivalent merge/split operations and all tree bracketings for fixed local input sets Primary local composition layer Factorization channels and recursive composition are native Needs an extra layer for overlap/gluing between different flow charts (A_\infty/C_\infty/L_\infty) algebra Controlled failure of strict associativity via higher products (m_n) or brackets (\ell_n) Primary defect layer Closest to how tree amplitudes already arise by homotopy transfer Abstract unless tied to explicit chart data Semistrict (2)- or (n)-category Coherence between different reassociation paths Audit / proof layer Makes overlap consistency and pentagon data explicit Heavy formalism for the first working prototype Braided / quasi-Hopf / weakly associative algebra Reassociation plus braiding or monodromy data Conditional extension Natural if overlap maps need both (R)- and (\Phi)-data Likely overkill before branch-cut/crossing issues are isolated Octonionic / division-algebra overlay Nonassociative labels and latent (SU(3)) selection rules Exploratory diagnostic Might reveal hidden local triplet structure Not a direct amplitude basis for QCD
The practical priority ordering is therefore clear. Start with a colored operad of bracketed flow patches for local compositions and a monoidal-category layer for overlap maps. Add an (A_\infty) correction layer when exact reassociation fails but the failure is controlled and local. Only after that should one promote the overlaps to a semistrict higher category or import quasi-Hopf braiding. This ordering follows directly from the fact that homotopy transfer already computes tree amplitudes, while monoidal and higher-categorical tools already have usable computational libraries. citeturn5view3turn22view9turn23view5
The octonionic clue from Cohl Furey is real but should be handled cautiously. Furey’s constructions show that complex octonions and the associated Clifford-algebraic machinery organize (SU(3)) representations and division-algebraic ladder operators in a way that mirrors chromodynamic structure, including three-generation patterns. That does not mean QCD amplitudes themselves should be rewritten octonionically. It does mean that nonassociative algebra can carry latent color selection rules, so octonionic associators are a plausible label set for where local bracketing defects ought to concentrate. citeturn23view6turn23view7turn22view13turn23view8
Targets and bases
Your initial target list is exactly right, but the first implementation should separate “boundary bookkeeping” tests from “subleading-color” tests. Tree six-gluon is the right place to learn whether the model is mathematically sound; one-loop four- and five-gluon are the right places to learn whether it is physically useful. The natural helicity progression is tree six-gluon MHV (\to) tree six-gluon NMHV (\to) one-loop four-gluon all-plus and MHV (\to) one-loop five-gluon all-plus (\to) one-loop five-gluon MHV/single-minus. That sequence moves from no integrals to finite rational integrals to full IR-sensitive integrated amplitudes. citeturn22view0turn31view0turn11search1 Target amplitude Why it is first-rate for this program What it tests best Best first helicity sectors Tree-level six-gluon full-color First nontrivial tree target with many cubic channels and bracketings but no loop integration Pure boundary/bracketing reconstruction, factorization, KK/BCJ/Jacobi coherence MHV, then NMHV One-loop four-gluon full-color Smallest integrated target with nontrivial color mixing and universal IR structure Transition maps between trace, flow, and cubic bases; Catani poles All-plus, then MHV One-loop five-gluon full-color First truly interesting loop target for subleading color and protected finite sectors Subleading-color coefficients, rational sectors, withheld-coefficient prediction All-plus, then MHV or single-minus Later stress test: two-loop five-parton / full-color splitting Modern data set with genuine nonplanar/full-color complexity Whether the framework scales beyond the prototype horizon Full-color benchmarks only after one-loop success
The reason to start tree level with six gluons, rather than four or five, is structural rather than phenomenological. Four points are too rigid, and five points are close to minimal. Six points are the first place where the bracketing complex is large enough to be informative while still being fully manageable analytically. At tree level, all-gluon amplitudes are known in compact color-ordered form, and DDM-type decompositions already isolate the independent ordered objects. That makes six-gluon tree amplitudes an ideal “unit test” for any augmented representation. citeturn22view0turn10search0
