CONVENTIONS for the weight-six record (cegm-x36, weight six)
===========================================================

Equation and section labels refer to the paper
"Factorization without clustering and the six-point Grassmannian string
integral" (M. D. Schwartz) in the version of 2026-09-28, in the format
Eq. (N) [label]: N the number printed in that version and label the
LaTeX label, which survives renumbering.  That version writes K(3,6) for the
integral and X(3,6) for the configuration space of six points in the plane
over whose positive part it is taken; this file below its header and the
data files, whose wording is kept from the first edition, write X(3,6) for
both.  References used in this record:
  Eq. (31) [eq:seed]        the generic rational point s* (the "seed point" here), Sec. 2.4
  Eq. (32) [eq:s0]          the base point s0, Sec. 2.4
  Eq. (33) [eq:basetower]   the expansion at s0 through weight eight; its c6 entry is the
                           exact A, B of this record at s0 (repeated in Table 2)
  Eqs. (93)-(94) [eq:Q5, eq:R5]  Q5, R5 at s*, Sec. 5.3
  Eq. (92) [eq:QRfun]       c5(s) = Q5(s) zeta(5) + R5(s) zeta(2) zeta(3), Sec. 5.3
  Eq. (95) [eq:ABfun]       c6(s) = A(s) pi^6 + B(s) zeta(3)^2, Sec. 5.3
  Eq. (98) [eq:p3]          the third printed point s~ (not a point of this record), Sec. 5.4;
                           its coefficients through weight six are Eq. (142) [eq:Bp3tower]
  Eq. (14) [eq:laurent]     K(3,6)(eps s) = sum_w eps^(w-4) c_w(s) and the homogeneity of c_w, Sec. 2.1
  Eq. (17) [eq:chamber]     the convergence region (all sixteen channel forms positive), Sec. 2.1
  Eq. (23) [eq:posint]      the positive parametrization, Sec. 2.2, with the matrix of
                           Eq. (20) [eq:matrix36], the monomial minors of Eq. (21) [eq:monomials]
                           and the ten polynomials P_t of Eq. (22) [eq:polys]
  Eq. (28) [eq:coneform]    the cone expansion, Sec. 2.3
  Eq. (85) [eq:WrCr]        c_w(s) = sum_r W_r(s) C_r over cone constants (the "census" here)
  Appendix B.4 [ssec:num-expansion]  how A, B were determined (543 coordinates, 596
                           conditions, 190 points) and tested (90 further points): this record
  Sec. 7.2 [ssec:count-law]  the seven-point reality scan and its three walls (prose, no
                           equation; referred to from Sec. 6.4 [ssec:count-reality])
  Sec. 7.4 [ssec:exp-beyond] the D4 cluster twist: paragraph "The D_4 cluster string
                           integral", Eq. (124) [eq:d4univ], Eq. (125) [eq:c4split]; the
                           deformation first appears at Eq. (40) [eq:d4edge], Sec. 3.1
Not printed in that version, and therefore reference values / supplementary
data of this record: the exact A, B at s1 and s2, and the whole twist-axis
material of item 3 (the functions c2(gamma), c3(gamma) of twist_gamma.json,
their poles, the twist charges); the paper prints the D4 twist family only
in the weight-four paragraph of Sec. 7.4.

The description strings inside the JSON data files are unchanged from the
first two editions and cite the numbering of an earlier version of the
paper (then titled "Grassmannian string integrals").  Correspondence,
earlier numbering -> numbering of 2026-09-28:
  "Eq. (11)" seed point              -> Eq. (31) [eq:seed]
  "Eq. (12)" expansion / census      -> Eq. (14) [eq:laurent]; Eq. (85) [eq:WrCr]
  "Eq. (16), (17)" Q5, R5 at s*      -> Eqs. (93)-(94) [eq:Q5, eq:R5]
  "Eq. (18)" A, B at s0, s1, s2      -> s0: the c6 entry of Eq. (33) [eq:basetower]; s1, s2: not printed
  "Eq. (19)" c6 = A pi^6 + B z3^2    -> Eq. (95) [eq:ABfun]
  "Eq. (20)" c5 = Q5 z5 + R5 z2 z3   -> Eq. (92) [eq:QRfun]
  "Eq. (21)" positive chart, P_t     -> Eq. (23) [eq:posint], Eq. (22) [eq:polys]
  "Eq. (24)" base point              -> Eq. (32) [eq:s0], Eq. (33) [eq:basetower]
  "Section 3", "3.4", "3.5"          -> Sec. 2.3 and Appendix B.4; Appendix B.4; Eq. (14) [eq:laurent], Sec. 2.1
  "Section 6.3" twist spectrum       -> not printed (supplementary data; the diagonal twist
                                        c = c' = gamma appears only at Eq. (125) [eq:c4split])
  "Section 8" reality walls          -> Sec. 7.2 [ssec:count-law]
  "Section 9.3", "Theorem 7",
  "Eq. (37)", "Eq. (38)"             -> Sec. 7.4 [ssec:exp-beyond] (the weights-two-and-three
                                        statement is made in the text, without a theorem number),
                                        Eq. (124) [eq:d4univ], Eq. (125) [eq:c4split]

