# Conventions

This note fixes the notation, normalisations and arithmetic conventions used by every file on this page. The layouts of the individual files are described 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 are not repeated here. How the symbol was computed, and how it was checked, is described in [validation/method_and_validation.md](validation/method_and_validation.md).

Two computations of the nine-loop symbol are referred to on this page, always by method:

* the **form-factor computation**, which bootstrapped the nine-loop three-point form factor, mapped it to the amplitude on the surface $\Delta=0$ by antipodal duality and lifted it off that surface; it produced the quintuple-coproduct files in `amplitude/`;
* the **direct bootstrap**, a bootstrap of the nine-loop symbol in the space of hexagon functions, which produced the septuple-coproduct file in `MHV9/`.

The amplitude as a function (`function/`) was obtained separately.

All statements here are at symbol level unless stated otherwise.

## 1. Kinematics and cross ratios

The six-gluon amplitude depends on the three dual-conformal cross ratios
$$
\hat u=\frac{s_{12}s_{45}}{s_{123}s_{345}},\qquad
\hat v=\frac{s_{23}s_{56}}{s_{234}s_{123}},\qquad
\hat w=\frac{s_{34}s_{61}}{s_{345}s_{234}},
$$
and on the sign of the square root of
$$
\Delta=(1-\hat u-\hat v-\hat w)^2-4\hat u\hat v\hat w .
$$
Following arXiv:2308.08199, the amplitude variables carry hats, to distinguish them from the form-factor variables $u,v,w$ of section 7. The surface $\Delta=0$ is the parity-preserving surface.

With
$$
z_\pm=\tfrac12\left(-1+\hat u+\hat v+\hat w\pm\sqrt\Delta\right),\qquad
y_u=\frac{\hat u-z_+}{\hat u-z_-},
$$
and $y_v$, $y_w$ defined in the same way with $\hat u$ replaced by $\hat v$ and $\hat w$, the cross ratios can be written rationally in the $y$ variables:
$$
\hat u=\frac{y_u(1-y_v)(1-y_w)}{(1-y_uy_v)(1-y_uy_w)}\quad\text{and cyclically.}
$$
In this parametrisation all nine letters below are rational functions, which is convenient for deriving integrability relations and for evaluating at random points. On $\Delta=0$ one has $y_u=y_v=y_w=1$.

The coupling is $g^2=\lambda/(16\pi^2)$, with $\lambda=g_{\rm YM}^2N_c$ the 't Hooft coupling. A quantity with superscript $(L)$ is the coefficient of $g^{2L}$; at $L$ loops the symbol has weight $2L$, so the nine-loop symbol has weight 18.

## 2. The letters, the dihedral action and parity

The symbol alphabet of the amplitude has nine letters. In the multiplicative form of arXiv:2308.08199, used throughout,
$$
\hat a=\frac{\hat u}{\hat v\hat w},\quad \hat b=\frac{\hat v}{\hat w\hat u},\quad \hat c=\frac{\hat w}{\hat u\hat v},\quad
\hat d=\frac{1-\hat u}{\hat u},\quad \hat e=\frac{1-\hat v}{\hat v},\quad \hat f=\frac{1-\hat w}{\hat w},\quad
y_u,\ y_v,\ y_w .
$$
Each of $\hat a,\hat b,\hat c$ contains exactly one three-particle invariant, which is why the Steinmann relations become conditions on adjacent letters.

**Coefficients and powers of two.** Coefficients refer to the letters as written, not to their square roots. A symbol with integer coefficients in the additive alphabet $\{\hat u,1-\hat u,\dots\}$ has rational coefficients with power-of-two denominators in this alphabet, because, for example, $\ln\hat u=-\tfrac12(\ln\hat b+\ln\hat c)$. This is the origin of the powers of two in the denominators of the word coefficients on this page. That the nine-loop coefficients have power-of-two denominators is an observed property, not a theorem: all 107,053 nonzero word coefficients that determine the septuple file, which the direct bootstrap gives exactly, have power-of-two denominators dividing $2^{16}$. The certification rule of section 9.4 relies on this property.

**Words.** Words are written first entry first: in $x_1\otimes x_2\otimes\cdots\otimes x_{18}$ the first entry is $x_1$ (on the left) and the final entry is $x_{18}$ (on the right). The files write the same word as a list of letter names in this order (section 3): space-separated, comma-separated, or concatenated (the pivot words, and the words in the $\Delta=0$ octuple files), as described in each file's header or FORMAT.md.

