# The nine-loop six-gluon MHV symbol: method and validation

This note describes how the symbol on this page was obtained and what was done to check it. Notation (kinematics, letters, the dihedral action and parity, the normalisation E and E^(1), the hexagon spaces H_n and their nested coproduct basis, coproducts, final entries and the tail spaces W_1 … W_5, the form-factor alphabet and spaces, antipodal duality, and the finite-field and reconstruction conventions) is defined in [conventions.md](../conventions.md). The layout of each data file is given in [amplitude/FORMAT.md](../amplitude/FORMAT.md), [form_factor/FORMAT.md](../form_factor/FORMAT.md), [form_factor/FORMAT_C5xB13.md](../form_factor/FORMAT_C5xB13.md) and [ope_data/README.md](../ope_data/README.md), and is not repeated here. Recorded outputs of checks are listed in [records/index.html](records/index.html).

Two computations of the nine-loop symbol are referred to in this note:

* **the form-factor computation**, which produced the quintuple-coproduct files in `amplitude/` together with the form factor, the data on Δ = 0, the flux-tube data, the eight-loop lift in `lower_loops/` and the sample file in `samples/`. Sections 2–4 describe it;
* **the direct bootstrap**, a bootstrap of the nine-loop symbol in the space of hexagon functions, which produced the septuple-coproduct file `MHV9septuples.txt` (in `MHV9/MHV9septuples.zip`).

## 1. Status

The symbol of E^(9) in quintuple-coproduct form was computed by the form-factor computation at two 31-bit primes, p1 = 2^31 − 1 = 2147483647 and p2 = 2147483629 (the form factor at three, p1, p2 and p3 = 2147483587). The form-factor computation is a single computation by a single pipeline, which gave the symbol modulo the two primes. The computation was carried out by Claude (Anthropic).

The amplitude is delivered modulo the two primes. Under the certification criterion of [conventions.md](../conventions.md) (a two-prime reconstruction of height below 2^30 with a power-of-two denominator), 1,014,476 of its 1,018,297 nonzero basis coordinates (99.62%) are certified rationals; the other 3,821 are not certified. The coefficient of every word in the distributed sample (20,630 words, 20,400 of them with nonzero coefficient) reconstructs to a certified dyadic rational. The form factor is certified over the rationals entry by entry.

The septuple-coproduct file `MHV9septuples.txt` (in `MHV9/MHV9septuples.zip`) comes from the separate direct bootstrap; its word coefficients were reconstructed exactly from five primes and checked at a sixth, and the file's header describes how. 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 [records/08_output.txt](records/08_output.txt). (The 107,053 are the nonzero ones among the 346,248 word coefficients that, together with the adjacency relations, determine the symbol in the direct bootstrap's representation: one word for each of its 38,472 weight-17 basis elements and each of the nine final letters, namely the basis element's pivot word, on which no other basis element is nonzero, followed by that letter.) The full comparison, one row per nonzero word coefficient, is recorded in [records/09_septuple_vs_quintuple_107053_words.txt.gz](records/09_septuple_vs_quintuple_107053_words.txt.gz).

The function-level files in `function/` were obtained separately. They are outside the scope of this note.

The symbol is supported by the agreement of the two computations and by the consistency checks of section 6.

## 2. The form factor

The first part of the form-factor computation bootstraps the nine-loop three-point form factor E_c^(9) of the chiral stress-tensor multiplet, a symbol of weight 18 in the six-letter alphabet a, b, c, d, e, f. No Feynman integral is evaluated: every step is a linear-algebra statement over a finite field.

### 2.1 The deep-back ansatz: why it is needed

Through eight loops the form factor was bootstrapped (arXiv:2204.11901) in the ansatz C_{L+1} ⊗ B_{L−1}: the coaction-restricted front space of weight L + 1 times the final-entry ("back") space of weight L − 1. At nine loops this is C_10 ⊗ B_8, with 6654 × 279 = 1,856,466 unknowns. The nested coproduct tables of both spaces are about ten per cent dense, so wherever the seam between front and back is placed, the linear system that sews the two together has of order 10^11 nonzero entries, too many to hold in the memory available. A letter-expanded sparse reformulation, validated through seven loops, leaves a dense Schur complement of the size of the dihedral-reduced ansatz and does not help at nine loops.

The remedy is to move the seam towards the front by making the back space deeper. By antipodal duality, the final w entries of the form factor are the first w entries of the amplitude on Δ = 0. The final-entry space of the form factor at weight w can therefore be predicted from the hexagon space H_w, which is cheap to build: restrict H_w to Δ = 0, relabel the letters by the antipodal letter map (â, b̂, ĉ, d̂, ê, f̂ → d, e, f, a, b, c) and reverse the words.

### 2.2 The deep-back ansatz: why it is sound

This prediction reproduces the published final-entry spaces exactly through weight 7, and at weight 8 it gives 282 dimensions against the true 279. The three extra relations at weight 8 are function-level relations: they are not implied by the symbol-level conditions used to build H_w (as noted near the end of section 5 of arXiv:2204.11901). The predicted spaces are therefore expected to be supersets of the true ones, which is all an ansatz needs: a superset can admit spurious solutions, which the constraints then remove, but it cannot exclude the true one. Chaining upward from the exact weight-8 space, the predicted final-entry spaces at weights 9–14 have dimensions

    491, 857, 1472, 2495, 4171, 6891,

identical at five primes (four 21-bit primes and p1). Each predicted B_s lies inside the one-letter extension of B_{s−1} (491 of 510, 857 of 876, 1472 of 1523, 2495 of 2551). That the predicted spaces at weights 9–13 contain the true ones at symbol level is an assumption at nine loops (section 5). With B_13 as the back space the ansatz becomes

    C_5 ⊗ B_13 :  108 × 4171 = 450,468 unknowns,  75,923 dihedral-invariant coordinates,

and the seam sits between weights 5 and 6 of the eighteen.

