https://en.wikipedia.org/wiki/High_voltage
The dielectric breakdown strength of dry air, at Standard Temperature and Pressure (STP), between spherical electrodes is approximately 33 kV/cm. This is only as a rough guide, since the actual breakdown voltage is highly dependent upon the electrode shape and size.
per Beckett drawing electrode spacing is 5/32 = .156" = 3.97 mm = .397 cm.
33 kV/cm x .397 cm = 13 KV min required to jump the gap, unless they are pointed which they are. But the original 10 KV transformer has been the standard for 60 years so it has to be a good reference value.
Also remember that in a oil fired furnace the ignition spark is on continuously with oil flow. If the ignition spark stops for just a fraction of a second the flame stops also. So i do not believe that a capacitor could be charged up and it would just get a spark every 1/4 of a second or so.
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Thinking about BradTheRad and SunnySkyGuy's post's, The answer is probably in wave shape and duty cycle.
Assuming a switching power supply, a rectangular wave is plausible and to get the power down you would have to cut back on the duty cycle. So maybe a 10 or 20% duty square wave?
Start with this assuming 100% duty.
6KV x 35mA = 210 VA - compared to their 35 VA claim.
At 90% efficiency 233 VA
So 50% duty would be 233 VA / 2 = 116 VA
25% duty would be 233 VA / 4 = 58 VA
12.5% duty would be 233 VA / 8 = 29 VA
So
35 VA / 233 VA = .150 = 15% duty
Does this sound realistic.