The meaning of a true connection implies the exclusion of anything foreign to that connection. 

אמר ר' אלעזר אמר ר' אושעיא אינו חייב אלא על דבר שעיקרו מדברי תורה ופירושו מדברי סופרים ויש בו להוסיף ואם 

הוסיף גורע ואין לנו אלא תפילין אליבא דרבי יהודה

Rabbi Elazar says that Rabbi Oshaya says: One is liable [as a zaken mamrei] only for issuing a ruling with regard to a matter whose essence, whose basic obligation, is from matters of Torah and whose explanation is from traditional rabbinic interpretations of the Torah and which includes the possibility to add to it, and if one added to it, one compromises his fulfillment of the mitzva and does not satisfy his obligation. And we have only the mitzva to don phylacteries that meets those criteria. And Rabbi Oshaya’s statement is in accordance with the opinion of Rabbi Yehuda, who says: A rebellious elder is liable only for a matter whose essence is from matters of Torah and whose explanation is from traditional rabbinic interpretations of the Torah.

והאיכא לולב דעיקרו מדברי תורה ופירושו מדברי סופרים ויש בו להוסיף ואם הוסיף גורע

The Gemara asks: But isn’t there the mitzva of lulav and the other species that one takes on the festival of Sukkot, whose essence is from matters of Torah, and whose explanation is from traditional rabbinic interpretations that establish the identity and the number of the four species enumerated in the Torah, and which includes the possibility to add other species to it, and if one added to it, one compromises his fulfillment of the mitzva and does not satisfy his obligation?

בלולב מאי סבירא לן אי סבירא לן דלולב אין צריך אגד האי לחודיה קאי והאי לחודיה קאי ואי סבירא לן דצריך אגד גרוע ועומד הוא

The Gemara rejects this possibility: That is not the case, as with regard to the mitzva of lulav, what do we hold? If we hold that fundamentally a lulav does not require binding of the species together in order to fulfill the mitzva, then adding an additional species is inconsequential, as these species with which he fulfills the mitzva stand alone and that additional species stands alone. It is as though he were holding the species of the mitzva and an additional unrelated item that does not affect fulfillment of the mitzva. And if we hold that a lulav requires binding of the four species together in order to fulfill the mitzva, fulfillment of the mitzva is already compromised from the outset. The rebellious elder is liable only when the object of the mitzva was as it should be and the addition compromised that object and disqualifies it. In this case, the object was never as it should be. [Sanhedrin 88b]

So we see that adding anything to a connection that is uncalled for compromises the connection. In the words of the Rogochover [צ"פ בירור הלכות שונות עמ' 55]:

"דעיקר האגד והקשר הוא שיהיו אלו המינים נפרדים מאחרים"

Another example [Sanhedrin 11a]:

ת"ר אין מעברין את השנה אלא

במזומנין לה מעשה ברבן גמליאל שאמר השכימו לי שבעה לעלייה השכים ומצא שמונה אמר מי הוא שעלה שלא ברשות ירד


by those who were invited by the Nasi, the president of the Great Sanhedrin, for that purpose. There was an incident involving Rabban Gamliel, who said to the Sages: Bring me seven of the Sages early tomorrow morning to the loft designated for convening a court to intercalate the year. He went to the loft early the next morning and found eight Sages there. Rabban Gamliel said: Who is it who ascended to the loft without permission? He must descend immediately.

אבל אמרו חכמים אין מעברין את השנה אלא במזומנין לה. 

But the Sages said: The year may be intercalated only by those who were invited for that purpose.

Wrote the Rogochover that if one univited person joins "עי"ז בטל כל הזימון" - the whole thing is cancelled. 

Another example of this phenomenon [3a]:

וכמה תהא שותפות של עובד כוכבים ותהא פטורה מן הבכורה אמר רב הונא אפילו אזנו מתקיף לה רב נחמן ולימא ליה שקיל אזנך וזיל



The Gemara asks: And according to the Rabbis, how much should the gentile’s partnership in the animal be in order for it to be exempt from being counted a firstborn? Rav Huna says: It is sufficient even if the gentile owns only its ear. Rav Naḥman objects to this: Let the animal have firstborn status, and let the priest say to the gentile: Take your ear and go, as a blemished firstborn animal belongs to the priesthood.

