Cosmically Normalized Six-Point Amplitudes at Nine Loops
Computer-readable files for the symbol and the function of the nine-loop six-gluon MHV amplitude
in planar N=4 super-Yang-Mills theory, in the conventions of arXiv:1903.10890 and arXiv:2308.08199
This page is laid out in the style of Lance Dixon's Cosmic page, which holds the six-, seven- and eight-loop amplitudes in this same format.
Status. The nine-loop symbol in quintuple-coproduct form below was computed by Claude (Anthropic) by the route of arXiv:2308.08199 carried one loop
higher: the nine-loop three-point form factor of the chiral stress-tensor multiplet (in the conventions of arXiv:2204.11901)
was bootstrapped, mapped onto the amplitude on the parity-preserving surface Δ = 0 by antipodal duality, and
lifted off that surface with the remaining ambiguities fixed by the behaviour at the origin and by the two-gluon flux-tube
(pentagon OPE) data at order T2. The form factor is certified over the rationals at three 31-bit primes; the amplitude
was computed separately at each of two 31-bit primes, with identical structure, and is delivered modulo the two primes; the
rational coefficient of an explicit word follows from the two residues wherever the two primes certify it, as they do for every word in the distributed sample.
The septuple-coproduct file (the section Nine-Loop Amplitude Defined via Coproducts below) comes from a separate direct bootstrap of the nine-loop symbol in the space of hexagon functions; the two representations agree on every coefficient compared. The amplitude as a function (the last section of this page) was obtained separately. The method, its assumptions and what remains untested are described in validation/method_and_validation.md, and the conventions in conventions.md; checks made on the files of this page are recorded in validation/records/. The programs of the computation are not distributed.
Nine-loop results by Claude (Anthropic).
Normalisation: E is the BDS-like normalised amplitude of arXiv:2308.08199 (at symbol level the same as the cosmic
normalisation of arXiv:1903.10890), g² = λ/(16π²), E(1) = −½(bd+cd+ce+ae+af+bf).
Letters a, b, c, mu (= d), mv (= e), mw (= f), yu, yv, yw are the hatted alphabet of arXiv:2308.08199; words are written
first entry first; the quintuple-coproduct files, the sample file and the installation check below write the same letters as ah, bh, ch, dh, eh, fh, yu, yv, yw. Details: README.md and conventions.md. Checksums: MANIFEST.sha256.
The eight files over 100 MB are hosted on Zenodo (DOI 10.5281/zenodo.22949278).
Nine-Loop Amplitude in Quintuple-Coproduct Form
The symbol as 424 quintuple coproducts Eα over the weight-13 extended-Steinmann hexagon symbol space (dimension 5,431), S[E(9)] = Σα Eα ⊗ Sα, in the representation of arXiv:2308.08199 one loop higher: the 424 × 5,431 coordinate matrix modulo each of the two primes (from the two residues, 1,014,476 of the 1,018,297 nonzero coordinates, 99.62%, reconstruct to certified rationals and 3,821 do not; conventions.md, section 9.4), the lift before the OPE step, with its five directions, and the lift at the second prime. Format: amplitude/FORMAT.md.
E9_symbol_complete_mod2147483647.npz (4.0MB), E9_symbol_complete_mod2147483629.npz (4.1MB),
E9_lift_quintuples_mod2147483647.npz (6.8MB)
(the directory: amplitude/)
Hexagon Symbol Space and Final-Entry Tails
The extended-Steinmann hexagon symbol space to weight 13 (14 at the first prime) as nested coproduct tensors at the two primes, which the coordinates above refer to; the same tensors at four 21-bit primes with their exact reconstruction through weight 9, the 41 pair relations and the restriction to Δ = 0 (in the tarball); and the exact MHV multiple-final-entry tails W1..W5 of dimensions 6/21/62/166/424, whose 424 quintuples are the Sα.
hexspace_p2147483647_w14.npz (438.3MB), hexspace_p2147483629.npz (144.9MB),
MHV_final_entry_spaces.txt (21.8MB), hexagon_space_w13.tar.gz (486.1MB)
(the directory: hexagon_space/)
Sample Coefficients
20,630 random weight-18 words (20,400 with nonzero coefficient, 230 with zero) with their coefficients modulo each of the two primes and as rationals, to check any other representation of the symbol against without computing anything. The coefficient of any word follows from the quintuple representation by thirteen matrix products (amplitude/FORMAT.md), or from the septuple file below with the coproduct tables of SixGluonAmpsAndCops.m.