The deep-back construction was validated end to end at eight loops, where the answer is known: the analogous ansatz C_5 ⊗ B_11 gives 142 → 43 → 27 → 0 parameters under the adjacency relations, the branch-cut condition, the strict collinear limit and the leading-logarithmic OPE, and reproduces exactly all 279 independent octuple final entries of the relation file of arXiv:2204.11901 (16,439,122 terms). Section 6.1 lists the lower-loop controls.

### 2.3 Sketch-and-verify

The large homogeneous systems are solved by sampling. The constraint families are sampled by random rank-one functionals (a front functional times a back functional), about 1.2 times as many rows as unknowns in all; one dense modular row reduction gives a candidate kernel; and the candidate is then verified exactly against every constraint family by residual elimination. A sampled system can only have a kernel that is too large, never too small, so a candidate that passes the exact verification is the true kernel, and one that fails is caught and reduced. The same sketch-and-verify procedure is used on the amplitude side (section 4.1), where every parameter fixed by sampling was obtained identically from at least two independent random draws.

### 2.4 The dihedral reduction

The form-factor ansatz is reduced to its dihedral-invariant coordinates in a cyclic eigenbasis of the front and back spaces. The prime must be congruent to 1 mod 3, so that the cube roots of unity exist in the field. In this basis the flip relates the two non-trivial cyclic sectors, and the 75,923 invariant coordinates split as 25,883 in the trivial cyclic sector and 50,040 in the other two.

### 2.5 The nine-loop cascade

At each of the three primes the stages are:

| stage | parameters |
|---|---|
| pair and triple adjacency across the seam: 91,171 random rank-one rows, one dense 91,171 × 75,923 modular row reduction | 319 |
| exact residual check of every pair and triple family on the 319-dimensional space | 319 |
| branch-cut condition at back slots 1, …, 13 (exact), successively 210, 148, 113, 91, 79, 71, 66, 64, 62, 61, 60, 60, 60 | 60 |
| exact verification of all 66 constraint families | 60 (0 violations) |
| strict collinear limit E_c^(9)\|_{v→0} = (1/9!)(−ln²v − ln²d)^9 | 44 |
| form-factor OPE, T² ln⁸T (leading logarithm, LL) | 1 |
| form-factor OPE, T² ln⁷T (next-to-leading logarithm, NLL) | 0 |
| form-factor OPE, T² ln⁶T (NNLL), held out | 0 of 135 equations violated |
| full strict-collinear check of the unique solution, all 2^18 components | passes |

The cascade is identical at the three primes, as are the word supports of all 279 octuple components and the coproduct spans. The L-th discontinuity condition of arXiv:2204.11901 was not imposed.

### 2.6 The form-factor OPE data

The near-collinear expansion of the form factor is controlled by the form-factor OPE of Sever, Tumanov and Wilhelm (arXiv:2009.11297, arXiv:2105.13367, arXiv:2112.10569). At order T² the contributing states are two-particle states: gluon, fermion and scalar pairs. The finite-coupling two-particle integrands were implemented as exact residue sums at weak coupling, in the parametrisation u = 1/(1 + S² + T²), v = T²/(1 + T²) of arXiv:2204.11901, organised by powers of g², ln T and ln y with y = S², from the flux-tube building blocks (the Beisert–Eden–Staudacher kernel, the source vectors and the Zhukowsky factors) expanded at weak coupling. The symbol-level projection (all zeta values set to zero) is applied from the start. At nine loops the implementation gives the T² ln⁸T, ln⁷T and ln⁶T coefficients as exact series through S^24, together with a closed form of the leading-logarithmic term. Only the two highest logarithms were used as input; the third was held out. The implementation reproduces all twenty published T² entries with k ≥ L − 3 through eight loops, where k is the power of ln T in the coefficient of T² ln^k T at L loops.

### 2.7 Reconstruction and output

The coefficient matrix c of E_c^(9) = Σ_{i,j} c_ij F_i ⊗ B_j, over the 108 front basis elements and the 4171 back basis elements, has 450,468 entries, of which 355,144 are nonzero. It was reconstructed from the three primes by Chinese remaindering and rational reconstruction. The product of the primes is about 9.9 × 10^27 and the reconstruction bound 7.0 × 10^13; every entry is certified in the sense that the reconstructed rational, of height at most 1.33 × 10^11, lies far below the bound (false-positive probability at most 4.4 × 10^−13 per entry). Every denominator is a power of two dividing 2^14. The explicit symbol follows from c by expanding the back basis: the 279 weight-10 octuple coproduct components, indexed by the independent octuple final entries in the order of arXiv:2204.11901, have 295,186,924 terms in all, between 478,389 (for aaaaaaaf) and 1,253,533 per octuple, with rational coefficients from the three primes (an expected number of spurious reconstructions of about 5 × 10^−4 over the whole symbol), a maximum absolute numerator of 132,843,110,400 and denominators dividing 2^14.

## 3. The antipodal map onto Δ = 0

Reversing every word of the form factor and relabelling the letters by the antipodal map (with the normalisation factors given in [conventions.md](../conventions.md)) gives the amplitude's symbol on Δ = 0, a symbol in the six letters â, …, f̂ organised by amplitude final entries. An amplitude k-tuple final entry on Δ = 0 is the flip of the form-factor symbol with the corresponding reversed and relabelled prefix removed, so the form factor's octuple final entries become the amplitude's octuple first entries, and the amplitude's triple final entries are read off from the form factor's triple first entries. The full symbol on Δ = 0, counted as in arXiv:2308.08199, has more than thirty billion terms; it is organised by ten weight-15 triple final entries, from which the others follow by dihedral symmetry, and the per-entry counts are in [delta0/](../delta0/index.html). At eight loops the same operation gives 1,671,656,292 terms, the count of arXiv:2308.08199. The Δ = 0 symbol is a linear image of the form factor, so it inherits the form factor's three-prime certification. It was stored modulo p1 as its 279 weight-10 octuple coproduct components, one file per form-factor octuple, and its 111 y-free quintuple coproduct components were projected onto the restriction of H_13 to Δ = 0 as the input of the lift. Here a y-free quintuple is one of the 111 quintuple final entries whose pivot word contains no y letter (layer 0 of the tail file; see [conventions.md](../conventions.md)); their tails can still contain words with y letters, which drop out on Δ = 0, where y_u = y_v = y_w = 1.