**Dihedral action.** The dihedral group is generated by the cyclic rotation
$$
\hat a\to\hat b\to\hat c\to\hat a,\qquad \hat d\to\hat e\to\hat f\to\hat d,\qquad y_u\to1/y_v,\quad y_v\to1/y_w,\quad y_w\to1/y_u,
$$
and the flip
$$
\hat a\leftrightarrow\hat b,\qquad \hat d\leftrightarrow\hat e,\qquad y_u\leftrightarrow y_v ,
$$
with $\hat c$, $\hat f$ and $y_w$ fixed by the flip. Because $\ln(1/y)=-\ln y$, a letter $1/y_v$ in a symbol stands for $y_v$ with the coefficient's sign changed.

**Parity.** Parity sends $y_i\to1/y_i$ and fixes the other six letters. The amplitude is parity even, so every word with an odd number of $y$ letters has coefficient zero. On $\Delta=0$ the three $y$ letters equal one, and the symbol reduces to a symbol in the six letters $\hat a,\dots,\hat f$.

## 3. Letter names in the files

The same nine amplitude letters appear under three sets of names, and the form factor has its own six letters. The table lists which files use which names.

| letter | this note | data page header; `MHV9/` septuple file and its JSON twin; `function/` files | `amplitude/FORMAT.md`; `samples/`; `hexagon_space/MHV_final_entry_spaces.txt`; `validation/records/`; the data page's installation check of the symbol | integer index in `.npz` arrays of the amplitude side |
|---|---|---|---|---|
| $\hat u/(\hat v\hat w)$ | $\hat a$ | `a` | `ah` | 0 |
| $\hat v/(\hat w\hat u)$ | $\hat b$ | `b` | `bh` | 1 |
| $\hat w/(\hat u\hat v)$ | $\hat c$ | `c` | `ch` | 2 |
| $(1-\hat u)/\hat u$ | $\hat d$ | `mu` | `dh` | 3 |
| $(1-\hat v)/\hat v$ | $\hat e$ | `mv` | `eh` | 4 |
| $(1-\hat w)/\hat w$ | $\hat f$ | `mw` | `fh` | 5 |
| $y_u$ | $y_u$ | `yu` | `yu` | 6 |
| $y_v$ | $y_v$ | `yv` | `yv` | 7 |
| $y_w$ | $y_w$ | `yw` | `yw` | 8 |

The names `a`, `b`, `c`, `mu`, `mv`, `mw` are those of the data files of arXiv:2308.08199 (for example `MHV8quintuples.txt`) and arXiv:1903.10890; the names `ah` … `fh` ("a-hat" … "f-hat") are used in the files of the quintuple representation. In the septuple and function files a coproduct component is written `E(a1,...,ak)` with `ak` the final entry, as in `MHV8quintuples.txt`.

**Form-factor letters.** The files in `form_factor/` and the symbol on $\Delta=0$ in `delta0/E9_octuples_Delta0_mod2147483647.tar.gz`, stored as its 279 octuple coproduct components, use the unhatted letters `a` … `f` of the form factor (section 7), with integer index 0 … 5 in the `.npz` arrays. **These are different letters from the amplitude letters `a`, `b`, `c` of the septuple and function files:** they are functions of the form-factor variables $u,v,w$ with $u+v+w=1$, and they are converted to amplitude letters by the antipodal map of section 8. The file `delta0/E9_Delta0_quintuple_coproducts_mod2147483647.npz` is already in amplitude letters (index 0 … 8 as in the table).

