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tubhya;mahyau Gayi

तुभ्यॱमह्यौ ङयि ONPANINI 72095

(Replace /yuSm /asm) with tubhya तुभ्य mahya मह्य before /Ge.

/yuSmad- + /Ge
→ /yuSmad- + /am by Geprathamay...
→ tubhya तुभ्य + ad अद् + /am by this rule
→ !**tubhyam तुभ्यम् "to thee"

/asmad- + /Ge → !**mahyam मह्यम्

As in

aham prahitas tubhyaM rAmeNa
अहं प्रहितस्तुभ्यं रामेण
"rAma राम sent me to you"

These two words are often replaced with /te'' and /me''.

What's that Gayi ङयि word in the sUtra?

It is just —

Ge- ङेॱ + /Gi
→ Gay ङय् + /Gi by ecoyavAyAvaH
→ !**Gayi ङयि "before /Ge"

But I read somewhere that there are no nounbases ending in e ए.

The "somewhere" meant that there are no commonly used nounbases ending in e ए. You are still free to make your own, like /pANini did with /Ge /ze. They work mostly like /go-.

Ge- ङेॱ + /Gas → !**Ges ङेस् by GasiGasozca, as in "GeryaH"

Ge- ङेॱ + /su → Ge ङे by svamornapuMs..., as in "Geprathamay..."

So if your name is Jay as in "Stephen Jay Gould", feel free to Sanskritize it as a nounbase je- जेॱ.

But my name is Jay Candrasekhar. Can I do that too?

Nope. If you're from India, then you're jaya- जयॱ, "victory", jayanta- जयन्तॱ "victorious", or something like that.