## 4. The lift off Δ = 0

### 4.1 The ansatz and the layers

The amplitude is sought in the quintuple-coproduct form S[E^(9)] = Σ_α E_α ⊗ S_α over the 424 allowed quintuple final entries, each E_α in H_13 (dimension 5431): 424 × 5431 = 2,302,744 unknown coordinates. The constraints are: agreement with the Δ = 0 data; the 41 pair relations at the junction between the last letter of E_α and the first letter of the tail S_α (inside E_α and inside S_α they hold by construction); parity; and dihedral symmetry. On Δ = 0, where y_u = y_v = y_w = 1, the tails collapse and the amplitude reduces to a sum over the 111 y-free quintuples. Matching the projected Δ = 0 data determines those 111 components up to the kernel of the restriction map on H_13, which is two-dimensional in the parity-even sector, so 222 parameters remain at this stage. The junction relations then carry the information to the other 313 quintuples (layers 1–5), whose pivot words contain y letters. The tails are organised in layers by their y content, and the relations that couple the last letter of E_α to the first letter of a tail in the next layer form a linear system for that layer's components.

**Dihedral reduction on the amplitude side.** Dihedral symmetry is imposed inside each layer by working in symmetry-adapted coordinates. On H_13 the trivial, sign and two-dimensional representations of the dihedral group occur with multiplicities 800, 623, 1423 in the parity-even sector and 201, 187, 387 in the odd sector; the reduction lowers the number of unknowns six- to twelve-fold. At six and seven loops the reduced and unreduced lifts give identical results.

Each layer's system is solved by sketch-and-verify (section 2.3): the sketch systems are assembled from random combinations of the junction relations, and every kernel is verified against the full relation set. The five layers at nine loops, at p1, are:

| layer | quintuples | unknowns | before symmetry | sketch | rank | free |
|---|---|---|---|---|---|---|
| 1 | 96 | 18,648 + 222 | 111,552 | 19,878 × 18,870 | 18,861 | 9 |
| 2 | 69 | 35,473 + 9 | 210,672 | 37,321 × 35,482 | 35,473 | 9 |
| 3 | 54 | 18,526 + 9 | 109,353 | 19,526 × 18,535 | 18,525 | 10 |
| 4 | 24 | 12,129 + 10 | 74,493 | 12,810 × 12,139 | 12,138 | 1 |
| 5 | 70 | 28,629 + 1 | 168,336 | 30,126 × 28,630 | 28,622 | 8 |

The "unknowns" column counts the symmetry-reduced coordinates of the layer plus the parameters carried in from the previous one. The 222 parameters from the Δ = 0 kernel are consumed in the first layer, and eight dihedral-invariant directions remain at the end; imposing the dihedral condition afterwards on the full symbol adds no equation (8 → 8), as it must.

At eight loops, where the weight-11 space has dimension 1887 (multiplicities 289, 214, 501 even and 68, 60, 127 odd), the Δ = 0 kernel is empty in the even sector, the five layers have ranks 6141, 12409, 6377, 4214, 9944 and leave 3, 2, 3, 0, 3 free directions, and three dihedral-invariant directions remain. Before the symmetry reduction the eight-loop ambiguity is 9 = 3 + 0 + 2 × 3 in the trivial, sign and two-dimensional representations, the nine of arXiv:2308.08199. At six and seven loops the lifts are unique once the origin condition is imposed (with ambiguities 1 and 3 before it), again as in arXiv:2308.08199.

### 4.2 The origin

Basso, Dixon and Papathanasiou (arXiv:2001.05460) conjecture that as û, v̂, ŵ → 0, ln E becomes a quadratic form in the logarithms. At symbol level only the one-loop terms survive, and the condition is that S[E^(L)] → S[(E^(1))^L / L!] in the limit. Imposed on the eight remaining directions (2418 equations at nine loops), it fixes three: 8 → 5. At eight loops the corresponding step, with 1449 equations, is 3 → 1; arXiv:2308.08199 likewise found that the origin fixes two of its three dihedral parameters and that the last direction vanishes at the origin.

The five directions left at nine loops, and the one at eight loops, vanish on Δ = 0, at the origin, in the strict collinear limit and in the near-collinear expansion at orders T^0 and T^1 at every power of ln T. Three of the nine-loop directions (those left after the leading- and next-to-leading-logarithmic T² step) and the eight-loop direction were also checked to vanish in the 2 → 4 multi-Regge limit to all logarithmic orders (in the Euclidean region and after one and two discontinuities in û); the other two nine-loop directions were not checked there. The directions are visible at order T² of the near-collinear expansion, and that is where they are fixed.

### 4.3 The hexagon flux-tube formulas and matching

The near-collinear expansion of the hexagon Wilson loop is organised by the excitations of the colour flux tube (Alday, Gaiotto, Maldacena, Sever and Vieira, arXiv:1006.2788; Basso, Sever and Vieira, arXiv:1303.1396, arXiv:1306.2058, arXiv:1402.3307, arXiv:1407.1736). In the variables T = e^{−τ}, S = e^{σ}, F = e^{iφ} of arXiv:1903.10890 (its eq. (3.23)),

$$
\hat u=\frac{F}{F+FS^2+ST+F^2ST+FT^2},\qquad
\hat v=\frac{FS^2}{(1+T^2)(F+FS^2+ST+F^2ST+FT^2)},\qquad
\hat w=\frac{T^2}{1+T^2}. \tag{4.1}
$$