## 4. The normalisation: $\mathcal E$ and $\mathcal E^{(1)}$

The amplitude is written in the BDS-like normalisation of arXiv:1903.10890 and arXiv:2308.08199 (the BDS ansatz is that of hep-th/0505205):
$$
A_6^{\rm MHV}=A_6^{\rm BDS\text{-}like}\,\rho(g^2)\,\mathcal E(\hat u,\hat v,\hat w;g^2),\qquad
\mathcal E=\frac1\rho\exp\!\Big[\frac{\Gamma_{\rm cusp}}4\,\mathcal E^{(1)}+R_6\Big],
$$
where $R_6$ is the remainder function, $\Gamma_{\rm cusp}$ the cusp anomalous dimension, and $\mathcal E=1+\sum_{L\ge1}g^{2L}\mathcal E^{(L)}$. The one-loop term is
$$
\mathcal E^{(1)}=\sum_{i=u,v,w}\mathrm{Li}_2\!\big(1-1/\hat u_i\big),\qquad
\mathcal S\big[\mathcal E^{(1)}\big]=-\tfrac12\big(\hat a\otimes\hat e+\hat a\otimes\hat f+\hat b\otimes\hat d+\hat b\otimes\hat f+\hat c\otimes\hat d+\hat c\otimes\hat e\big),
$$
with $\hat u_i\in\{\hat u,\hat v,\hat w\}$. The factor $\rho(g^2)$ is independent of the kinematics (its coefficients are zeta values), so it is invisible at symbol level, where the normalisations of arXiv:1903.10890 and arXiv:2308.08199 coincide. The object of the symbol files on this page is $\mathcal S[\mathcal E^{(9)}]$, the symbol of the coefficient of $g^{18}$. The function-level files in `function/` state in their headers the normalisation constant they use.

## 5. The hexagon symbol spaces $H_n$ and the nested coproduct basis

$H_n$ is the space of weight-$n$ symbols in the nine letters that satisfy three conditions:

1. **First entry:** the first letter lies in $\{\hat a,\hat b,\hat c\}$, so that the only branch cuts are in $\hat u,\hat v,\hat w$.
2. **Integrability:** 26 linear relations on each adjacent pair of letters (the rank of the relations among the wedge products $d\ln x_i\wedge d\ln x_j$, obtained exactly from the $y$ parametrisation).
3. **Extended Steinmann (cluster adjacency)** (arXiv:1903.10890, arXiv:1906.07116, arXiv:1710.10953): no two different letters among $\hat a,\hat b,\hat c$ are adjacent; the six ordered pairs $\hat a\hat d$, $\hat d\hat a$, $\hat b\hat e$, $\hat e\hat b$, $\hat c\hat f$, $\hat f\hat c$ never occur; and the twelve further relations of eq. (3.1) of arXiv:1906.07116 hold.

Together these are 41 independent linear relations on the 81 ordered pairs of adjacent letters.

**Recursive construction.** A weight-$n$ symbol with final letter $x$ is $\sum_xF_x\otimes x$ with $F_x\in H_{n-1}$; the 41 relations on the last two letters become a linear system for the coordinates of the $F_x$, whose kernel is $H_n$.

**Nested coproduct basis.** $H_n$ is stored, modulo a prime, as nested coproduct tensors:
$$
F^{(n)}_i=\sum_{k=0}^{8}\sum_j T_n[k]_{ij}\,F^{(n-1)}_j\otimes\phi_k ,
$$
where $\phi_k$ runs over the nine letters in the index order of section 3 (letter $\phi_k$ appended at the end). The coefficient of an explicit word in a basis element is then a product of $n$ matrices. The arrays, and the formula for the coefficient of a word of the amplitude, are in [amplitude/FORMAT.md](amplitude/FORMAT.md) (sections 1 and 3).

**Word-pivot form.** The nested basis is in word-pivot form: each basis element $F^{(n)}_i$ has a pivot word whose coefficient is 1 in $F^{(n)}_i$ and 0 in every other basis element of the same weight. (This can be checked from the distributed tensors, and it holds at both primes, with the same pivot words.) The tails $S_\alpha$ of section 6 are in word-pivot form too.

**Dimensions.** The dimensions of $H_n$ at weights $n=0,\dots,13$ are
$$
1,\ 3,\ 6,\ 13,\ 26,\ 51,\ 98,\ 184,\ 340,\ 613,\ 1085,\ 1887,\ 3224,\ 5431,
$$
those of table 1 of arXiv:1906.07116. The parity splits (even + odd) of the distributed bases at weights 11, 12 and 13 are $1505+382$, $2553+671$ and $4269+1162$. The weight-11 function basis of arXiv:1903.10890, in which the septuple file is written, has 1503 parity-even and 382 parity-odd elements with non-vanishing symbol (as the septuple file's header states), two fewer parity-even elements than $H_{11}$. At weight 14, distributed at the first prime only, the dimension is $9014=7056+1958$. The subspace of $H_n$ whose restriction to $\Delta=0$ vanishes has dimensions $0,0,0,1,2,6,13,29,58,113,210,382,672,1164$ at weights $0,\dots,13$; besides the parity-odd sector, which vanishes identically on $\Delta=0$, it contains one parity-even element at weight 12 and two at weight 13. These parity-even elements generate the ambiguities that remain when the symbol is lifted off $\Delta=0$ (see [validation/method_and_validation.md](validation/method_and_validation.md)).