For the loop targets, one-loop all-plus amplitudes are especially helpful because they are finite and rational, so they isolate structural issues without the full burden of transcendental function spaces. Henn, Power, and Zoia used precisely this finiteness/rationality to expose hidden conformal structure in the one-loop all-plus sector. In contrast, MHV sectors bring back the universal infrared poles predicted by Catani, making them the correct second-stage test of whether the bracketed framework respects factorization and IR universality. citeturn31view0turn11search1
The basis choice should also be staged. At tree level, use cubic color-dressed / DDM data as the source of bracketing and trace/color-flow as target projections. At one loop, use primitive amplitudes plus trace structures as the authoritative integrated representation, and treat color-flow charts as the boundary-carrying intermediary. The recent literature now spans all of these bases: DDM at tree level, color-flow formulations equivalent to usual QCD to all orders, loop-level full-color constructions from unitarity cuts, compact one-loop color decompositions, and recent work systematizing trace-vs-structure-constant relations at higher loop order. citeturn10search0turn23view1turn23view0turn24search0turn26view0 Basis Natural object Keeps local boundary data? Keeps local bracket data? Where it should be used Trace basis Cyclic words and products of traces Partly No Integrated amplitude output and comparison to literature Color-flow basis Oriented double lines, products of Kronecker deltas Yes Not by itself Local boundary charts and overlap geometry Cubic color-dressed basis Trivalent graphs with (f^{abc}) color factors Indirectly Yes Source of bracket trees, Jacobi relations, BCJ diagnostics Primitive amplitudes Gauge-invariant ordered loop building blocks Partly Partly One-loop chart coefficients and test data
The central move is to replace each ordinary basis element (C) by an augmented element ((C,T)), where (T) is the rooted binary tree or bracketing forest inherited from the cubic or color-flow source chart. Translation between bases should then proceed in five explicit steps.
- Generate the cubic or generalized-unity cut graph (g) and its color factor (c_g), keeping vertex incidence and propagator-channel labels. citeturn23view0turn27view0
- Resolve every adjoint edge into double-line or color-flow data, using a color-flow representation equivalent to standard QCD, and record the local boundary components this creates. citeturn23view1turn5view5
- Attach to each connected boundary component a local rooted binary tree (T) that records the order in which the component was assembled from trivalent patches. For tree amplitudes this is directly the factorization tree; at one loop it becomes a bracketing forest plus a loop insertion tag. This is the step current bases usually omit. citeturn6view0turn5view3
- Project the augmented chart ((\Gamma,T)) to trace or primitive bases, but do not discard (T). Instead, store the projection map (P_{(\Gamma,T)\to \lambda}) so that every trace or primitive coefficient remembers which local boundary chart produced it. citeturn23view0turn24search0turn26view0
- Fit or derive overlap maps between charts as matrices (\Phi_{\alpha\beta}(s,\epsilon)). These are the associators of the prototype. They should be sparse, local, and compatible with Jacobi, factorization, and color identities. citeturn23view3turn23view0turn23view2
Prototype architecture
The proposed prototype is best thought of as an associahedra sheaf over color-flow configuration space. Each color-flow patch carries a bracketing fiber; overlaps carry explicit transition maps; the physical amplitude is a global section that is independent of the chosen local chart. This is a genuine analytical program, not just a new data structure, because the overlap maps themselves are physical objects constrained by cuts, poles, and color identities. citeturn6view0turn23view0turn5view3flowchart LR subgraph Base["Color-flow chart space"] U1["Uα: local flow patch"] U2["Uβ: local flow patch"] U3["Uγ: local flow patch"] end subgraph Fibers["Bracketing fibers"] K1["Kα: local associahedron"] K2["Kβ: local associahedron"] K3["Kγ: local associahedron"] end U1 --> K1 U2 --> K2 U3 --> K3 K1 <-->|"Φαβ"| K2 K2 <-->|"Φβγ"| K3 K1 <-->|"Φαγ"| K3 G["Global section = full amplitude"] --- K1 G --- K2 G --- K3
A minimal algebraic implementation is to define, for each chart (U_\alpha),
[
\mathcal A_\alpha=\sum_{i} b_{\alpha,i}(s,\epsilon)\,e_{\alpha,i},
]
where (e_{\alpha,i}) are local primitive/chart basis elements. On overlaps,
[
\mathcal A_\beta=\Phi_{\alpha\beta}\,\mathcal A_\alpha .