The framed Wilson loop W is related to E by an exact formula (arXiv:1903.10890, eq. (3.24)),

$$
\mathcal W=\rho\,\mathcal E\,\exp\Big[\tfrac14\Gamma_{\rm cusp}\big(X-\mathcal E^{(1)}\big)\Big],\qquad
X=-\mathrm{Li}_2(1-\hat u)-\mathrm{Li}_2(1-\hat v)+\mathrm{Li}_2(\hat w)+\ln^2(1-\hat w)-\ln(1-\hat w)\ln\frac{\hat v}{\hat u}-\ln\hat u\ln\hat v+\zeta_2, \tag{4.2}
$$

and its expansion is W = 1 + T(F + F^{−1}) W_[1] + T²[(F² + F^{−2})(W_[2] + W_[1,1]) + w_{2,0}] + O(T³). Here W_[a] = ∫ du/(2π) μ_a(u) T^{γ_a(u)} S^{i p_a(u)} is the contribution of the a-gluon bound state, with the finite-coupling measure, energy and momentum of arXiv:1407.1736; W_[1,1] is the continuum of two same-helicity gluons, whose Born-level integrand is eq. (54) of arXiv:1402.3307; and w_{2,0} is the helicity-zero sector, to which scalar pairs, fermion pairs, gluon–antigluon pairs and the effective twist-two fermion excitation contribute (arXiv:1402.3307). The exact expansions X|_{T^0} = 0 and E^(1)|_{T^0} = −2 ln²T − ½ ln²y − 2ζ_2 (with y = S²) hold, and the X and E^(1) terms are equal in the T^1 and T²F^{±2} sectors (and, up to a piece common to both, in T²F^0). Hence in every sector through order T²

$$
\mathcal E\big|_{T^aF^b}=\mathcal E\big|_{T^0}\cdot\mathcal W\big|_{T^aF^b},\qquad
\mathcal E\big|_{T^0}=\rho^{-1}\exp\Big[\tfrac14\Gamma_{\rm cusp}\big(-2\ln^2T-\tfrac12\ln^2y-2\zeta_2\big)\Big]. \tag{4.3}
$$

This is the form the symbol-level comparison uses.

The bound-state contributions were evaluated as exact residue sums at weak coupling, with the measures, energies and momenta of the a ≥ 2 bound states obtained from the gluon ones by fusion, in two separate implementations: one exponentiates the integrand symbolically, the other expands pole by pole in Laurent series graded in g², with a modular mode for long series. Closed forms in the harmonic polylogarithms G(a_1, …, a_n; y), a_i ∈ {0, −1}, were fitted to modular series (through y^3200 at LL and NLL, y^5200 for the nine-loop NLL block, y^10200 at NNLL) by exact linear solves, and re-expanded against exact rational series through y^20 or y^40. The single-gluon closed forms are those of Papathanasiou (arXiv:1310.5735), which the series reproduce. The data used at nine loops are: the T^1 blocks at leading and next-to-leading logarithm (ln⁸T, ln⁷T; all powers of ln T through six loops); the T²F^{±2} blocks of W_[2] at LL, NLL and NNLL in closed form; the N³LL block W_[2] + W_[1,1] at ln⁵T as an exact series through y^40, in which the Born-level two-gluon continuum contributes exactly [ln T (E_1(u) + E_1(v))]^{L−4}/(L−4)! times its Born integrand; and the T²F^0 block at leading logarithm, to which only the effective twist-two excitation contributes. The distributed blocks are described in [ope_data/README.md](../ope_data/README.md).

### 4.4 The three near-collinear probes, and fixing the last five directions

The near-collinear expansion of a symbol in quintuple-coproduct form was implemented in three ways:

1. a full non-commutative probe, used for the vanishing statements of section 4.2;
2. an abelianised probe organised by characters, used to fix parameters;
3. a direct y-series carrier that gives the exact zeta-free series of each coefficient in y = S², used to match the N³LL data, which are available only as a series.

With the data of section 4.3 the five directions are fixed in three steps, at p1:

* the LL and NLL blocks of T²F^{±2} give 23 equations of rank 2 (5 → 3);
* the NNLL block gives 13 equations of rank 2, identical at two random evaluation points (3 → 1);
* the last direction, whose ln T content stops at ln⁵T, is fixed by matching W^(9)_[2] + W^(9)_[1,1] in the ln⁵T sector: 24 equations of rank 1 with zero residuals (1 → 0). This is the sector in which the Born-level two-gluon continuum first contributes.

In summary:

    2,302,744 → 8 →(origin) 5 →(T²F^{±2}, LL+NLL) 3 →(NNLL) 1 →(N³LL) 0.

With nothing left free, all nine-loop T²F^{±2} blocks at ln^{5,6,7,8}T, the T^1 blocks at LL and NLL and the T²F^0 block at LL equal the OPE predictions (section 6.1).

At eight loops the one direction left after the origin contributes to T²F^{±2} at NLL. Its coefficient was fixed from two independent random points (rank 1, zero residuals), verified at a third on every LL and NLL block, and reproduced independently by the NNLL closed form and by the N³LL series match; from its values at the two primes it reconstructs to t = −7839775/(2^13 · 3^4) in the normalisation of the direction chosen in the computation (only the first-prime eight-loop lift is distributed).

### 4.5 The second prime, and rational reconstruction

The entire amplitude-side chain, from the projection of the Δ = 0 data through the five layers, the origin and the three OPE steps, was run a second time at p2, with the hexagon space and the tails rebuilt at that prime. The structure is identical: the same five ranks and free counts, 8 → 5 at the origin, ranks 2, 2, 1 at the three OPE steps, the same remaining direction at each step, and an identical pattern of zeros among the 2,302,744 coordinates. Of these, 1,018,297 are nonzero at both primes, and every one of the 424 quintuple components is nonzero. A bug found during the computation, the factor 4^{−L} of the antipodal map applied as a floating-point product that is exact only for p1 = 2^31 − 1, was fixed before the p2 results were produced.

Rational reconstruction from the two primes (product about 4.6 × 10^18, bound |n|, d < 2^30) gives a fraction within the bound for 1,014,740 of the nonzero coordinates and none for the other 3,557. The modular files are the primary data (the reconstructed rationals are not distributed as a file); which reconstructed coordinates are certified is given in section 6.2.

