The new Republic of Sol was a much larger logistical arrangement than the Imperium was; after all, the Imperators had always held a loose hand on Planetary Governors, as the same test stood for all; each governor had the talent to lead the Imperium, it was one of the contingencies built into the Imperium by design, and in the event of the death of the HaMaadimi dynasty control reverted to the other Imperators. In principle any Imperator could operate independently, although in practice central control for efficiency's sake was employed; naval administration for example, was handled centrally, as were target export materials Planetary Governors were expected to send.
The Republic however, was intended to be structured very differently, and with much greater complexity. It was an experiment in liberty. Executive, Judicial and Legislative functions of governance were all divided and assigned to different structures in an effort to safeguard controls - after all, the Imperium relied on the assumption of someone like Imperator HaMaadimi leading. The Republic may or may not have that.
The Legislature was assigned to the Priests and accepted by the people, as per the Law of Moses. The Senate handled most of the Executive functions of state, as per the Reipublicae Solis' ancient predecessor. The Judicial functions had a system of primary and appeals courts, juries, settlements for trial by combat, restitution arbiters and prisons. The issue of money was reserved to pure public meetings of the Senate to determine rate of economic growth (
or decline, although when the economic growth can be quantified in moons each year, declines become rare) and the issuing of currency to match. Any attempt to impose a central bank was considered high treason, as were the taking out of non-fully costed loans by the Senate; borrowing, except for the purpose of investment that would bring in higher returns than the costs invoked, was banned.
The Priesthood operated the legislative code on the basis of applying the ancient law given to desert-dwelling nomads and applying it to an interstellar civilisation, and therefore there was an extreme level of debate to clarify each point. But equally, with millions of clergy able to do it full time, eventually an exhaustive two hundred volume codification of an expanded Torah was agreed.
The Senate was composed of a multiple tiered system of Council of Imperators, Optimates and Vox Populi. The Council of Imperators was in effect the head of state and was open to all those who passed the Imperator's standard, with two Consuls chosen from among the Imperators by the Optimates and the Vox Populi. The Council of Imperators, by their nature of how they attain the position, possess the vision to see the goal to aim towards and the ability to make goals into reality. Leadership of the army and navy also belonged to Imperators, with the restriction that no Imperator who held military Imperium could come within 30 AU of Sol (
that is, sunward of Neptune) in much the same way no general or governor possessing military Imperium of old was allowed to cross the Rubicon with their army in tow.
Based very much on Cicero's idealised definition of the Optimates, the Optimates section of the Senate was for "the responsible, self-controlled, dutiful, loyal, self sacrificing" - or, to use terms more familiar to Stellaris, for Specialists. The Optimates therefore was a forum for debate of experts in their fields as it pertains to the apparatus of state.
Vox Populi, is obviously The Voice of the People, in which all citizens of the Reipublicae Solis may vote for one candidate who represents their constituency and additional candidates based on population sizes.
In short, the executive function of state is decided by the Imperators, subject to the veto of the people if it is applied, with the skilled, the knowledgable and the wise responsible for putting the plans of the Council of Imperators into detailed action.
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Handling the Logistics of this was the purpose for which Luna would be terraformed under Rivkah's command, in order to be reshaped into a capital Ecumenopolis suitable for hosting the vast apparatus of state required to run the Reipublicae Solis.