Source for the requirement for an action to be within "toch k'dei dibbur" of a warning for Capital Punishment

Question

In order for a person to be killed by Beis Din via a capital punishment, there needs to be:

• a) 2 witnesses (Deuteronomy 19:15) that saw him transgress a sin that is liable to death.
• b) a warning ("hasra'ah") that the action the person is about to commit is liable to death (Sanhedrin 40a-b).

Within this requirement of warning, in order for a person to eligible for capital punishment the warning must be immediately followed by the action within the amount of time of "toch k'dei dibbur" (see Sanhedrin 40b)-- ie several seconds of time.

As the Rambam in Hilchos Sanhedrin 12:2 writes:

וְצָרִיךְ שֶׁיַּעֲבֹר וְיַעֲשֶׂה תֵּכֶף לַהַתְרָאָה בְּתוֹךְ כְּדֵי דִּבּוּר

He must commit the transgression immediately after receiving the warning, within the time to offer a salutation.

---

Being as this is literally a life or death matter:

where do we learn the source for this toch k'dei dibbur requirement by capital punishment?

Answer 1

In Kesubos 33a Rava says that warning has to be done toch kedei dibbur (within several seconds) so that we know the perpetrator did not forget the warning and act erroneously:

אמר רבא תדע ניתרי בהו אימת ניתרי בהו מעיקרא אמרי אישתלין

Rashi and Yad Rema in Sanhedrin 40b quote this reason as well:

המית בתוך כדי דיבור כדי שאילת שלום להתראה דאי לאחר כדי דיבור מצי אמר אשתלאי:

Rovo's reasoning is: If there is a longer gap than toch kedei dibbur between the warning and action, we must give the perpetrator the benefit of the doubt that he forgot acted erroneously and therefore cannot be liable to death since we say in Kesubos 15a:

ספק נפשות - דיני נפשות להקל דהא והצילו העדה כתיב- במדבר לה-
Death sentences must be judged leniently
(i.e we cannot implement death penalty unless 100% sure it was done on purpose) when possible
since it is written"The Congregation (Sanhedrin) shall save" (Bamidbar 35,25)

Shitta Mekubetzes explains

ספק דיני נפשות להקל דהא והצילו העדה כתיב וצריך לחפש לו זכות כל מאי דאפשר. וזהו שכתב רש"י
We should find any merit whatsoever to stop him from being liable to death, (and in this case the possibility of having forgotten will save him from death penalty as only a known meizid transgressor can be punished)

Answer 2

Toch K'dei dibbur is the definition of "immediate", and all activity within toch kdei dibbur is considered simultaneous. Anything which happens after this time is grouped together as "at a later time". We do not have any "medium-term" time frame. Therefore, if the act was within Toch Kdei Dibbur, it was "at the same time/immediate" to the warning, while if that period has elapsed, it is "later", and we cannot connect the hasraa to the act.

The source, as you wrote, is the Gemara in Sanhaedrin.The Gemara does not bring a source, which means that it is following the general definition of "immediate".