That the reconstruction fails for some coordinates is a feature of the basis, not of the amplitude: the nested coproduct basis of H_13 has entries of height up to 3 × 10^12, while the coefficients of random words are small. The coefficient of an explicit weight-18 word x_1 ⋯ x_18 follows from the distributed files by thirteen matrix–vector products,

    c(x_1 ⋯ x_18) = Σ_α S_α[x_14 ⋯ x_18] · ( E_α · T_13[x_13] T_12[x_12] ⋯ T_1[x_1] · [1] )   (mod p),

and the two residues determine the rational whenever the two-prime reconstruction is certified, that is, has height below 2^30 and a power-of-two denominator ([conventions.md](../conventions.md)). The power-of-two condition lowers the chance that a random residue pair passes the test from about 0.2–0.3, measured on shifted controls (889 of 3,000 and 190 of 1,000 pass the height test alone; [records/oracle_validation_results.json](records/oracle_validation_results.json)), to about 2^−26, roughly the number of fractions with |n| < 2^30 and a power-of-two denominator (about 2^36) divided by p1 · p2 (about 2^62); none of the 4,000 shifted controls passed it. On 20,400 random words with nonzero coefficient, every coefficient reconstructs from the two primes with a power-of-two denominator dividing 2^16 and a numerator below 6.9 × 10^7; the form-factor computation also reports that in a further sample of 6,000 y-free words drawn from the Δ = 0 files, one coefficient had height at least 2^30 and is not certified; that sample is not among the records on this page.

## 5. Assumptions

**Theorems and definitions, not assumptions.** The symbol calculus; integrability as the condition for a tensor to be a symbol; dihedral symmetry and parity on both sides; the first-entry condition; the MHV final-entry condition and the six double-final-entry relations that follow from the Q̄ equation; the strict collinear limits; and the finite-field linear algebra with its verification (every sampled kernel verified exactly against the full constraint set; identical counts at several primes).

**Shared with the published bootstraps.**

* The hexagon-bootstrap hypothesis, that the nine-letter alphabet and the extended-Steinmann adjacency conditions persist at nine loops, and its form-factor counterpart, the six-letter alphabet with the adjacency restrictions of arXiv:2204.11901. These are tested only indirectly, by the held-out flux-tube data.
* The coaction principle on the form-factor side, in the form used by arXiv:2204.11901: the restriction of the weight-4 front space to the 48-dimensional span of the six-loop coproducts.
* The exponentiated behaviour at the origin at symbol level (arXiv:2001.05460), imposed on the lift.
* The flux-tube OPE: the finite-coupling single-gluon and bound-state data and the Born-level two-gluon continuum on the amplitude side, and the two-particle form-factor OPE, on both sides expanded at weak coupling beyond the published data points. The two-gluon bound-state data at L ≥ 4 and the Born-level continuum are predictions of the framework, validated against every published data point but not independently beyond them; the five directions fixed last rest on them alone, since no other sector of the near-collinear expansion or other limit that was checked sees those directions (in the multi-Regge limit, three of the five were checked and are invisible there; the other two were not checked).
* Antipodal duality at nine loops, used in both directions: forwards, to obtain the amplitude on Δ = 0, and backwards, in the form of the predicted final-entry spaces.
* The MHV multiple-final-entry relations of arXiv:2308.08199 through the quintuple level, which define the tail spaces W_k: the quintuple-coproduct ansatz assumes that the nine-loop amplitude lies in H_13 ⊗ W_5.

**Specific to the form-factor computation.** The duality-predicted final-entry spaces at weights 9–13 are assumed to be symbol-level supersets of the true spaces. This is supported by the exact reproduction of the eight-loop form factor from C_5 ⊗ B_11, by the internal consistency of every stage of the nine-loop cascade and the held-out NNLL sector, and by E_c^(9) filling 487 of the 491 predicted weight-9 directions; but "unique within the ansatz" never excludes a solution outside the ansatz, and the surplus of matched held-out data is the only hedge. The predicted back-space tables were generated separately at each prime, with identical dimensions, pivots and nonzero counts; their consistency across primes is established downstream, by the identical cascades and the successful three-prime reconstruction, and a direct cross-prime check of table entries was made at weight 9 only. Random sampling was used to assemble the large systems, with exact verification afterwards; the rational reconstruction of the 295 million octuple coefficients is certified probabilistically, that of the coefficient matrix entry by entry.

## 6. Validation

Section 6.1 lists the checks that the form-factor computation made on its own results as it ran; its own recorded output is [records/oracle_validation_results.json](records/oracle_validation_results.json). (In the file name of this record, "oracle" means the program that evaluates word coefficients from the quintuple-coproduct files by the formula of section 4.5.) Section 6.2 lists checks made on the distributed files, each with the record of its output. Every check is at symbol level, and at p1 only unless stated otherwise.