]
The first coherence condition is a Čech-style cocycle condition on triple overlaps,
[
\Phi_{\beta\gamma}\Phi_{\alpha\beta}=\Phi_{\alpha\gamma}.
]
The second is an elementary pentagon condition whenever five reassociation paths exist between the same bracketings. The third is pole compatibility:
[
\operatorname*{Res}{s_I=0}\Phi{\alpha\beta}
\Phi^{(L)}{\alpha\beta}\otimes \Phi^{(R)}{\alpha\beta},
]
meaning that reassociation must commute with factorization into lower-point charts. These are the key equations that make the model nontrivial. citeturn6view0turn32view0turn11search1
The prototype should store four principal classes of objects: Object Required fields Function Chart external labels, flow graph, rooted trees, loop tags, primitive coefficient vector local boundary/bracketing datum Move source chart, target chart, move type, sparse matrix, kinematic domain elementary reassociation or basis change Defect violated relation, support charts, residue data, gauge-shift metadata stores where associativity/coherence fails Section family of chart vectors plus glue maps candidate full amplitude
The architecture below is the right dataflow for a first working code base.flowchart TD A["Known amplitudes or generalized cuts"] --> B["Generate cubic / color-flow graphs"] B --> C["Attach local rooted trees and loop tags"] C --> D["Project to primitive / trace bases"] D --> E["Solve for associator matrices Φαβ"] E --> F["Finite-field reconstruction of rational entries"] F --> G["Coherence tests: pentagon, Jacobi, factorization, IR"] G --> H["Global section / reconstructed amplitude"] G --> I["CK-defect and octonionic diagnostic layers"]
In code, the fastest route is a hybrid stack. Use FORM or Mathematica/Python for color algebra and symbolic transformations; use Catlab for typed wiring-diagram objects and serialized composition data; use finite-field linear algebra to determine overlap maps pointwise; and use homotopy.io only for small-instance proofs that the generated move complexes really satisfy the intended coherence relations. This is precisely where existing category-theory tooling is already strong enough to be useful without forcing the entire project into a proof-assistant workflow. citeturn22view8turn22view9turn23view5turn29search7
Diagnostics, duality, and octonions
The tests should be chosen so that each one isolates one possible failure mode. Tree-level tests tell you whether the boundary/bracketing model is mathematically self-consistent. One-loop tests tell you whether it survives integration, trace mixing, and infrared universality. Beyond that, modern full-color five-parton and splitting results provide a stretch goal for later validation on genuinely nonplanar structures. citeturn22view0turn11search1turn5view0turn34search0turn23view9 Target Observable or coefficient to test Why it matters Tree 6g Exact reconstruction in DDM/trace/color-flow projections Confirms that local charting is not losing information Tree 6g Residues on all two- and three-particle poles Confirms boundary data are compatible with factorization Tree 6g KK and BCJ relations in bracketed form Confirms overlap maps know about color algebra One-loop 4g Single- and double-trace coefficients Smallest test of integrated full-color basis changes One-loop 4g Catani (\mathbf I^{(1)}) pole structure Confirms IR universality after gluing charts One-loop 5g all-plus Finite rational coefficients Cleanest subleading-color benchmark with no IR poles One-loop 5g MHV/single-minus Pole structure plus finite remainder Tests whether the method survives generic integration data Later extension Full-color two-loop 5-parton or 2-loop splitting data Stress test against known nonplanar/full-color modern results
The color-kinematics interpretation is especially important because it turns “associator defects” into physically meaningful objects. BCJ says that color factors of cubic graphs satisfy Jacobi relations and asks for kinematic numerators obeying the same algebra. In a bracketed chart model, it is more natural to expect
[
n_s-n_t-n_u=\Delta_\Phi ,
]