1. Kinematic coordinates
------------------------
The X(3,6) integral depends on the twenty generalized Mandelstam invariants
s_ijk (i<j<k in 1..6) subject to the six conservation constraints
sum over all triples containing i of s_ijk = 0, i = 1..6.  Every point in
this record is given by the fourteen INDEPENDENT coordinates, in this order:
  135, 136, 145, 146, 156, 235, 236, 245, 246, 256, 345, 346, 356, 456
The other six are determined by the constraints:
  s_123 = s_145 + s_146 + s_156 + s_245 + s_246 + s_256 + s_345 + s_346 + s_356 + 2*s_456
  s_124 = s_135 + s_136 + s_156 + s_235 + s_236 + s_256 + s_345 + s_346 + 2*s_356 + s_456
  s_125 = -s_135 - s_145 - s_156 - s_235 - s_245 - s_256 - s_345 - s_356 - s_456
  s_126 = -s_136 - s_146 - s_156 - s_236 - s_246 - s_256 - s_346 - s_356 - s_456
  s_134 = -s_135 - s_136 - s_145 - s_146 - s_156 - s_345 - s_346 - s_356 - s_456
  s_234 = -s_235 - s_236 - s_245 - s_246 - s_256 - s_345 - s_346 - s_356 - s_456
All coordinates are exact rationals (strings "p/q").  c_w(s) is the
alpha'-expansion coefficient at weight w of Eq. (14) [eq:laurent],
K(3,6)(eps s) = sum_w eps^(w-4) c_w(s), computed as the finite sum over cone
constants of Eq. (85) [eq:WrCr], in the normalization of Eqs. (92)-(95); it
is homogeneous, c_w(lambda s) = lambda^(w-4) c_w(s) (Eq. (14)), so scaling a
point scales its A, B, Q5, R5 by lambda^2 and lambda respectively.

Named points: s* is the generic rational point of Eq. (31) [eq:seed]; s0 is
the base point of Eq. (32) [eq:s0], s* with every coordinate rounded to the
nearest third (its exact A, B are the c6 entry of Eq. (33) [eq:basetower]);
s1 = s0 with s_136 lowered by 1/3; s2 = s0 with s_245 raised by 1/3
(reference_points.json lists all four; the paper prints A, B at s0 only,
the values at s1, s2 being reference values of this record).

2. The two-leg convention and what "cert" means
-----------------------------------------------
Every c6 value in this record is the census total sum_k w_k(s) V_k, where
w_k(s) are exact rational weights and V_k are the numeric values of the
weight-six integrand classes.  The V_k were computed by two independent
tailored quadrature legs at different precisions and orders, run as
separate campaigns; legA and legB are the two census totals, one per leg,
printed to 80 digits.
  cert = floor(-log10(|legA - legB| / |legA|)),
the number of leading digits on which the two legs agree; the record's
values are agreed between the two legs to cert digits (64 to 76 across the
283 points; 68 at s*, s0, s1, s2).  agree = floor(-log10(|A pi^6 + B zeta(3)^2 - legA| / |legA|))
is the number of digits on which the exact two-term form reproduces legA.
The held-out battery passes a point when agree >= min(30, cert - 2).
Digits beyond cert in legA or legB are not agreed between the legs and are
printed only so the arithmetic can be repeated exactly.