### 6.1 Checks made during the form-factor computation

* **Lower-loop reproductions, form factor.** The letter-basis re-bootstrap at two to five loops reproduces the full published symbols (12, 636, 11,208 and 263,880 terms), with the parameter cascade of table 6 of arXiv:2204.11901 at every stage except after the discontinuity condition, where it leaves two parameters fewer (15 and 36 at four and five loops against 17 and 38); the later stages and the final symbols agree, so the two extra conditions are true ones. At six, seven and eight loops the same machinery (front spaces of weight 7 and 6 at six and seven loops, C_5 ⊗ B_11 at eight) reproduces the 279 independent octuple final entries of the relation file of arXiv:2204.11901 (cascades 86 → 25 → 13 → 1 → 0, 43 → 16 → 0, and 142 → 43 → 27 → 0 from C_5 ⊗ B_11; at eight loops 16,439,122 terms, with none of the equations at the lower logarithmic orders (between 111 and 183 per order) violated). The front, coaction-restricted and back space dimensions of arXiv:2204.11901 are reproduced.
* **Lower-loop reproductions, hexagon side.** The dimensions of H_n through weight 13 are those of arXiv:1906.07116 (the symbol-level parity split at weight 11, 1505 even and 382 odd, was counted in the computation); the tail spaces have dimensions 6, 21, 62, 166, 424; the flip reproduces the eight-loop term count 1,671,656,292 and passes every check available on Δ = 0 through eight loops (first and last entries, extended Steinmann, the âd̂-type adjacency, integrability, dihedral invariance), as does the flip of the nine-loop form factor. The six- and seven-loop lifts are unique after the origin condition. At eight loops the lift has three dihedral-invariant directions, of which the origin fixes two, and the ambiguity before symmetry reduction is 9 = 3 + 0 + 2 × 3, the counts stated in arXiv:2308.08199; the restriction of the eight-loop lift to Δ = 0 equals the flip of the published eight-loop form factor on all 111 y-free quintuples; and the last direction is fixed by the NLL two-gluon data, with the value verified at a third random point and reproduced by the NNLL closed form and the N³LL series.
* **Flux-tube data.** The form-factor OPE implementation reproduces all twenty published T² entries with k ≥ L − 3 through eight loops. The hexagon OPE implementation reproduces the weak-coupling expansions of the bound-state energies, momenta and measures of arXiv:1407.1736; the single-gluon closed forms of arXiv:1310.5735 (all 21 entries through six loops through S^17, and the seven- to nine-loop LL and NLL entries through S^41); the one-loop relations W^(1)_[1] = X|_{T^1} and W^(1)_[2] = X|_{T²F²}; the three-loop T² terms of the remainder function published with arXiv:1308.2276 at every power of ln T through y^10, which probes the two-gluon measure at next-to-next-to-leading order; and the entries of table 1 of arXiv:1402.3307 through four loops in the helicity-zero sector, including the one published entry that probes both the N³LL bound state and the Born-level two-gluon term. The two implementations agree with each other through y^20. On the amplitude side, the near-collinear expansions of the published one- to five-loop symbols reproduce the T^1 and T²F^{±2} blocks at every power of ln T that the available data determine, and those of the six- and seven-loop lifts reproduce every available block (T^1 at all powers of ln T; T²F^{±2} from LL to N³LL including the two-gluon term; T²F^0 at LL).
* **Held-out data at nine loops.** Form factor: the NNLL sector T² ln⁶T, not used as input, is satisfied (0 of 135 equations violated). Amplitude: with nothing free, the T^1 blocks at LL and NLL and the T²F^0 block at LL, none of which entered the fixing, equal the OPE predictions; these blocks do not see the five OPE-fixed directions and so test everything before them (the Δ = 0 data, the lift and the origin step). Within the fixing steps, the 60 equations of the three T²F^{±2} steps have total rank 5 and no residual, so 55 of them are held-out predictions in the sectors ln^{5,6,7,8}T at S² and S⁴. At eight loops the NNLL closed form and the N³LL series each reproduce the value of the parameter that the NLL data fixed.
* **Second prime.** The amplitude-side chain at p2, with the hexagon space, tails and Δ = 0 projection rebuilt, gives identical ranks and free counts, the same surviving directions and an identical zero pattern over the 2,302,744 coordinates, which excludes an error confined to one prime. The form factor is certified at three primes entry by entry. The two-prime reconstruction of the amplitude coordinates is described in section 4.5.
* **Exact constraints.** The junction, parity and dihedral relations hold exactly on every coordinate of the particular solution of the lift and of each remaining direction, and every remaining direction vanishes on Δ = 0.
* **Symmetries and vanishing conditions on words** ([records/oracle_validation_results.json](records/oracle_validation_results.json)). Word coefficients evaluated from the quintuple representation were tested on random nonzero words generated by a random walk through the support, modulo p1. Of 1000 words with the first letter changed to one outside {â, b̂, ĉ}, all have coefficient zero; of 1000 with the last letter changed to one of â, b̂, ĉ, all vanish (the MHV final-entry condition); of 1000 with an extended-Steinmann pair (âb̂ or one of its images) inserted at a random slot, all vanish; of 1000 with an odd number of y letters, all vanish (parity); and of 1000 fully random words, all vanish. On 1000 random nonzero words (799 containing y letters, with even y counts up to twelve), the cyclic rotation, the flip and the parity map y_i → 1/y_i each return the same coefficient in every case (all nonzero words have an even number of y letters, so on them the parity image is the identity and that comparison is trivial; the substantive parity test is the vanishing of the 1000 odd-y words), and the p1 and p2 residues of all 1000 reconstruct to certified rationals. These conditions were imposed on the ansatz, so this tests the assembly of the result rather than its physics.
* **The restriction to Δ = 0** (same record). The coefficient of a y-free word in the full symbol must equal its coefficient in the flipped form factor. Sampling 80 of the 279 octuple files of the stored Δ = 0 symbol (84,955,130 terms read), 3000 random nonzero y-free words agree with the full symbol in every case, and 600 y-free words absent from their octuple file give zero; the 3000 coefficients reconstruct from the two primes to certified rationals. Only words whose last eight letters, in form-factor language, form one of the 279 pivot octuples can be checked this way. Since the full symbol was built from the Δ = 0 data, this is a consistency test of the lift and of the word evaluation, not an independent check of the form factor.
* **The sample file.** All 20,400 nonzero words of `samples/E9_sample_coefficients.txt` reconstruct from the two primes to certified rationals (largest height 68,111,961). The evaluation of all 20,630 words against the quintuple representation is listed in section 6.2.
* **Multi-Regge limit.** The super-leading-logarithmic identity (the ln^{≥L} classes of Disc_u R^(L) vanish) holds for the lifted six-, seven- and eight-loop symbols, a check of Regge factorisation on the output of the lift.
* **Invisibility of the residual directions.** The five directions left after the origin at nine loops (one at eight) vanish on Δ = 0, at the origin, in the strict collinear limit and at orders T^0 and T^1 at every power of ln T; they are visible only at order T². Three of the nine-loop directions (those left after the leading- and next-to-leading-logarithmic T² step) and the eight-loop direction were also checked to vanish in the multi-Regge limit to all logarithmic orders; the other two nine-loop directions were not checked there.

### 6.2 Checks made on the distributed files

Each of these compares the files of this page with each other or with published results; the recorded output is named.