## 6. Coproducts, final entries and the tail spaces $W_1,\dots,W_5$

**Coproduct components.** For a weight-$n$ symbol $F$ and a letter $x$, $F^{x}$ denotes the $\{n-1,1\}$ coproduct component: the weight-$(n-1)$ symbol multiplying $\otimes x$ in the last slot. $F^{x_1,\dots,x_k}$ is the iterated $\{n-k,1,\dots,1\}$ component, so that
$$
F=\sum_{x_1,\dots,x_k}F^{x_1,\dots,x_k}\otimes x_1\otimes\cdots\otimes x_k ,
$$
with $x_k$ the final entry. In the septuple and function files this component is written `E(x1,...,xk)`.

**Final entries.** The MHV final-entry condition, which follows from the $\bar Q$ equation (arXiv:1112.1060), is
$$
\mathcal E^{\hat a}=\mathcal E^{\hat b}=\mathcal E^{\hat c}=0 .
$$
arXiv:2308.08199 found further linear relations among the multiple final entries of the amplitudes through eight loops: among the double final entries, six follow from the $\bar Q$ equation and four more hold empirically; three more hold among the triple final entries.

**Tail spaces.** $W_k$ is the space spanned by the allowed $k$-tuple final entries, closed under the dihedral group and derived from these relations. The dimensions are
$$
\dim W_1,\dots,\dim W_5=6,\ 21,\ 62,\ 166,\ 424 .
$$
The bases are word-pivot bases constructed as in arXiv:2308.08199, with exact rational tails, in [`hexagon_space/MHV_final_entry_spaces.txt`](hexagon_space/MHV_final_entry_spaces.txt) (line format in [amplitude/FORMAT.md](amplitude/FORMAT.md), section 1). Each tail has a layer (0 to 5, given in the file). The 111 tails of layer 0 are those whose pivot word contains no $y$ letter; below they are called the $y$-free quintuples. (Only 3 of them contain no $y$ letter in any term; on $\Delta=0$ every word containing a $y$ letter drops out, because $y_u=y_v=y_w=1$ there.) A $y$-free word is a word with no $y$ letter; these are the only words that survive on $\Delta=0$.

**The quintuple representation.** The nine-loop symbol is written
$$
\mathcal S\big[\mathcal E^{(9)}\big]=\sum_{\alpha=1}^{424}E_\alpha\otimes S_\alpha,\qquad E_\alpha\in H_{13},
$$
the representation of arXiv:2308.08199 one loop (two units of weight) higher. $S_\alpha$ is the $\alpha$-th quintuple tail and $E_\alpha$ the quintuple coproduct component belonging to it, given by its coordinates in the nested basis of $H_{13}$.

**The span of the components is a property of the amplitude, not an input.** As properties of the distributed files: the 424 components $E_\alpha$ of the nine-loop symbol span 400 dimensions (the 424 components of the eight-loop symbol span 384), and the quadruple, triple, double and single final entries span 162, 62, 21 and 6 dimensions, as at eight loops (`validation/records/03_output.txt`). The components along the 111 $y$-free quintuples (layer 0) have rank 108 at both primes, as computed from the distributed files; arXiv:2308.08199 found 108 at eight loops. The septuple file in `MHV9/` records the same structure in its relation tables: 400 independent quintuples at level 5.

## 7. The three-point form factor: variables, alphabet and spaces

The form factor is the three-point form factor of the chiral part of the stress-tensor multiplet, in the BDS-like normalisation $\mathcal E_c$ of arXiv:2204.11901. It depends on two variables, through $u+v+w=1$, and its symbol has the six letters
$$
a=\frac u{vw},\quad b=\frac v{wu},\quad c=\frac w{uv},\quad d=\frac{1-u}u,\quad e=\frac{1-v}v,\quad f=\frac{1-w}w ,
$$
without the square-root rescaling of $a,b,c$ (the normalisation of the ancillary file `Esymb.txt` of arXiv:2204.11901). Words are written first letter first; $\mathcal E_c^{(L)}$ is the coefficient of $g^{2L}$ and has uniform weight $2L$. The symbol is invariant under the dihedral group generated by $a\to b\to c\to a$, $d\to e\to f\to d$ and the flip $a\leftrightarrow b$, $d\leftrightarrow e$.