where (\Delta_\Phi) measures the failure of the chosen local reassociation data to match a strict BCJ-compatible numerator assignment. If (\Delta_\Phi=0), you have a genuinely color-dual chart system. If (\Delta_\Phi\neq0) but is exact in the homotopy complex or removable by generalized gauge shifts, then the obstruction is bookkeeping, not physics. If (\Delta_\Phi) survives all such reductions, it is a genuine double-copy obstruction. Mogull and O’Connell’s two-loop five-gluon analysis—where extra loop-momentum dependence was needed to reconcile cuts with graph symmetries—is exactly the sort of phenomenon this defect language is meant to localize. citeturn9search0turn9search1turn23view2turn9search2turn6view1
The octonionic/division-algebra layer should be added only after the baseline chart model works. Baez’s overview emphasizes the octonions precisely because they are nonassociative yet tied to Clifford algebras, spinors, and exceptional symmetry. Furey’s program then shows that complex octonions and their ladder-operator structures naturally realize (SU(3))-like data. The practical proposal is therefore modest and testable: assign to each trivalent local patch an octonionic label or a division-algebraic charge sector, define a local associator score (\sigma_v) from those labels, and ask whether nonzero (\sigma_v) correlates with charts carrying subleading-color coefficients or nonzero CK defects. If there is no correlation, discard the overlay. If there is, you have found a nontrivial selection rule for where reassociation complexity lives. citeturn23view8turn23view7turn23view6turn22view13
A useful medium-term stretch test is not immediately another amplitude, but a limit observable. The 2026 full-color two-loop splitting-amplitude result is important because it cleanly identifies collinear-factorization-violating contributions and shows how they cancel in color-summed squared amplitudes. Once the one-loop prototype exists, the right question is whether your associator defects localize precisely in the sectors that later become CFV terms at higher loops. That is a much sharper validation test than a generic amplitude comparison. citeturn23view9turn11search5
Workflow and resources
The workflow should distinguish between a reorganization prototype and a from-scratch amplitude-generation prototype. The reorganization prototype starts from known analytic amplitudes and asks whether the bracketed chart machinery compresses or clarifies them. The from-scratch prototype starts from cuts or graphs and attempts to reconstruct both amplitudes and associator maps directly. The former is what you should build first. Finite-field reconstruction and numerical-unitarity tools become central only once the chart algebra works at the reorganization level. citeturn23view3turn27view0turn5view0
The software stack below matches that division of labor. The factual tool capabilities come directly from their papers or official documentation: FiniteFlow provides multivariate rational reconstruction and dataflow graphs; FIRE performs IBP reduction, including modular arithmetic; pySecDec provides numerical sector decomposition and integrand libraries; FORM provides large-scale symbolic algebra; Caravel supplies numerical- unitarity infrastructure; Catlab supports monoidal/string-diagram representations; homotopy.io supports finitely presented semistrict higher categories; and Mathematica/Wolfram Language remains the most convenient orchestration layer for symbolic prototyping. citeturn23view3turn22view6turn23view4turn22view8turn27view0turn22view9turn23view5turn29search7 Task Preferred tools Why Color algebra, basis projection, symbolic simplification FORM, Mathematica/Wolfram Language Efficient symbolic handling of very large expressions Bracketing graphs, wiring diagrams, typed compositions Catlab.jl Native support for monoidal/string-diagram data Small coherence proofs and semistrict move validation homotopy.io Direct graphical manipulation of higher-categorical proofs Pointwise fitting of overlap maps FiniteFlow, Mathematica or Python front-end Sparse finite-field reconstruction of rational map entries One-loop integral checks pySecDec Independent numerical verification at benchmark kinematics From-scratch one-/two-loop cut construction Caravel, generalized unitarity stack Native numerical-unitarity modules and finite-field support IBP for extension beyond the first prototype FIRE Standard reduction route once needed
