PROOF POSITIVE that the Sabbath "IS" on Saturday:

Because the Romans associated the Jewish Sabbath with their Saturday.

And we follow Yeshua’s Example of the 1^st Century who kept “this” Saturday Sabbath.

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Re a recurring weekly Sabbath, examining the records of Roman historians and other writers can be helpful. By the admission of Lunar Sabbath keepers, and clear historical record, the Romans did not follow the moon phases when determining the weeks, or even the months for that matter.

For this reason, we can look into the writings of Roman historians and other writers to see if the Romans associated the Jewish Sabbath with their "Saturday/Day of Saturn." If they did associate the Sabbath with their Day of Saturn in the first century or before, this would be undeniable evidence that the Jewish week and the Roman week were both kept on the same recurring weekly cycle.

As we will see in this section, there is no doubt that Roman historians and other writers explicitly and repeatedly consider the timing of Jewish Sabbath keeping to be linked with the seventh day of the Roman week, which was a repeating seven day cycle, independent of the moon phases. 

70 - 84 CE (AD)

Frontinus, a Roman Soldier who lived from c. 40 CE to 103 CE, wrote book on military strategy called Strategematicon in 84 A.D. In it, he writes:

"The deified Augustus Vespasian attacked the Jews on the day of Saturn, a day on which it is sinful for them to do any business." Frontinus Stratagem 2.1.17. 

The original Latin version of this book has "Saturnis" for Saturn, confirming that the Romans associated the Sabbath day with their "day of Saturn." It is this association that resulted in the 7th day of the modern week being called "Satur-day." 

Since this book was written a mere 14 years after Vespasian's (Titus') destruction of Jerusalem, this is this is obviously very strong historical evidence directly from a first century eyewitness, tying in the Sabbath with "Saturn's day (Saturday)."

63 BCE - 229 CE

Cassius Dio, a Roman Historian who lived from ca. 155 to 229 CE, using the historical annals of the Roman empire, wrote about 3 battles which the Roman empire had with the Jews. 

The first battle was during a time when the Hyrcanus II and Aristobulus II, two brothers who were the offspring of the Maccabees, were engaged in a dispute over who would rule. The Romans, through the actions of Pompey, came in and settled the dispute, siding with Hyrcanus. While speaking of Pompey's battle, the Sabbath is mentioned. 

The setting is 63 BCE: 

"Most of the city, to be sure, he took without any trouble, as he was received by the party of Hyrcanus; but the temple itself, which the other party had occupied, he captured only with difficulty. 
For it was on high ground and was fortified by a wall of its own, and if they had continued defending it on all days alike, he could not have got possession of it.
As it was, they made an excavation of what are called the days of Saturn, and by doing no work at all on those days afforded the Romans an opportunity in this interval to batter down the wall. 
The latter, on learning of this superstitious awe of theirs, made no serious attempts the rest of the time, but on those days, when they came round in succession, assaulted most vigorously. 
Thus the defenders were captured on the day of Saturn, without making any defense, and all the wealth was plundered. 
The kingdom was given to Hyrcanus, and Aristobulus was carried away." Cassius Dio Roman History 37.16.1-4

So the Romans took advantage of the fact that the Jews would not work on the Sabbath. When was the Sabbath? Again, the weekly Sabbath coincides with the Roman "days of Saturn." 

The second battle listed by Cassius Dio occurred in 36 BCE, is the one that resulted in the very first King Herod coming to power: 

"The Jews, indeed, had done much injury to the Romans, but they suffered far more themselves. 
The first of them to be captured were those who were fighting for the precinct of their god, and then the rest on the day even then called the day of Saturn. 
And so excessive were they in their devotion to religion that the first set of prisoners, those who had been captured along with the temple, obtained leave from Sosius, when the day of Saturn came round again, and went up into the temple and there performed all the customary rites, together with the rest of the people. 
These people Antony entrusted to a certain Herod to govern; but Antigonus he bound to a cross and flogged,— a punishment no other king had suffered at the hands of the Romans,— and afterwards slew him." Cassius Dio Roman History 49.22.4-6

Note that Cassius Dio reports the Jews as keeping "customary rites" at the temple on "the day even then called Day of Saturn." This indicates that the Sabbath wasn't only called the "Day of Saturn" during his lifetime, but it was called the "Day of Saturn" back in 36 BCE, well before Yahushua was born in Bethlehem.

