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# Exact arithmetic and certification for T626
Write z = sqrt(3)/pi. All computations before final evaluation take place in
Q[z], and all results have degree at most one. This attachment derives the
method; it is not a claim that numerical agreement proves the formulas.
## Axis seeds without a conjecture
Cserti (2000), equation (62), gives
t(m,n) = (1/pi) integral_0^(pi/2)
[1-exp(-|m-n| x) cos((m+n)y)] / [sinh(x) cos(y)] dy,
cosh(x) = 2/cos(y) - cos(y).
Set m=n=k, substitute u=cos(y), and use
sinh(x) cos(y) = sin(y) sqrt(4-cos(y)^2). Consequently
t(k,k) = (1/pi) integral_0^1
[(1-T_(2k)(u))/(1-u^2)] / sqrt(4-u^2) du.
The quotient is an even polynomial with integer coefficients. The numerator
vanishes at both u=1 and u=-1. Its exact quotient and zero remainder are
checked for every seed. The Chebyshev polynomials are constructed using
T_0=1, T_1=u, T_(j+1)=2u T_j-T_(j-1).
Let M_j = integral_0^1 u^(2j)/sqrt(4-u^2) du. Then M_0=pi/6. Integrating the
derivative of u^(2j-1) sqrt(4-u^2), for j>=1, gives
sqrt(3) = 4(2j-1) M_(j-1) - 2j M_j.
Thus M_j/pi is computed exactly in Q[z]. No quadrature error or guessed
P-finite recurrence enters the generator. Rotation invariance gives
t(k,0)=t(k,k), furnishing all axial seeds including t(0,0)=0.
## Off-axis values
For a nonzero triangular displacement v, the resistance difference is harmonic:
6t(v) equals the sum at its six neighbors. Apply this at (m-1,n-1), solving
for t(m,n), to get the interior recurrence in Mathar (2022), equation (52).
At the edge n=1, reflection and rotation combine the two outer terms, giving
t(m,1)=3t(m-1,0)-t(m-1,1)-(t(m,0)+t(m-2,0))/2, m>=2.
Reduction to the wedge m>=2n>=0 makes this a terminating recursion: every term
either has smaller m or the same m and smaller n. The sole boundary exception
m=n=1 reduces to the known axis seed. These exact harmonic equations and the
independently integrated axis seeds determine every stored triangular value.
The known small coefficients are
t(1,0)=1/3,
t(2,0)=8/3-4z,
t(2,1)=-2/3+2z,
t(3,1)=-5+10z.
## Honeycomb normalization
With positive combinatorial Laplacian, the Fourier matrix is
[[3, -conjugate(f)], [-f, 3]], f=1+exp(ik1)+exp(i(k1+k2)).
Its determinant is 2(3-cos(k1)-cos(k2)-cos(k1+k2)), the triangular Laplacian
symbol. The diagonal inverse entries are 3/det and the BA entry is f/det.
Computing the voltage difference yields
h_A(m,n)=3t(m,n),
h_B(m,n)=t(m,n)+t(m+1,n)+t(m+1,n+1).
These are Cserti (77)-(78). They refer to the explicit edges A_u--B_u,
A_u--B_(u-a1), A_u--B_(u-a1-a2), so the displacement signs are fixed.
The tests additionally invert the two-by-two Fourier matrix on finite tori,
without using the triangular-to-honeycomb conversion. Its zero-wave-vector
pseudoinverse has diagonal 1/12 and off-diagonal -1/12; omitting that optical
mode would fail the exact finite-torus adjacent-edge control.
## Error claims
A result a+b*z with b=0 is returned as an exact rational. Otherwise a and b
are embedded as Arb balls and evaluated with certified sqrt(3) and pi. The
working precision for 100 requested digits is 461 bits. The 128-bit guard is
validated by the enclosure widths and cancellation recorded for every entry
in `numerical-evidence.json`. Decimal serialization is independently checked:
the complete computed ball must lie inside the interval denoted by the last
written place, not merely have a close midpoint.
Independent mpmath quadrature at 125 and 150 digits checks every entry using
the integral, with known origin, adjacent and short-displacement values as
controls first. Finite-torus Fourier solves check all entries at side lengths
64, 128 and 256 and include exact Foster-theorem controls for the adjacent
resistances. Those numerical checks corroborate the method; certification
comes from the exact derivation and Arb enclosure.
The skill record is the archived matching local/live manifest at
`agents/runs/20261030T125612Z-build-skills/manifest.json`, copied in the
attachments as `skill-manifest.json`. Both skills have SHA-256
`28ecbeeccb0c5e5d6eb541b5a5b0f88aec47b154374d909085116d34f5629199`.