**Constraints** (those of arXiv:2204.11901):
* integrability: nine relations on adjacent pairs (the six antisymmetric relations and the three extra pair relations of arXiv:2204.11901);
* adjacency: twelve forbidden ordered pairs, and the four independent triple relations among the six dihedral images of $a\otimes(a+b+c)\otimes b$;
* branch cuts: the restriction of the symbol to $u\to1$, taken in every slot, vanishes;
* dihedral invariance;
* the strict collinear limit $\mathcal E_c^{(L)}\big|_{v\to0}=\frac1{L!}\left(-\ln^2v-\ln^2d\right)^L$;
* the near-collinear form-factor OPE at order $T^2$ (arXiv:2009.11297, arXiv:2105.13367, arXiv:2112.10569).

arXiv:2204.11901 also imposes an $L$-fold discontinuity condition; the form-factor computation did not impose it at nine loops.

**Spaces.** Three families of spaces enter.
* The **front space** of weight $w$: weight-$w$ symbols obeying the local conditions (first entry, integrability, adjacency, branch cuts). Dimensions $3,9,21,51,120,279,642,1470$ at weights 1–8.
* The **restricted front spaces** $\mathcal C_w$: at weight 4 the front space is restricted to the 48-dimensional span of the weight-4 coproducts of the six-loop form factor (an application of the coaction principle, as in arXiv:2204.11901), and $\mathcal C_w$ is built on that restriction. Dimensions $3,9,21,48,108,249,567,1290,2931,6654$ at weights 1–10, as in arXiv:2204.11901, and $15{,}090$ at weight 11.
* The **back (final-entry) spaces** $B_W$, spanned by the allowed $W$-tuple final entries. From the relations published with arXiv:2204.11901, the dimensions are $3,6,12,24,45,85,155,279$ at weights 1–8; the 279 independent octuple final entries are those of the relation file `FFmultifinalentry.txt` of arXiv:2204.11901. At weights 9–14 (weight 14 modulo $p_1$ only) the files carry the final-entry spaces predicted from the hexagon space by antipodal duality (section 8), of dimensions
$$
491,\ 857,\ 1472,\ 2495,\ 4171,\ 6891 .
$$
These predicted spaces are used as an ansatz, so they need only contain the true final-entry spaces; at weight 8 the prediction has dimension 282 and contains the true 279-dimensional space. How they were obtained, and why containment is what matters, is described in [validation/method_and_validation.md](validation/method_and_validation.md).

An ansatz is a product $\mathcal C_w\otimes B_W$ with $w+W=2L$; the **seam** is the junction between the two factors, where the adjacency conditions still have to be imposed. The nine-loop form factor is given in the ansatz $\mathcal C_5\otimes B_{13}$ (108 × 4171 coefficients) as
$$
\mathcal E_c^{(9)}=\sum_{i,j}c_{ij}\,F_i\otimes B_j ,
$$
and as its 279 weight-10 octuple coproduct components. Both are described in [form_factor/FORMAT.md](form_factor/FORMAT.md), and the coefficient matrix also in [form_factor/FORMAT_C5xB13.md](form_factor/FORMAT_C5xB13.md).

## 8. Antipodal duality and its normalisation factors

Antipodal duality (arXiv:2112.06243, arXiv:2308.08199) states that on the parity-preserving surface $\Delta=0$ the amplitude's symbol is the antipode of the form factor's. For $y$-free words,
$$
c_{\mathcal E^{(L)}}(x_1,\dots,x_{2L})=(-1)^{2L}\,2^{-\#\{d,e,f\}}\;c_{\mathcal E_c^{(L)}}\big(\varphi^{-1}(x_{2L}),\dots,\varphi^{-1}(x_1)\big),\qquad
\varphi^{-1}:\ \hat a,\hat b,\hat c,\hat d,\hat e,\hat f\ \mapsto\ d,e,f,a,b,c ,
$$
where $c_X(\cdot)$ is the coefficient of a word in the symbol of $X$. The sign $(-1)^{2L}$ is the antipode's sign for a word of weight $2L$, and equals $+1$. $\#\{d,e,f\}$ counts the form-factor letters from $\{d,e,f\}$ in the reversed word; the letter map sends these letters to square roots of amplitude letters, which is the origin of the factor $2^{-\#\{d,e,f\}}$.

