This is a profoundly insightful question, because you are cutting straight to the philosophy of science and asking for the structural vocabulary of paradigm shifts.

When you view scientific paradigms as "factoring systems"—different mathematical or logical frameworks that can produce equivalent outputs from the same inputs—you are stepping into the realm of underdetermination (the idea that evidence alone is rarely sufficient to pick a single unique theory).

In this specific situation, philosophers, historians of science, and systems theorists use a very precise set of terms for the moving parts. Here is how they refer to the elements of your scenario:

1. The "Roughly Equivalent Results" (The Output)
This is called Empirical Equivalence (or observational equivalence). When two radically different systems produce the same predictions within the margin of error, they are said to be empirically equivalent. In your Copernican example, both systems had high empirical equivalence relative to the naked-eye data of the 16th century. (Note: Tycho Brahe’s later data actually broke this equivalence, which is why Kepler won out).

2. The "Different Systems of Logic/Factoring" (The Frameworks)
These are referred to as Incommensurable Frameworks (a term coined by Thomas Kuhn and Paul Feyerabend). However, because you use the word "factoring," mathematicians would call them Non-Isomorphic Representations or Alternative Axiomatizations. They are different formalisms that map the same phenomena. Crucially, they are not directly translatable word-for-word; you cannot simply "convert" Ptolemy's epicycles into Copernicus's heliocentrism without changing the very definition of what "motion" and "rest" mean.

3. The "Extra" Elements that Break the Tie (The Deciders)
Since empirical equivalence means data cannot choose between them, scientists rely on Non-Empirical Theory Choice Criteria (also called Epistemic Virtues). These are the "tie-breakers." They include:

• Simplicity (often formalized as Parsimony or Occam’s Razor).

• Fecundity (the ability to predict entirely new classes of phenomena that the old system wasn't designed to handle—Copernicus eventually won on this, thanks to Newton).

• Coherence (how well the system fits with adjacent fields, like physics).

4. The "Factoring" Itself (The Mathematical Act)
If you strictly use the math metaphor, the act of switching from one factoring to another is called a Change of Basis (in linear algebra) or a Gauge Transformation (in physics). When two systems produce equivalent results, they are said to be Covariant—the underlying reality stays the same, but the coordinate system you use to describe it changes.

5. The "Unfactorable" Element (The Core)
Every paradigm has a central, non-negotiable core that cannot be changed without destroying the system. In philosophy of science (via Imre Lakatos), this is called the Hard Core of a research programme. Surrounding it is a Protective Belt of auxiliary hypotheses that can be adjusted to absorb new data. In your factoring metaphor, the hard core is the "prime" element that cannot be factored out; it is the ontological commitment (e.g., "The Earth moves" vs. "The Earth is stationary").

6. The "Residue" or "Noise"
When you factor reality through Paradigm A vs. Paradigm B, there is always leftover complexity that doesn't fit neatly. This is called Anomalous Data or Residual Phenomena. In Ptolemy's time, these residuals were things like the slight variation in the brightness of Mars (which epicycles struggled with). In modern physics, this is the gap between General Relativity and Quantum Mechanics—they are empirically equivalent in their own domains, but the "residue" at the middle is where physicists look for cracks.

---

The Most Important Term for Your Exact Question:

When people point out that the old system "returns more accurate results" during a revolution, they are invoking a concept called Pragmatic Conservatism—the logical principle that, all else being equal, we should stick with the established system because it has a proven track record.

But the deeper philosophical term for what you are describing—two different factorings producing equivalent results—is Duhem-Quine Thesis (underdetermination). It states that a hypothesis can never be tested in isolation, because you are always testing the entire factoring system (the paradigm) against reality. Therefore, when the results are equivalent, the choice between factorings is not a choice about data, but a choice about metaphysics (what you believe reality is) and aesthetics (what you believe a beautiful explanation looks like).

So, to summarize your "parts":
You have an Empirically Equivalent output, produced by two Incommensurable Formalisms, each with a Hard Core and a Protective Belt. The choice between them is settled not by accuracy, but by weighing Epistemic Virtues like fecundity and coherence, until new Anomalous Data eventually breaks the equivalence for good.