E9_sample_coefficients.txt (1.8MB)
(the directory: samples/)
Nine-Loop Three-Point Form Factor
The symbol of the nine-loop form factor of the chiral stress-tensor multiplet (BDS-like normalisation and alphabet of arXiv:2204.11901), from which the amplitude on Δ = 0 follows by the antipodal map: the certified rational coefficient matrix over the weight-5 front basis and the weight-13 final-entry basis (108 × 4,171) with the three per-prime solutions, the exact front basis, the final-entry tables at weights 9–13 predicted from the hexagon space by duality (9–14 at the first prime), the exact octuple basis, and the explicit symbol as the 279 rational weight-10 octuple coproduct components (295,186,924 terms); the coproduct spans, the parameter cascade, and the nine-loop form-factor OPE terms that closed it. Format: form_factor/FORMAT.md.
E9_C5xB13_rational_3primes.npz (2.4MB), C5_basis_rational.txt (461KB),
E9_octuples_rational.npz (1.32GB), B8_octuple_basis_exact.txt (25.3MB)
(the directory: form_factor/)
Amplitude on Δ = 0
The antipodal image of the form factor: the nine-loop symbol on the parity-preserving surface Δ = 0, stored modulo the first prime as its 279 weight-10 octuple coproduct components, one file per form-factor octuple (295,186,924 terms in all; counted as in arXiv:2308.08199, the symbol written out in full as weight-18 words would have 30,024,320,034 terms with nonzero coefficient modulo the first prime, from the per-entry counts in delta0/E9_Delta0_counts.json), in the form-factor alphabet and normalisation, to be mapped to amplitude letters by the antipodal map of conventions.md, section 8; the symbol's projection onto the 111 y-free quintuple final entries (those whose pivot word contains no y letter; conventions.md, section 6), which is the input of the lift; and the term counts.
E9_octuples_Delta0_mod2147483647.tar.gz (1.97GB),
E9_Delta0_quintuple_coproducts_mod2147483647.npz (1.8MB), E9_Delta0_counts.json (1KB)
(the directory: delta0/)
Flux-Tube Data
The near-collinear (pentagon OPE) data of the amplitude side, generated from the finite-coupling flux-tube formulas and validated against every published block: the single-gluon blocks, the two-gluon bound state from leading to next-to-next-to-next-to-leading logarithm, the Born-level two-gluon continuum, and the helicity-zero sector; closed forms and exact series. Described in ope_data/README.md.
(the directory: ope_data/)
Eight-Loop Control
The eight-loop lift produced by the same programs, before and after its one remaining direction was fixed by the two-gluon data. A check made on these files compares the closed lift with the symbol of the published eight-loop amplitude MHV8quintuples.txt of arXiv:2308.08199 on 1,000 random words with nonzero coefficient; all 1,000 agree modulo the first prime (validation/records/04_output.txt).
E8_lift_OPEfixed_mod2147483647.npz (1.2MB), E8_lift_quintuples_mod2147483647.npz (1.2MB)
(the directory: lower_loops/)
Validation
The method and every check, with pointers: validation/method_and_validation.md; the conventions: conventions.md. The recorded outputs of checks made on the files of this page (the sample file evaluated from the quintuple representation at both primes; the multiple-final-entry structure, rank 400 for the quintuples and 162/62/21/6 below, at both primes; the eight-loop lift against the symbol of the published eight-loop amplitude on 1,000 random nonzero words; the form-factor files against one another; the symmetry and vanishing tests; the septuple file against the quintuple representation on 2,000 words, and on all 107,053 nonzero word coefficients that determine the septuple file) are in validation/records/; of these, validation/records/oracle_validation_results.json is the form-factor computation's own record of its symmetry, vanishing and Δ = 0 checks.
Installation check: the coefficient of the word bh bh fh dh dh eh dh fh bh dh fh ah ah fh eh fh dh eh is 829521918 modulo the first prime, 1173913588 modulo the second, and −105757/65536 as a rational, a line of the sample file; and the rank of the 424 quintuple coproducts is 400 at both primes.