3. The cluster twist: (c, c') and gamma (Sec. 7.4; supplementary data)
-------------------------------------------------------------------
Sec. 7.4 [ssec:exp-beyond], paragraph "The D_4 cluster string integral"
(the deformation first appears at Eq. (40) [eq:d4edge], Sec. 3.1): the
X(3,6) integral, K(3,6) in the paper, is the c = c' = 0 member of the
type-D4 cluster string integral of Arkani-Hamed, He and Lam
(arXiv:1912.08707; their eq. (6.12) and their Section 9.3),
I_{3,6} = I_{D4}(c = c' = 0).  The D4 integrand is the X(3,6)
integrand times two extra factors P1^c P2^c', where P1 and P2 are the two
additional cluster variables.  In the chart used throughout the record
(the positive parametrization of Eq. (23) [eq:posint], matrix of Eq. (20)
[eq:matrix36], in which the minors <145>, <156>, <345>, <456> are the
monomials a, ab, c, acd in the four chart variables a, b, c, d (Eq. (21)
[eq:monomials]) and the other ten non-constant minors are, up to monomial
factors, the ten polynomials P_t of Eq. (22) [eq:polys]), the two
factors are the subtraction-free polynomials with exponent vectors (in
a, b, c, d)
  P1 (c-exponent)  : [[0, 1, 1, 0], [0, 0, 1, 1]]
  P2 (c'-exponent) : [[2, 1, 0, 0], [2, 0, 0, 0], [1, 1, 1, 0], [1, 0, 1, 0], [1, 0, 0, 0]]
each monomial with coefficient +1, i.e.
  P1 = b*c + c*d
  P2 = a**2*b + a**2 + a*b*c + a*c + a
In terms of the minors of the same chart (verified symbolically when this
file was generated) P1 = <134><256> - <234><156> and
P2 = <136><245> - <126><345>, where <134>, <234> and <126> are three of the
six constant minors, each equal to 1 in this chart: P1 and P2 are the two
cluster variables of Gr(3,6) that are not minors, the
X = (134)(256) - (234)(156) and Y = (136)(245) - (126)(345) of Sec. 7.4
(Arkani-Hamed, He and Lam, arXiv:1912.08707, eq. (6.12)), Y the image of X
under the cyclic shift i -> i+1 of the six labels.  The exponents c, c'
enter exactly as the s_t enter the
factors P_t^{s_t} of Eq. (23); the convergence region in (s, c, c') is
fixed by the same positivity conditions on the cone exponents kappa as for
X(3,6) (Eq. (17) [eq:chamber], Sec. 2.1), with the two extra letters
included in the fan.

The twist axis (the "diagonal twist" c = c' = gamma of Sec. 7.4, at which
Eq. (125) [eq:c4split] is evaluated) is the one-parameter family
c = c' = gamma at fixed kinematics s; (c, c') enter the tropical exponents
kappa linearly, as the s_ijk do.  The exact rational functions
c2(gamma)/zeta(2) and c3(gamma)/zeta(3) shipped in twist_gamma.json were
computed at the split-locus kinematic point at which Eq. (125) is
evaluated, printed in Sec. 7.4 just before that equation (its fourteen
coordinates are in that file and agree with the printed ones; at gamma = 0
they give c2/zeta(2) = -170563/720 and c3/zeta(3) = 14456777/43200).
The weight-two coefficient along the axis is zeta(2) times a rational
function of gamma of numerator degree six over a denominator of degree seven,
and the weight-three coefficient is zeta(3) times a rational function of
numerator degree seven over the same denominator; both have simple poles at
gamma in
  {-3/2, -9/10, -1/2, -1/3, 1/4, 1/3, 16/15}
(each pole is a twist at which one channel exponent kappa(gamma) vanishes).
The twist charge of a kappa-vanishing channel is the integer slope of its
kappa in gamma, kappa(gamma) = kappa(0) + (charge) gamma; thirteen of the
fan's sixteen rays carry such a kappa-vanishing channel, with charges in
{-3, -2, -1, 1, 2}, and the thirteen channels vanish at twelve distinct
twists, seven of which are the poles above.  None of this twist-axis
material (the two rational functions, their poles, the charges) is printed
in the version of 2026-09-28 of the paper, which prints the D4 twist family
only in the weight-four paragraph of Sec. 7.4; it is supplementary data of
this record, in the conventions stated here.  The statement of Sec. 7.4
that the weight-two and weight-three coefficients of I_{D4} are rational
multiples of zeta(2) and zeta(3) for all s and all (c, c') (made in the
text, without a theorem number; the three two-dimensional constants are
Eq. (124) [eq:d4univ]) and the weight-four values of Eq. (125) [eq:c4split]
at gamma = 1/6 and gamma = -1/4 use these same (c, c').

4. The seven-point reality scan of Sec. 7.2: segments
--------------------------------------------------
See wall_segments.json: u(t) = (1 - t) uA + t uB, t in [0, 1]; the 28
coefficients are ordered as the non-constant minors <ijk> of the 3x7 matrix
(minor_order), and the three brackets [t_lo, t_hi], each proven by interval
arithmetic, carry the real counts at both ends.  The walls are named in the text of
Sec. 7.2 [ssec:count-law] by the coarse-scan positions 0.0824, 0.1730,
0.3441; the paper attaches no equation to them (Sec. 6.4
[ssec:count-reality] refers to that scan).