A realistic minimal resource envelope is surprisingly modest if you do not regenerate amplitudes from scratch. Tree six-gluon and one-loop four-gluon reorganizations can be done on a laptop or a modest workstation: 8 CPU cores and 16–32 GB RAM are enough. One-loop five-gluon full-color reorganization with finite-field fitting of overlap maps is more comfortable on a 16–32 core workstation with 64–128 GB RAM. Only when you move to cut-based reconstruction or two-loop stress tests do you need cluster-style resources. citeturn23view3turn27view0turn23view4turn22view6
The one place where I would be conservative is IBP. For the first one-loop prototype, do not let IBP dominate the agenda. For four- and five-gluon one-loop amplitudes the real question is whether the bracketed representation yields better structural organization, not whether you can rediscover scalar boxes and pentagons the hard way. Use FIRE and pySecDec as verification tools and for future-proofing, but not as the main engine until the chart algebra has earned that extra complexity. citeturn22view6turn23view4
Risks and research plan
The biggest failure mode is conceptual rather than technical: the bracketed framework may collapse to a verbose rewrite of known Jacobi/shuffle/trace identities without producing either compression or prediction. That risk is real, and it is why the project needs explicit go/no-go criteria. A second failure mode is overformalization: higher categories or quasi-Hopf structures can swallow the problem before the first numerical check is made. A third is that octonionic labels may be mathematically beautiful but physically irrelevant for amplitudes. The mitigation is to stage the work so that each extra layer is added only after the previous layer has produced measurable value. citeturn23view0turn5view3turn23view7
The plan below is deliberately prioritized toward early falsifiability. Phase Milestone and deliverable Success criterion Phase one Tree 6g chart engine: repository that generates cubic graphs, color-flow charts, rooted-tree annotations, and projections to DDM/trace/color-flow forms Exact reconstruction of known six-gluon tree amplitudes at random kinematics and correct residues in every physical factorization channel Phase two Associator solver: machine-readable overlap matrices (\Phi_{\alpha\beta}) for the tree target, plus automated pentagon/Jacobi tests All elementary coherence tests pass; overlap maps are sparse and local Phase three One-loop 4g full-color prototype: charted reconstruction of single-/double-trace amplitude pieces Reproduces known coefficients and Catani pole structure; no chart dependence remains in the reconstructed amplitude Phase four One-loop 5g full-color benchmark: all-plus first, then MHV/single-minus Correct subleading-color coefficients and successful prediction of a held-out subset of coefficients or helicity sectors from fitted overlap data Phase five CK-defect module: compute (\Delta_\Phi) on Jacobi triplets and compare to generalized-gauge freedom Defects either vanish, become homotopically exact, or localize to a small, interpretable set of charts Phase six Octonionic overlay evaluation Keep only if it correlates with nontrivial defects or subleading-color sectors; otherwise remove from the core framework Phase seven Stretch validation against modern full-color five-parton or splitting data Prototype scales without architectural change and passes at least one modern nonplanar/full-color stress test
The clearest hard criterion is this: by the end of the one-loop five-gluon phase, the framework should do at least one of the following better than ordinary representations. It should either reduce the number of genuinely independent coefficient functions, predict withheld coefficients from local overlap data, or localize color-kinematics obstruction to a sharply defined subset of charts. If it does none of those, the program has likely not bought enough structure to justify itself. If it does even one convincingly, then the next step is obvious: export the whole architecture to modern full-color two-loop data rather than inventing a more ornate formalism. citeturn5view0turn34search0turn23view9
iHack, therefore iBlog 