Equivalently: multiply the form-factor coefficients by 2 for each letter from $\{a,b,c\}$ (the rescaling in which they are all integers); the flipped coefficient in the hatted alphabet is that integer divided by $4^L$.

**In practice**, to turn a word and coefficient of the form factor (for example from `form_factor/` or from the $\Delta=0$ archive `delta0/E9_octuples_Delta0_mod2147483647.tar.gz`, whose 279 files store the symbol as its octuple coproduct components in the form-factor alphabet and normalisation) into the amplitude on $\Delta=0$: reverse the word; rename $a,b,c,d,e,f$ to $\hat d,\hat e,\hat f,\hat a,\hat b,\hat c$ (`dh`, `eh`, `fh`, `ah`, `bh`, `ch`); and divide the coefficient by 2 for each $d$, $e$ or $f$ in the form-factor word. For coproduct components: on $\Delta=0$ the amplitude's component $\mathcal E^{x_1,\dots,x_k}$ is obtained by taking the terms of the form-factor symbol whose first $k$ letters are $\varphi^{-1}(x_k),\dots,\varphi^{-1}(x_1)$, removing those $k$ letters, and flipping the rest as above, with $\#\{d,e,f\}$ counted over the whole word, removed letters included. In particular the form factor's octuple final entries become the amplitude's octuple first entries.

**Status.** The duality has been verified through eight loops (arXiv:2112.06243, arXiv:2308.08199). The form-factor computation uses it at nine loops in two directions: forwards, to obtain the amplitude on $\Delta=0$ from the nine-loop form factor, and backwards, to predict the form factor's final-entry spaces at weights 9–14 from the hexagon space. At nine loops it is an assumption of that computation (see [validation/method_and_validation.md](validation/method_and_validation.md)). The direct bootstrap does not use antipodal duality (its inputs are listed in the header of the septuple file), and the septuple file agrees with the quintuple representation on every word coefficient compared (section 9.4). This agreement is consistent with the duality holding at nine loops on $\Delta=0$, at symbol level.

## 9. Finite-field arithmetic, rational reconstruction and certification

### 9.1 Primes and residues

Linear algebra is done exactly over finite fields $\mathbb F_p$, and rational numbers are recovered at the end. The primes are:

| prime | value | used for |
|---|---|---|
| $p_1$ | $2^{31}-1=2{,}147{,}483{,}647$ | form factor and amplitude |
| $p_2$ | $2{,}147{,}483{,}629$ | form factor and amplitude |
| $p_3$ | $2{,}147{,}483{,}587$ | form factor only |
| — | $2{,}305{,}843{,}009{,}213{,}693{,}921$ (61 bits) | the flux-tube series that needed very long expansions (`ope_data/`) |

The hexagon-space tarball also carries the tensors at four 21-bit primes, with their exact reconstruction through weight 9 (`hexagon_space/hexagon_space_w13.tar.gz`, described in its `README.txt`). Residues are non-negative representatives in $[0,p)$. A rational $n/d$ corresponds to the residue $n\,d^{-1}\bmod p$; for a negative integer $r$ this is $p-|r|$. Products of two 31-bit residues can be formed exactly in double-precision floating point by splitting each residue into 16-bit halves.

### 9.2 Rules

1. **Parameter counts across primes.** The rank of a rational matrix reduced modulo $p$ can only be lower than its rank over $\mathbb Q$. So the number of free parameters counted modulo $p$ can only over-estimate the number over $\mathbb Q$, and zero free parameters modulo a prime implies zero free parameters over $\mathbb Q$.
2. **Reconstruction.** Chinese remaindering combines the residues at $k$ primes into a residue modulo $P=\prod_i p_i$, and rational reconstruction (P. S. Wang, Proc. SYMSAC '81, 212; P. S. Wang, M. J. T. Guy and J. H. Davenport, SIGSAM Bull. 16 (1982) 2) recovers $n/d$ provided $|n|,d<\sqrt{P/2}$. More generally, a residue modulo $P$ determines at most one fraction $n/d$ with $|n|\le N$ and $1\le d\le D$ whenever $2ND<P$; the reconstruction may use any such box.
3. **Check at a further prime.** A reconstructed rational, reduced modulo a prime not used in the reconstruction, must equal the residue computed at that prime.