* **The sample file against the quintuple representation.** The 20,630 words of `samples/E9_sample_coefficients.txt`, evaluated from the quintuple-coproduct files at both primes: all 20,630 agree, 230 of them zero in both ([records/02_output.txt](records/02_output.txt); the record prints a per-prime line for p2 and a summary line for both primes).
* **Multiple-final-entry structure and zero pattern.** From the quintuple-coproduct files at both primes: 1,018,297 nonzero coordinates, all 424 quintuple components nonzero, rank of the quintuple final entries 400; quadruple, triple, double and single final entries of rank 162, 62, 21 and 6; identical zero patterns at the two primes; and the complete symbol lies in the span of the lift before the OPE step and its five directions ([records/03_output.txt](records/03_output.txt)).
* **Eight-loop control against the published amplitude.** The distributed eight-loop lift, with its last direction fixed by the NLL two-gluon data (`lower_loops/`), against the published eight-loop symbol of arXiv:2308.08199: on 1000 random weight-16 words with nonzero coefficient (random walks respecting the first-entry condition and the forbidden adjacent pairs, with the last five letters among the 384 independent quintuple final entries of the published file), the published coefficient, computed exactly from the quintuple file of arXiv:2308.08199 with the coproduct tables of its basis and reduced modulo p1, agrees in all 1000 cases ([records/04_output.txt](records/04_output.txt)). This tests the amplitude side of the method end to end at the highest loop order where the answer is known, including the eight-loop flux-tube data that closed it.
* **The form-factor files.** The rational coefficient matrix, reduced modulo each of the three primes, against the three per-prime solutions: 0 mismatches of 450,468 entries (355,144 nonzero) at each prime; every denominator a power of two up to 2^14; maximum absolute numerator 132,843,110,400; the 279 octuple components with 295,186,924 terms in all ([records/05_output.txt](records/05_output.txt)).
* **Symmetries and vanishing conditions, repeated.** On 500 further random nonzero words: first letter moved outside {â, b̂, ĉ}, last letter set to one of â, b̂, ĉ, an extended-Steinmann pair inserted, and one non-y letter replaced by a y letter (odd parity) all give zero in 500 of 500 cases; the cyclic, flip and parity images all give the same coefficient in 500 of 500 cases (on these words, all with an even number of y letters, the parity image is the identity, so that part of the test is trivial) ([records/07_output.txt](records/07_output.txt)).
* **Certified coordinates.** Reconstructing each nonzero coordinate from its two residues gives a fraction within the symmetric bound (height below 2^30) for 1,014,740 coordinates. The certification rule also requires a power-of-two denominator, which 264 of those 1,014,740 lack, unlike all the other 1,014,476, so the rule does not certify those 264. Under the certification rule (height below 2^30 and a power-of-two denominator), 1,014,476 of the 1,018,297 nonzero coordinates (99.62%) are therefore certified; the 264 together with the 3,557 coordinates the reconstruction did not recover make 3,821 uncertified coordinates, all in 140 quintuples. This follows directly from the two modular files and has no separate record.
* **The septuple file against the quintuple representation.** 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 [records/08_output.txt](records/08_output.txt). The full comparison, with the exact coefficient from the septuple file and the two residues and the two-prime rational from the quintuple representation for every nonzero word coefficient, is [records/09_septuple_vs_quintuple_107053_words.txt.gz](records/09_septuple_vs_quintuple_107053_words.txt.gz).

  For these word coefficients the two-prime reconstruction can be compared with exact values. The quintuple representation gives each coefficient modulo the two primes p1 = 2^31 − 1 and p2 = 2147483629; for the 3,401 septuple word coefficients whose numerators are at least 2^30 in absolute value, the symmetric bound |n|, d < 2^30 does not recover the rational number (it returns no value for 3,092 and a wrong value for 309). The exact coefficients of the direct bootstrap all have power-of-two denominators at most 2^16 and numerators below 2^37 in absolute value; taking a bound shaped to these sizes, |n| ≤ 2^44 and 1 ≤ d ≤ 2^16 (for which the reconstruction is unique, since 2 · 2^44 · 2^16 < p1 · p2), the quintuple representation's two residues reconstruct to the direct bootstrap's exact value for all 107,053 nonzero word coefficients, including those 3,401. The bound was chosen from the sizes of the direct bootstrap's exact values, so this is a consistency check between the two computations, not a reconstruction from the two residues alone. The exact values and both residues are in the columns of record 09, so this can be checked from it. This says nothing directly about the 3,821 uncertified basis coordinates, which remain uncertified; a third prime could certify them.

## 7. Structural findings

These are properties of the nine-loop symbol and of the form factor as given by the distributed files.

* **Leading-logarithmic form-factor data no longer suffice.** At eight loops the leading-logarithmic form-factor OPE data fix the last parameter of the deep-back ansatz (142 → 43 → 27 → 0). At nine loops one parameter survives them and is fixed only by the next-to-leading logarithm (section 2.5).
* **The last amplitude directions need order T².** The five directions that survive the lift and the origin at nine loops (one at eight) are fixed only at order T² of the near-collinear expansion, the last of them only in the sector where the Born-level two-gluon continuum first contributes (section 4.4).
* **The quintuple final entries grow while the lower ones do not.** The 424 quintuple final entries allowed by the pair relations span 400 dimensions at nine loops, against 384 at eight loops (arXiv:2308.08199), so the nine-loop symbol satisfies 24 independent linear relations among its quintuple final entries, against 40 at eight loops. The quadruple, triple, double and single final entries span 162, 62, 21 and 6 dimensions, the same counts as at eight loops; in particular the quadruple final entries again span only 162 of the 166 allowed, although no relation beyond those defining W_4 was imposed. One reading of stable counts at low depth with a growing count at the deepest level is that the low-depth relations hold at every loop order and the deepest ones saturate only as the loop order grows; the counts are in [records/03_output.txt](records/03_output.txt).
* **Predicted form-factor final-entry spaces at weights 9–14.** Restricting the hexagon spaces H_9 … H_14 to Δ = 0 and mapping them by antipodal duality predicts symbol-level form-factor final-entry spaces of dimensions 491, 857, 1472, 2495, 4171 and 6891 at weights 9–14. This bears on the question, left open in arXiv:2204.11901, of the final-entry spaces beyond weight 8. These predicted spaces are expected, and at weights 9–13 assumed (section 5), to contain the true ones; at weight 8 the same construction gives 282 against the true 279. The nine-loop form factor fills 487 of the 491 predicted weight-9 directions, as the eight-loop form factor fills 279 of the predicted 282 at weight 8. The extended-Steinmann hexagon space at weight 14 has dimension 9014 = 7056 parity-even + 1958 parity-odd.
* **Coproduct spans of the form factor.** The {n, 1, …, 1} coproduct components of E_c^(9), for n = 1, …, 10, span
  3, 9, 21, 48, 108, 242, 517, 809, 487, 279
  dimensions (eight loops: 3, 9, 21, 48, 108, 242, 466, 279). So the 1674 weight-9 components whose nine-letter final strings end in one of the 279 independent octuple final entries (6 × 279 = 1674) span 487 dimensions. E_c^(9) therefore implies 1674 − 487 = 1187 relations among them. The predicted weight-9 final-entry space has 491 dimensions (1183 relations), so E_c^(9) satisfies four relations beyond the predicted ones and fills all but 4 of the predicted directions.

