No
As the official Rules Of The Senate say in section VI (Quorum):
- A quorum shall consist of a majority of the Senators duly chosen and sworn.
- No Senator shall absent himself from the service of the Senate without leave.
- If, at any time during the daily sessions of the Senate, a question shall be raised by any Senator as to the presence of a quorum, the Presiding Officer shall forthwith direct the Secretary to call the roll and shall announce the result, and these proceedings shall be without debate.
- Whenever upon such roll call it shall be ascertained that a quorum is not present, a majority of the Senators present may direct the Sergeant at Arms to request, and, when necessary, to compel the attendance of the absent Senators, which order shall be determined without debate; and pending its execution, and until a quorum shall be present, no debate nor motion, except to adjourn, or to recess pursuant to a previous order entered by unanimous consent, shall be in order.
Thus there must be a majority of the US Senate, that is at least 51 Senators, present to do any business, and in particular to pass any bill or resolution. The scenario described in the question, where less than a majority can control the action of the Senate, thus cannot occur.
MY understanding is that when any Senator rises to question whether a quorum is present, in addition to the reading of the names on the Senate floor, lights flash indicating a quorum call in the office of each Senator, and if the Senator is present and did not expect a business session, that Senator would be likely to get to the floor, and if the senator is not in the office but even one staff member is, that staffer would be very likely to call the Senator wherever s/he might be.
Moreover, section VII of the Rules provides that:
- Until the morning business shall have been concluded, and so announced from the Chair, or until one hour after the Senate convenes at the beginning of a new legislative day, no motion to proceed to the consideration of any bill, resolution, report of a committee, or other subject upon the Calendar shall be entertained by the Presiding Officer, unless by unanimous consent: Provided, however, That on Mondays which are the beginning of a legislative day the Calendar shall be called under rule VIII, and until two hours after the Senate convenes no motion shall be entertained to proceed to the consideration of any bill, resolution, or other subject upon the Calendar except the motion to continue the consideration of a bill, resolution, or other subject against objection as provided in rule VIII, or until the call of the Calendar has been completed.
This ensures that the Senate cannot simply take up a bill the moment it convenes.
Moreover, there is a public Legislative Calander. This calendar:
- Displays time and date the Senate is next scheduled to convene
The CRS report "The Senate’s Calendar of Business" says in relevant part:
The Senate’s Calendar of Business lists bills, resolutions, and other items of legislative business that are eligible for floor consideration. When a Senate committee reports a bill, it
is said to be placed “on the calendar.” It is not in order for the majority leader or any other Senator to move that the Senate proceed to the consideration of a measure that is not on the calendar, though the majority leader could ask unanimous consent to do so.
...
The Senate’s other calendar, the Executive Calendar, lists treaties and nominations—which constitute the Senate’s executive business—that are available for floor action. Both of these
documents are published each day the Senate is in session and distributed to Senators’ personal offices and to all committee and subcommittee offices.
...
The front cover of the Calendar of Business gives the dates on which each session of the current Congress convened and adjourned sine die and the number of days the Senate actually has met
during each session. It also shows the date and time at which the Senate is next scheduled to convene.
...
Also included in the Calendar of Business are the following:
- calendars for the current month and year, showing the days on which the Senate met and the anticipated dates of future nonlegislative periods;
...
- "bills and joint resolutions read the first time” and awaiting the start of the next legislative day when they will be read by title for a second time; after this second reading, each such measure probably will be placed directly on the calendar
under the provisions of Rule XIV instead of being referred to committee.
Note that this means that no bill can be acted on on the same day that it is introduced, and every Senator's office is notified of bills that had a first reading. Thus no bills can be snuck in and acted on without every Senator having notice that the bill is coming up.
Also, as I understand it, a Senate meeting cannot be scheduled without notice of the date having been given in the Calendar, or else to every Senator's office.
So the sort of "private Calendar" and "private meeting" suggested in the question would not work, unless the rules are first changed.