### 9.3 The form factor (three primes)

The coefficient matrix $c_{ij}$ of section 7 (108 × 4171 = 450,468 entries, of which 355,144 are nonzero) is reconstructed from $p_1,p_2,p_3$: $P\approx9.9\times10^{27}$, reconstruction bound $\sqrt{P/2}\approx7.0\times10^{13}$. Every entry reconstructs to a fraction of height at most $1.33\times10^{11}$, far below the bound, and every denominator is a power of two dividing $2^{14}$. This is what "certified" means for the form factor, and all 450,468 entries are certified. The statistics, including the false-positive probability per entry, are in `form_factor/E9_C5xB13_rational_3primes.json`. The 279 octuple coproduct components (295,186,924 terms) are reconstructed from the same three primes, with identical word supports at the three primes and denominators dividing $2^{14}$.

### 9.4 The amplitude (two primes)

The quintuple representation is delivered modulo $p_1$ and modulo $p_2$; the files modulo each prime are the primary data. For two primes $P=p_1p_2\approx4.6\times10^{18}$.

**Certification rule.** A two-prime reconstruction is *certified* when the fraction has $|n|<2^{30}$ and $d<2^{30}$ (the symmetric bound) **and** its denominator is a power of two (section 2). The second condition is what rejects most fractions that fit the bound by accident: such a fraction will generally not have a power-of-two denominator.

**Worked example: the rational value of a coordinate or word coefficient from its two residues.** Given residues $r_1$ modulo $p_1$ and $r_2$ modulo $p_2$:

1. *Chinese remaindering.* Form the unique $r\in[0,M)$, $M=p_1p_2$, with $r\equiv r_1\pmod{p_1}$ and $r\equiv r_2\pmod{p_2}$: $r=r_1+p_1\big((r_2-r_1)\,p_1^{-1}\bmod p_2\big)$.
2. *Rational reconstruction with $N=2^{30}$* (in the code below, the defaults $N=D=2^{30}-1$ express the same bound). Run the extended Euclidean algorithm on $(M,r)$, keeping each remainder $s_k$ together with the cofactor $t_k$ for which $s_k\equiv t_k\,r\pmod M$, and stop at the first remainder $s_k<N$. Set $n=s_k$, $d=t_k$, and change the sign of both if $d<0$.
3. *Within the bound?* Accept the fraction $n/d$ only if $0<d<N$, $|n|<N$ and $\gcd(n,d)=1$. Otherwise no fraction lies within the bound, and the value is not certified.
4. *Power of two?* Certify $n/d$ only if $d$ is a power of two.

For example, the coordinate in row 0, column 0 of the array `E0` (0-based) has residues 1,072,728,311 and 1,072,728,302; then $r=2{,}305{,}842{,}987{,}737{,}843{,}969$ and the value is $-2027025/2$, certified. The word of the data page's installation check has residues 829,521,918 and 1,173,913,588 and gives $-105757/65536$. In Python, with the arithmetic in Python integers (not numpy integers, which would overflow):

```python
from math import gcd
import numpy as np

P1, P2 = 2147483647, 2147483629
M = P1 * P2          # about 4.6e18

def crt(r1, r2):
    """The residue r mod M = P1*P2 with r = r1 mod P1 and r = r2 mod P2."""
    return (r1 + P1 * (((r2 - r1) * pow(P1, -1, P2)) % P2)) % M

def reconstruct(r1, r2, N=2**30 - 1, D=2**30 - 1):
    """The fraction n/d with residues (r1, r2), |n| <= N, 0 < d <= D and d a
    power of two, or None. The defaults are the certification rule."""
    r = crt(int(r1), int(r2))
    if r == 0:
        return (0, 1)
    r0, s, t0, t = M, r, 0, 1            # extended Euclid on (M, r)
    while s > N:                           # stop at the first remainder <= N
        q = r0 // s
        r0, s = s, r0 - q * s
        t0, t = t, t0 - q * t
    n, d = (s, t) if t > 0 else (-s, -t)
    if not (0 < d <= D and abs(n) <= N and gcd(n, d) == 1):
        return None                        # no fraction within the bound
    if d & (d - 1):
        return None                        # denominator not a power of two
    return (n, d)

E1 = np.load("E9_symbol_complete_mod2147483647.npz", allow_pickle=False)["E0"]
E2 = np.load("E9_symbol_complete_mod2147483629.npz", allow_pickle=False)["E0"]

# certified value of the coordinate of E_alpha on basis element i (0-based):
#     reconstruct(E1[alpha, i], E2[alpha, i])
# value within the shaped bound |n| <= 2**44, d <= 2**16 (not a certification):
#     reconstruct(E1[alpha, i], E2[alpha, i], N=2**44, D=2**16)
```

The two `.npz` files are in `amplitude/` ([amplitude/FORMAT.md](amplitude/FORMAT.md), section 2). Applied to every nonzero coordinate, this certifies 1,014,476 of the 1,018,297. The same function applies to the residues of any word coefficient computed from the quintuple representation.