## 8. What remains untested

* Everything in this note is at symbol level; zeta-valued terms and the function are outside its scope.
* The five OPE-fixed directions have no check independent of the two-gluon flux-tube data, since no other sector of the near-collinear expansion or other limit that was checked sees them (in the multi-Regge limit, three of the five were checked and are invisible there; the other two were not checked).
* The multi-Regge limit and the coaction principle were not tested quantitatively at nine loops. The super-leading-logarithmic identity was checked only on the six- to eight-loop lifts; at nine loops three of the five OPE-fixed directions were checked to be invisible in this limit (the other two were not checked there), so such a test could not check those three. No quantitative comparison with the all-orders BFKL prediction was made at nine loops. The coaction principle was not tested on the amplitude.
* The rational form of 3,821 of the 1,018,297 nonzero basis coordinates (0.38%) is not certified by the two primes. For the word coefficients that determine the septuple file, the two residues reconstruct to the direct bootstrap's exact values under the shaped bound of section 6.2; for the basis coordinates themselves there is no such comparison, and a third prime could certify them.
* The duality-predicted final-entry spaces at weights 9–13 are assumed to contain the true ones at symbol level; "unique within the ansatz" never excludes a solution outside it (section 5).
* Every check is at one prime unless stated.
* The form-factor computation is a single computation by a single pipeline, which gave the symbol modulo the two primes. 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 [records/08_output.txt](records/08_output.txt).
* The number of terms of the fully expanded nine-loop symbol in nine letters was not counted; its restriction to Δ = 0 has more than thirty billion terms.
* The programs of the form-factor computation are not distributed. The sample file in `samples/` and the two representations of the symbol on this page are there to be compared against.

## 9. Reproducing the computation

The form-factor computation has four parts that can be repeated independently, each a linear-algebra statement with a target at lower loop order. All linear algebra is modulo a 31-bit prime; the flux-tube series that need very long expansions were generated modulo the 61-bit prime 2,305,843,009,213,693,921. The nine-loop systems need a machine with about 100 GB of memory for the form-factor seam and about 50 GB for the lift; every statement can be tested first at low weight.

1. **The spaces.** The hexagon space H_n: the inputs are the 41 pair relations as vectors in Q^81, of which the 26 integrability relations are the left null space of the wedge products d ln x_i ∧ d ln x_j evaluated at random rational points of the y parametrisation (see [conventions.md](../conventions.md)); the statement is the recursion from H_1 = span{â, b̂, ĉ} to H_13; the targets are the dimensions of table 1 of arXiv:1906.07116 and the parity splits. The form-factor spaces: the nine integrability relations, the pair and triple adjacency restrictions and the branch-cut condition; the targets are the dimensions 3, 9, 21, 51, 120, 279, 642, 1470 (front), 3, 9, 21, 48, 108, 249, 567, 1290, 2931, 6654 (coaction-restricted) and 3, 6, 12, 24, 45, 85, 155, 279 (back) of arXiv:2204.11901. The predicted back spaces: restrict H_w to Δ = 0, reverse and relabel; the targets are the published back spaces through weight 7, 282 at weight 8, and 491, 857, 1472, 2495, 4171, 6891 at weights 9–14.
2. **The form-factor seam.** The unknowns are c_ij with i over C_5 and j over B_13; the equations are the pair and triple adjacency relations across the seam (rows sampled by random rank-one functionals, then verified exactly; section 2.3), the branch-cut condition in the thirteen back slots, dihedral invariance, the strict collinear limit and the OPE rows; the targets are the cascades 142 → 43 → 27 → 0 at eight loops with C_5 ⊗ B_11 and 319 → 60 → 44 → 1 → 0 at nine loops, and, at eight loops, the 279 independent octuple final entries of the relation file of arXiv:2204.11901. For the form-factor OPE data, the two-particle residue sums of section 2.6 must reproduce the twenty published T² entries with k ≥ L − 3 through eight loops.
3. **The flip.** The antipodal map applied to the octuple components, with the tail relations of arXiv:2204.11901 used to expand the non-pivot octuples; the targets are the eight-loop count 1,671,656,292 and the Δ = 0 checks of section 6.1.
4. **The lift.** The unknowns are the 424 × 5431 coordinates; the equations are the projection of the Δ = 0 data onto the 111 y-free quintuples, the 41 pair relations at the junction layer by layer, parity, dihedral symmetry, the origin, and the OPE rows; the targets are the eight-loop chain of section 4.1, whose result can be compared with the published eight-loop amplitude as in [records/04_output.txt](records/04_output.txt), and the nine-loop ranks and free counts of the table there. For the hexagon OPE data, the residue sums of section 4.3 must pass the validations listed in section 6.1.

A reader with any other representation of the nine-loop symbol can compare it with the files here word by word, through the sample file or through the word-coefficient formula of section 4.5 applied to the quintuple-coproduct files.