Next, he records that the 70 CE Jerusalem destruction was on the Sabbath, which he once again calls the "day of Saturn:" 

70 - 229 CE

Thus was Jerusalem destroyed on the very day of Saturn, the day which even now the Jews reverence most. 
From that time forth it was ordered that the Jews who continued to observe their ancestral customs should pay an annual tribute of two denarii to Jupiter Capitoline. 
In consequence of this success both generals received the title of imperator, but neither got that of Judaïcus, although all the other honours that were fitting on the occasion of so magnificent a victory, including triumphal arches, were voted to them. Cassius Dio Roman History 65.7.2

So his report is that the Jews kept the Sabbath on the "day of Saturn" from 63 BCE up until his day, no later than 229 CE. His report also agrees with Frontinus' account of the 70 CE battle. 

c. 100 CE

The Historian Cornelius Tacitus (ca. 56CE – ca. 117CE), after suggesting that Jews kept the Sabbath out of laziness, also associated the Sabbath with the Roman idol, Saturn: 

They are said to have devoted the seventh day to rest, because that day brought an end to their troubles. Later, finding idleness alluring, they gave up the seventh year as well to sloth. 
Others maintain that they do this in honor of Saturn; either because their religious principles are derived from the Idaei, who are supposed to have been driven out with Saturn and become the ancestors of the Jewish people; or else because, of the seven constellations which govern the lives of men, the star of Saturn moves in the topmost orbit and exercises peculiar influence, and also because most of the heavenly bodies move round their courses in multiples of seven. From The Histories, Book V

Again, the fact that a pagan associated Sabbath keeping with Saturn demonstrates that the Roman week's day of Saturn (Satur-day) was concurrent with what Yahweh calls the Sabbath day. Tacitus is an eyewitness from the first century who has no "axe to grind" in regards to when the Sabbath should be observed. He wrote this a mere 30 years after the destruction of the temple in Jerusalem. 

28 BCE to 1 BCE

Tibullus, a Latin Poet who lived from 54 BCE - 19 BCE, references the Sabbath in one of his Elegies. The Poet is quoted by J. Hugh Michael in his paper entitled "The Jewish Sabbath in the Latin Classical Writers." In this Journal article, he says:

"Tibullus, again, gives us a glimpse of the influence of the Sabbath on the Romans. The Poet, sick in a foreign land, complains of his loneliness; neither mother nor sister has he there to nurse him; nor is Delia with him--she who had inquired of all the gods before permitting him to leave the city. Says Tibillus:

All promised a return; yet did nothing stay her from looking back in tears and terror on my journey. Yea, even I her comforter, after I had given my parting charge, sought still in my disquiet for reasons to linger and delay. Either birds or words of evil omen were my pretexts, or there was the [Christian] holy day of Saturn [=Saturday] to detain me. (Book I, Eleg. iii 13-18 in Postgate's translation in the Loeb Classics.)

The day of which the poet speaks is of course the Saturday, or, as Postgate puts it, "the Jewish Sabbath, on which no work was to be undertaken," and the implication is clear that unwillingness to set off on a journey on the Sabbath day was no unusual thing among the inhabitants of Rome, for Tibullus is enumerating the common reasons for the postponement of journeys of which he was only too ready to avail himself." The Jewish Sabbath in the Latin Classical Writers. J. Hugh Michael Victoria College, Toronto, Canada. The American Journal of Semitic Languages and Literatures, Vol. 40, No. 2. (Jan., 1924), pp. 117-124. 

It is interesting that Roman's would have any concern at all about traveling on the Sabbath. J. Hugh Michael references another writing by the Poet Ovid, who lived from 43 BCE to 17 CE:

Still more impressive is the witness of Ovid. In the Ars Amatoria he instructs the young Roman who is desirous of finding an object for his affections [a wife] how he should set about his search. 
The quest need not take him far a field: there is no lack of suitable damsels in Rome. 
The poet even specifies the parts of Rome where the quest of the amorous youth is most likely to be crowned with success: he should not neglect '(Adonis lamented of Venus, or the seventh day observed as holy by the Syrian Jew." 
The first part of this direction can only mean that the youth should visit the Temple of Venus when her grief for Adonis was commemorated on the anniversary of his death. 
Similarly the second part must mean that he should attend the Sabbath [Saturday] services held in the Jewish synagogues. 
Is it possible to attach any other meaning to the advice that the youth should not avoid the seventh day observed by the Syrian Jew? 
And what meaning is there in the counsel unless Roman maidens were wont to attend those services? 
It is not easy to think that it was the poet's intention that the young Roman should become enamored of a Jewess! The Jewish Sabbath in the Latin Classical Writers. J. Hugh Michael Victoria College, Toronto, Canada. The American Journal of Semitic Languages and Literatures, Vol. 40, No. 2. (Jan., 1924), pp. 117-124

The work that J. Hugh Michael references, Ars Amatoria, was written in approximately 1 BCE. Thus, it is evident that even the Romans were associating the seventh day of their Roman week with the Sabbath. It is also historical evidence that they were observing a 7 day week at that time. But was Sabbath keeping really this tied into their culture?