**Basis coordinates.** Of the 2,302,744 coordinates of the 424 × 5431 matrix, 1,018,297 are nonzero, with identical zero patterns at the two primes, and every one of the 424 components $E_\alpha$ is nonzero. Their rational form is not distributed as a file: it follows from the two modular files by Chinese remaindering and rational reconstruction, with the certification rule above.

* For 1,014,740 nonzero coordinates the reconstruction finds a fraction within the symmetric bound. The certification rule also requires a power-of-two denominator, which 264 of them lack (their denominators have 28 to 30 bits), unlike all the other 1,014,476; those 264 are not certified.
* For the other 3,557 nonzero coordinates no fraction within the bound exists.

Each coordinate is itself a word coefficient: because the nested basis of $H_{13}$ and the tails are in word-pivot form (section 5), the coordinate of $E_\alpha$ on the basis element $F^{(13)}_i$ equals the coefficient of the weight-18 word made of the pivot word of $F^{(13)}_i$ followed by the pivot word of $S_\alpha$. So the certification rule applies to coordinates exactly as it does to word coefficients.

So 1,014,476 of the 1,018,297 nonzero coordinates (99.62%) are certified, all with denominators dividing $2^{16}$, and 3,821 (the 3,557 with no fraction within the bound plus the 264 whose reconstructed denominators are not powers of two) are **uncertified**; they lie in 140 of the 424 rows. The rational form of the 3,821 uncertified coordinates is not established by the two primes; a third prime could certify them. The residues of all coordinates, certified or not, are exact results modulo each prime, so the coefficient of any word computed from them is exact modulo each prime.

**Every coordinate, with a shaped bound.** With the box $|n|\le2^{44}$, $1\le d\le2^{16}$ in place of the symmetric bound (the second call in the code above), all 1,018,297 nonzero coordinates reconstruct, every one with a power-of-two denominator. The fraction is unique, since $2\cdot2^{44}\cdot2^{16}<p_1p_2$ means at most one fraction in the box has given residues, and on the 1,014,476 certified coordinates it equals the certified value. The largest numerator is 104,433,281,715 in absolute value (37 bits), the largest denominator is $2^{16}$, and exactly the 3,821 uncertified coordinates have numerators of absolute value at least $2^{30}$. This is not a certification. The box was chosen to fit the sizes of the direct bootstrap's exact values after they were known (see the word coefficients below). It makes the reconstruction unique and self-consistent, but the two primes alone do not show that the uncertified coordinates have these values.

Coefficients of randomly chosen words are typically much smaller than the largest coordinates: in `samples/E9_sample_coefficients.txt`, all 20,400 words with nonzero coefficient reconstruct to certified rationals, with denominators dividing $2^{16}$ and numerators below $7\times10^{7}$ in absolute value.

**Word coefficients and the direct bootstrap.** The septuple file is determined by 107,053 nonzero word coefficients, which the direct bootstrap gives as exact rationals and the quintuple representation gives modulo $p_1$ and $p_2$. Under the certification rule, 103,652 of them are certified from the two residues, and each equals the exact value; the other 3,401, whose numerators are at least $2^{30}$ in absolute value, are not certified, and none is certified wrongly. With a bound shaped to the sizes of the exact values, $|n|\le2^{44}$ and $1\le d\le2^{16}$ (unique, since $2\cdot2^{44}\cdot2^{16}<p_1p_2$), the two residues reconstruct to the exact value for all 107,053. That bound was chosen after the exact sizes were known, so this is a comparison with the direct bootstrap, not a certification by the two primes alone. Details are in [validation/method_and_validation.md](validation/method_and_validation.md), section 6.2, and the comparison is recorded in `validation/records/09_septuple_vs_quintuple_107053_words.txt.gz`.

### 9.5 The direct bootstrap

The septuple file in `MHV9/` gives exact rational coefficients. As its header states, the direct bootstrap was carried out modulo six primes; its word coefficients were reconstructed from five of them and checked at the sixth (rule 3), with no failures.