Nine-Loop Amplitude Defined via Coproducts
The {11,1,1,1,1,1,1,1} septuple coproducts of the nine-loop MHV amplitude in terms of weight 11 hexagon functions, with the nested final-entry relations, in the format of MHV8quintuples.txt. Together with the basis file below this is a complete definition of the symbol (232MB unzipped): MHV9septuples.zip (66.5MB)
Its exact coefficients come from a separate direct bootstrap of the nine-loop symbol in the space of hexagon functions; that computation's word coefficients were reconstructed exactly from five primes and checked at a sixth. The septuple file and the quintuple representation agree on every coefficient compared: all 107,053 nonzero coefficients that determine the septuple file, 3,401 of them modulo the two primes only. A 2,000-word check is recorded in validation/records/08_output.txt. The full comparison, one row per coefficient, is recorded in validation/records/09_septuple_vs_quintuple_107053_words.txt.gz.
The same content as JSON (258MB unzipped); its tables for levels 1–4, marked "external": true, are Dixon and Liu's relations from MHV8quintuples.txt (arXiv:2308.08199): E9_exact_levels.zip (67.4MB)
The weight 11 hexagon functions are those of the Cosmic page; nothing beyond weight 11 is needed: SixGluonAmpsAndCops.zip
(the directory: MHV9/)
Nine-Loop Amplitude as a Function
The same 8,028 septuple final entries as full weight-11 hexagon functions: the symbol part of MHV9septuples.txt plus the exact zeta-valued completion (786,352 beyond-the-symbol coefficients over 6,305 of the labels, on the 251 weight-11 basis functions with vanishing symbol), in the format and with the relation tables of MHV9septuples.txt. Together with the constants below, the weight-11 basis as functions (SixGluonAmpsAndCops.m, with its values at (1,1,1)) and the six weight-11 constants n1,…,n6 of eq. (10.5) of arXiv:2308.08199, this is a complete definition of E(9) as a function, with no free parameter: the weight-18 function is rebuilt from its weight-11 septuple components by integrating up one weight at a time, with the constants of integration at each step taken from the tables below and the relation tables applied to the non-independent tails. On the lines (1,v,v), (u,u,1), (u,1,1) and (u,0,0) every function reduces to harmonic polylogarithms, as in sections 5 and 11 of that reference; at a generic point a numerical evaluation of hexagon functions is needed. The 251 basis functions with vanishing symbol are YE(11,1504)–YE(11,1718) and YO(11,383)–YO(11,418); the weight-11 constants of the amplitude enter through the values of the basis functions at (1,1,1) with the six constants n1,…,n6 given in the file header. The function was obtained separately; it has been computed once, and there is no second, independent computation of it. Beyond the assumptions of the symbol it rests on one more, stated in the file header: every relation among the septuples that holds at symbol level is taken to hold at function level (the relations that follow from integrability and the Qbar equation do; the empirical higher-level ones need not). The file (296MB unzipped):
MHV9septuples_function.zip (93.9MB)
The 72 constants of integration at weights 12 to 18, the values Ea1…ak(1,1,1) of the coproduct components for k = 0,…,6, exact over a stated basis of multiple zeta values (1,173 rational coefficients, zeros included; the convention is that of EZMHVcoproducts111.txt), one representative tail per constant; and the same constants as the full table of every independent tail (1,548 entries) in the style of EZMHVcoproducts111.txt. Both contain E(9)(1,1,1) itself, which is also given alone:
MHV9constants.txt (53KB), MHV9coproducts111.txt (154KB), E9_111.txt (2KB)
Installation check: the value of the amplitude at (1,1,1) given in MHV9constants.txt evaluates to 673807654.7340448746895848… (its coefficient of ζ15,3 is 743185511/5), and E(9)(1,1,1)/E(8)(1,1,1) = −14.1114787…; in MHV9septuples_function.txt the first listed septuple E(a,a,a,a,a,a,mv) carries 200 of the zero-symbol basis functions, among them YE(11,1522) with coefficient 12609/512, while its coefficient of YE(11,1) is −53582371/16384, as in MHV9septuples.txt.
(the directory: function/)
16 September 2026.