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6914 bytes, as of the version from 2026-11-09 09:32 (current). Recorded here, not run.
# Computation and checks for T686
Skill manifest: `agents/runs/20261109T084515Z-build-skills/manifest.json`.
Both archived skill versions have SHA-256
`886390324dd1971e381a43cf3b46b822ab0bc67d62f31d8481e29ec79556746b`.
## Meaning and precision
The generator computes H_(n,k), with independent unit phases, normalized Haar
measure, natural logarithms, signed powers, and no factorial divisor.
Rotational invariance identifies the magnitude with the radius of an n-step
uniform planar walk. Consequently H_(n,k) is W_n^(k)(0), and equals the
integral of p_n(x) log(x)^k over 0<x<n. Zeros cause integrable logarithmic
singularities, not regularized values.
The requested output is 100 significant digits. Complete computations are
repeated at 150 and 220 decimal working digits. `numberdb.agreeing` retains
the agreement interval. Arithmetic inside finite series uses Arb, but the
omitted tails are not bounded here. Thus the declared rigour is
`heuristic (agreement-checked)`. Neither a small Arb arithmetic radius nor
agreement between truncations is a proof of the error in the infinite sum.
## Three summands
Use the analytic generating function in BBSW Theorem 4.1, equation (23):
W_3(s) = sqrt(3)/(2*pi) * 3^(s+1) * Gamma(1+s/2)^2/Gamma(2+s)
* 3F2(1+s/2,1+s/2,1+s/2;1,3/2+s/2;1/4).
Truncated polynomials in s retain degrees 0 through 8. Successive
hypergeometric terms are generated by their rational ratio, and summed
through term 2*w for working precision w. The logarithm of the gamma
prefactor has coefficient log(3)-1 at s and
(-1)^j*((2^(1-j)-1)*zeta(j)+1)/j at s^j for j>=2. Its exponential is built
coefficient by coefficient. Multiplication gives the Taylor series of W_3;
the coefficient of s^k is multiplied by k!, because H_(3,k) is the derivative,
not the Taylor coefficient. The arithmetic field has bits(w)+96 bits.
## Four summands
Use BSWZ's third-order density equation (11), written as
(x^7-20*x^5+64*x^3) p''' + (6*x^6-60*x^4) p''
+ (7*x^5-32*x^3+64*x) p' + (x^4-64) p = 0.
The integral is divided into [0,1], [1,2], [2,3], [3,4]. Its density is
represented by convergent Frobenius series, avoiding direct 3F2 evaluation
near its argument 1.
* On [0,1], use p_4(x)=sum_j (r_j-s_j log(x))*x^(2*j+1).
The coefficients follow BSWZ equations (26),(30), with
s_0=3/(2*pi^2), r_0=s_0 log(8), and zero coefficients at index -1.
The homogeneous recurrence for s_j is the recurrence of the Domb numbers
rescaled as in equation (27). Integration uses
integral_0^1 x^(a-1) log(x)^k dx = (-1)^k k!/a^(k+1).
* At x=2, the ODE has exponents 0,1,1/2. The two analytic basis series A,B
have (constant,linear) coefficients (1,0) and (0,1); S is the half-power
basis with leading coefficient 1. The density on the left has finite
derivative, so contains no half-power term. Fix its value at 2 using BSWZ
(32), and fix its other analytic coefficient by its value at 1 from (47).
Thus C(u)=p_4(2)*A(u)+slope*B(u), with
slope=(p_4(1)-p_4(2)*A(-1))/B(-1). For u>0 the density is
C(u)-4/pi^2*sqrt(u)*S(u). The half-power coefficient follows from the
right derivative asymptotic in BSWZ Remark 3. It is independently checked
against Theorem 8's 3F2 density inside the interval.
* At x=4, put u=4-x. The density is sqrt(u)*F(u), where the ODE generates F
from its leading coefficient sqrt(2)/pi^2. This leading coefficient is
established by BSWZ Theorem 5. We do not use the additional printed
coefficients there; the ODE determines subsequent terms directly.
The analytic series about 2 and 4 use 4*w terms. Each interval has distance
at most 1 from its center while the nearest other singularity is distance 2.
The zero-centered series uses 2*w terms at squared argument at most 1/4.
These ratios motivate the cutoffs; they are not substituted for a rigorous
tail proof. Multiply by the truncated Taylor series of log(2+u)^k or
log(4-u)^k, integrate integer powers and half-integer powers explicitly, and
add the four pieces. Pairing [1,2] and [2,3] leaves twice the even coefficients
of C(u)*log(2+u)^k. The other denominators are j+3/2.
The arithmetic field uses bits(3*w+50)+96 bits to allow for pessimistic
interval propagation in the recurrence. In the 150/220 runs the largest
arithmetic radii were about 1.19e-346 and 1.36e-468. These radii describe the
finite arithmetic only. The precision comparison and independent checks,
not those radii alone, support the heuristic 100-digit output.
## Validation and its limits
`checks.py` first tests quadrature controls with known answers: total radial
mass 1, squared-radius expectation 3 or 4, and Smyth's first moments. It then
compares every three-term moment with a separate 130-digit mpmath quadrature
of the elliptic density, using Carlson's symmetric elliptic integral and
factored complementary arguments to avoid cancellation near x=1. Each
comparison must meet 115 decimal places relative to max(1,abs(value)).
For four terms, equation (70)'s polylogarithm/zeta closed form checks H_(4,2)
to 120 decimal places. Direct mpmath 3F2 evaluations check the piecewise
series at x=1/2,3/2,5/2,7/2 to 115 places. Numerical quadrature of the
piecewise density checks every moment through order 8 to 40 places; this
last check shares the density recurrence, but independently checks the
coefficient products and elementary antiderivatives. There is no claim of
a second independent 100-digit integration for all higher four-term moments.
BBSW Example 9.2's four printed three-term values agree to their printed
precision. The moment matrices [H_(n,i+j)] for 0<=i,j<=4 are positive
definite, and all stored even moments are positive. `dry_run.py` checks all
14 entries and measures their written length. After submission, every
stored value must agree with the recomputation within its written last-place
unit, and `verify(sample=None)` must pass. Numerical evidence records actual
checks; this document does not certify omitted tails.
## Resolved source discrepancy
BSWZ arXiv:1103.2995v2, Theorem 5, prints a negative coefficient for
(4-x)^(3/2). The PDF and HTML agree on that printing. Put u=4-x in their
Theorem 7, equation (31). Its hypergeometric argument is O(u^3), so its
hypergeometric factor is 1+O(u^3). Expanding the elementary prefactor gives
p_4(4-u) = sqrt(2)/pi^2 * sqrt(u)
* (1 + 3*u/16 + 23*u^2/512 + O(u^3)).
The coefficient is therefore positive. The ODE recurrence independently gives
F[1]/F[0]=3/16; `checks.py` tests it. Theorem 5's leading coefficient and
quadratic coefficient agree with this expansion, but its linear sign does
not. The generator uses the leading coefficient only and computes all
subsequent coefficients from the ODE. Direct hypergeometric evaluations and
the mass/first/second-moment controls confirm that choice. This is an
algebraically resolved coefficient discrepancy, not an unsettled definition
or a choice between numerical answers.