Manuscript I-B-G@2 · I-B-G-v2.tex · section · 3b4d700f

Reported high-frequency consistency check

Retained numerical data from the supplied draft › Reported high-frequency consistency check · lines 2732–2786

At $μ = π /2, β = 0$ , with $(r, r^{'}, Δ φ) = (1, 1.5, 2.9)$ , the supplied distance is

d = R_{0} = 2.22583034991387.

For a unique shortest geometric term, the closed resolvent formula and the $K_{0}$ asymptotic give

e^{s d} s R_{2 D, 0} (s) ⟶ \frac{1}{2 π} \frac{π}{2 d} .

This exact leading constant uses the direct $K_{0}$ contribution and the strictly longer additional arrivals. It does not follow from causal support alone.

The supplied draft gives the limiting value $0.133700810575215$ and the following eighty-digit comparison:

$s$	$e^{s d} s R_{2 D}$ , modes	$e^{s (d + 0.05)} s R_{2 D}$	Reported mode–closed relative discrepancy
5	0.175771933447982	0.225695630087555	$2.3 \times 1 0^{- 64}$
10	0.154928248627405	0.255433498944321	$3.2 \times 1 0^{- 74}$
20	0.137892303526610	0.374830142960743	$9.2 \times 1 0^{- 67}$
40	0.133685066614491	0.987806456803755	$7.7 \times 1 0^{- 52}$
80	0.133607527312649	7.29472382177342	$3.8 \times 1 0^{- 7}$

The high- $s$ loss is retained, not suppressed. The supplied account attributes it to cancellation in the mode sum and states that an earlier twenty-five-digit run lost all digits at $s = 40$ . These are reported diagnostics. Neither the apparently successful rows nor the failed row prove or refute Theorem thm:speed.

Displayed formulas in this passage

unlabeled · line 2737
$d = R_{0} = 2.22583034991387.$
unlabeled · line 2742
$e^{s d} s R_{2 D, 0} (s) ⟶ \frac{1}{2 π} \frac{π}{2 d} .$