איתמר רב חסדא אמר דבר שעושה אותו נבלה ורבא אמר דבר שעושה אותו טריפה


It was stated that the amora’im engaged in a dispute concerning this question: Rav Ḥisda says that if the gentile is a partner in an item, i.e., a part of the body, that renders the animal a carcass, meaning if that limb were removed the animal would die immediately, the animal is exempt from the obligations of firstborn status. And Rava says it is exempt if the gentile is a partner in an item that if removed renders the animal a tereifa, meaning the animal will die but not immediately.

במאי קמיפלגי בטריפה חיה למאן דאמר דבר שעושה אותו טריפה קסבר טריפה אינה חיה ולמאן דאמר דבר שעושה אותו נבלה אבל טריפה חיה


The Gemara asks: With regard to what principle do they disagree? The Gemara answers that they disagree about whether a tereifa can live for an extended period of time. According to the one who says that a gentile’s partnership in an item that renders the animal a tereifa exempts him from the obligations of firstborn status, it is because he holds that a tereifa cannot live, and therefore the gentile owns an essential portion of the animal. And according to the one who says that the gentile must be a partner in an item that renders the animal a carcass, this is because he holds that it cannot live without this part, but a tereifa can live for an extended period of time without those missing limbs.

אמרוה רבנן קמיה דרב פפא הא דרב הונא ורב חסדא ורבא לא פליגי הא בו הא באמו


The Sages said before Rav Pappa: That which Rav Huna said, that even if the gentile’s share of the animal is only its ear it does not have firstborn status, and that which Rav Ḥisda and Rava said, that the animal does not have firstborn status only if the gentile is a partner in the primary limbs of the body on which its life is dependent, do not disagree. This ruling of Rav Huna is referring to it, i.e., the fetus, while that ruling of Rav Ḥisda and Rava is referring to its mother.

אמר להו רב פפא מאי שנא בו דבעינן כל בכור וליכא אמו נמי בעינן (שמות לד, יט) כל מקנך תזכר וליכא אלא לא שנא


Rav Pappa said to them: What is different about the case of the fetus, where owning a portion of its ear exempts the Jew from the obligations of firstborn status? It is because we require the fulfillment of the verse “all the firstborn,”i.e., that all of it be owned by the Jew, and that is not the case. If so, with regard to its mother as well we require the fulfillment of the verse: “From all your livestock you shall take the males” (Exodus 34:19), which also indicates that all the livestock, including the mother, must belong to the Jew, and this is not the case. Rather, there is no difference, and they disagree both in the case of the fetus and in that of the mother.

מתקיף לה מר בר רב אשי מאי שנא מנפלים דאף על גב דלאו בני חיותא נינהו קדשי דאמר מר (שמות יג, יב) פטר שגר בהמה שגר בבהמה


Mar bar Rav Ashi objects to this: Why does gentile partnership in an essential limb of the fetus exempt the Jew from the obligations of firstborn status? In what way is this animal different from non-viable newborns, which are sacred even though they are not viable? As the Master said in reference to the verse: “Every firstborn that emerges from [sheger] an animal” (Exodus 13:12), any fetus that dwells [shegar] inside an animal, i.e., inside the mother’s womb, even one that is non-viable, is also sanctified as a firstborn.

התם כיון דלא עריבו בהו חולין קרינא בהו בבהמה כל בכור הכא כיון דעריבו בהו חולין לא קרינא בהו כל בכור


The Gemara responds: There, in the case of non-viable newborns, since there is no non-sacred element mixed in with them, we apply to the animal the phrase “all the firstborn.” But here, in the case where a gentile owns a portion of the fetuses, since a non-sacred element is mixed in with them, we do not apply to them the phrase: “all the firstborn.”

We learn that something that is independent, even though it has no viability, remains in the framework of kedusha. But something that involves a foreign property, such as ownership of a Gentile, becomes disqualified.   

[עפ"י משנת רבינו יוסף הרוגוצ'ובי עמ' 182-183]

This OF COURSE, has ramifications in Avodas Hashem as well. If anything not related to Hashem is important to a person, then it impedes his or her relationship to Hashem. 

The same applies to marriage. In order for it to remain sacred and stable, there can be no other relationships that might compromise the bond.