119 CE

The Sabbath does appear to be closely entwined to the weekly cycle of the first century Romans. This is evidenced by the writings of Suetonius. Suetonius (ca. 69CE - 130CE) was a Roman historian who wrote "Lives of twelve Caesars" in 119 CE. While describing the life of Tiberius Caesar (14 - 37 CE), he wrote::

"The grammarian Diogenes, who used to lecture every Sabbath at Rhodes, would not admit Tiberius when he came to hear him on a different day, but sent a message by a common slave of his, putting him off to the seventh day. When this man waited before the Emperor's door at Rome to pay his respects, Tiberius took no further revenge than to bid him return seven years later." Suetonius The Life of Tiberius 32.2

Rhodes was a major schooling center for Roman families. It's interesting that the seventh day is referred to as the "Sabbath" (Latin: sabbatis) in Roman literature, especially when it is written by a pagan historian who was even the Roman Emperor's secretary for a time (Wikipedia entry on Suetonius). 

It appears that J. Hugh Michael's references to the Roman poets, as well as this reference from Suetonius, is a strong indication that the seventh day Sabbath had become more tied into Roman society than most people realize. Possibly this is one of the reasons Josephus said:

"The masses have long since shown a keen desire to adopt our religious observances; and there is not one city, Greek or barbarian, nor a single nation, to which our custom of abstaining from work on the seventh day has not spread and where our fasts and the lighting of lamps and many of our prohibitions in the matter of food are not observed. Apion 2:282-283

In consideration of these things, it appears to me that the Romans picked up the recurring seven day week from the Jews, not the other way around. For 5 centuries or longer the Romans had kept an 8 day week with the days being named by the letters of the alphabet A through H. They did not keep a seven day week until they came into more contact with Sabbath keeping Jews in the early centuries BCE. 

The fact that they named the days of the week after various planetary idols doesn't prove that they originated the recurring 7 day week. The naming of the days after planetary idols appears to have originated with either the Babylonians or the Hindu civilization, not the Romans. 

So if you are a Lunar Sabbath keeper, the obvious question for you is, "If the Jews in the first century kept the Lunar Sabbath, and only later adopted what you call the 'Roman week,' why do the first century Romans keep saying the Jewish Sabbath is on their "day of Saturn (Saturday)?" The obvious answer to the obvious question is that the first century Jews, including Yahushua the Messiah and His disciples, kept the same recurring seven day week as the Romans. It is historical fact that the same week and day sequence exists to this day and has not changed. Yahweh has preserved His holy Sabbath!

Conclusion

    Historically, Lunar Sabbath keepers believe that the Lunar Sabbath was observed by 1st century Judaism, but was lost some time after the destruction of the temple in Jerusalem. But even if one rejects all the evidence to the contrary, it would seem quite impossible for every Sabbath keeper scattered all throughout the world to simultaneously drop a supposed "Lunar Sabbath," all without a trace of evidence of such a dramatic change. The clear historical record is that the Jews of the first century kept the Sabbath just as we do, on a recurring seven day week.

Scripturally, you can look from Genesis to Revelation, but you will not find a single commandment to keep the "Lunar Sabbath." Just as Yahweh has ensured that His inspired word was not lost in antiquity, He also protected His holy Sabbath. Adam kept it, Noah kept it, Abraham, Isaac, and Jacob kept it. Joshua kept it, the righteous judges and kings kept it. The prophets of Yahweh kept it, the returning exiles who were very meticulous about the Sabbath kept it, and we know that Yahushua and His disciples kept it, proving that Yahweh preserved His Sabbath up until the 1st century CE. The same is true today. Yahweh has preserved His word, His Name, and His Sabbath for us today. Let's forsake any teachings that would draw anyone away from the example He set in Genesis chapter 1 and 2.