Measured: the cap costs every digit at this point

Exploratory. Nothing here is a result (see hunts/README.md).

The two numbers

abs(epstein_completed(mpc('0.8','85.7'), (2, 1, 3), dps=D)), mpmath 1.4.1, this tree's zeta/epstein.py:

Raw values, real and imaginary parts included, in results.json.

Reading

The D = 20 value is 23 orders of magnitude too large, with no correct digits at all, not a value with fewer digits. D = 60 and D = 80 agree to all twelve digits printed, so the reference is stable under refinement and the gap is the low-precision run's, not the reference's.

The mechanism is visible in the code. epstein_completed builds Lambda_Q(s) as first + second/sqrt(d) + 1/(sqrt(d)(s-1)) - 1/s (zeta/epstein.py:960-971), a sum of terms of size O(1) at this point, and at s = 0.8 + 85.7i the answer is 1.6e-58. Reaching it requires roughly 58 digits of cancellation. At dps = 20 plus the internal guard there are not 58 digits to cancel, so what comes out is the round-off floor of the largest term rather than the function. This is the standard signature: the value does not improve as precision rises, it is replaced.

A second route, in sanity_stirling.py, confirms which of the two is the function. Lambda_Q(s) = (sqrt(d)/pi)^s Gamma(s) zeta_Q(s) with d = 5.75, and |(sqrt(d)/pi)^s Gamma(s)| = 2.6400016e-58 at this s (Stirling gives 3.2768565e-58 for |Gamma(s)| against 3.2768529e-58 exact). So the two candidate magnitudes imply

• D = 60: |zeta_Q(0.8 + 85.7i)| = 0.613, an ordinary value for a Dirichlet series just left of its abscissa of convergence;
• D = 20: |zeta_Q(0.8 + 85.7i)| = 1.8e+23, which no Epstein zeta function of a positive definite form takes at height 85.7.

That check touches neither lattice sum, so it is independent of the route it is checking, and it points the same way as the refinement test.

What this does and does not say

It says: at one point off the critical line at height 85.7, evaluating epstein_completed at dps = 20 returns noise, and zeta/epstein.py:1091-1093 is the one place in the file that forces that precision regardless of what the caller asked for.

It does not say that any battery verdict is wrong. count_zeros_box winds an argument, and a winding number can survive magnitudes being wrong when the phase is not, or can be defended by the box being placed where the function is O(1). That is a separate measurement and this hunt did not run it. It is recorded as a thread in HANDBACK.json, not answered here.

Reproduce

.venv/bin/python hunts/dps_cap/measure.py
.venv/bin/python hunts/dps_cap/sanity_stirling.py