The Separator Tests That Protect the Registry from Bad Claims
"Some cross-domain comparisons share actual mathematical structure. When that happens, predictions in one domain force predictions in the other. That's not analogy anymore. That's something else."
When you say "gravity is like sin," three different people hear three different things.
The physicist hears a metaphor. Cute, maybe, but not load-bearing. The theologian hears a devotional illustration — useful for sermons, not for proofs. The ninth-grader hears something that might be true but has no way to check.
All three are right to be cautious. Most cross-domain comparisons ARE just metaphors. "The economy is like a machine" — that's analogy. It helps you think, but it doesn't constrain anything. You can't use it to predict which part of the economy will break next Tuesday.
But some cross-domain comparisons are not metaphors. Some of them share actual mathematical structure. The same equation governs both systems. When that happens, predictions in one domain force predictions in the other. That's not analogy anymore. That's something else. And then there's a level above that: not just the same equation in two places, but the same underlying thing producing both domains. That's identity.
The question is: how do you tell which level you're at? That's what this document does.
Think of it as a measurement resolution problem. At low resolution, two signals look the same (analogy). At higher resolution, shared spectral structure (correspondence). At full resolution, the transfer function is identical (isomorphism). And sometimes you discover both signals come from the same source (identity).
Think of it as the difference between illustration and revelation. A sermon illustration is Level 1. When the actual mathematics of gravitational attraction and the actual theology of grace share the same formal properties — that's not illustration anymore. That's the creation revealing its Creator through its structure.
Your friend group and a flock of birds both move in patterns (Level 1). Both follow the same three rules (Level 2). Both governed by the same equations — one predicts the other (Level 3). Someone proves those equations come from a single deeper law that generates BOTH systems (Level 4). Same source. Two outputs.
Each level has a precise definition, a test, and examples from this registry. The levels are not a spectrum — they are discrete thresholds. An ISO is either above a threshold or below it.
Shared language, imagery, or intuitive resemblance between two domains. The comparison helps you think but does not constrain anything. Removing the poetic language leaves nothing structural behind.
Analogy is where the search begins. It is never where the proof ends. Many genuine isomorphisms were first noticed as analogies — the level is not an insult, it is an honest starting position.
A real structural echo between two domains. Shared patterns, shared topology, partial mathematical overlap. Maybe one-way explanatory usefulness. The mapping is not yet complete — it doesn't fully preserve the important relations in both directions.
A structure-preserving mapping between two domains. The relations, transformations, and dynamics that govern one system are preserved under the mapping to the other. All four tests are required. Failing any one keeps it at Level 2.
Not "same form in two places" but "same thing." A single generative structure produces both domains as outputs. You don't have two systems that happen to match — you have one system that necessarily generates both physics and theology as projections.
Maxwell did this for electricity and magnetism. Before Maxwell: two separate forces that happened to interact. After Maxwell: one field, two projections depending on your reference frame. The unification didn't just show similarity — it showed the two descriptions were incomplete fragments of a single complete description. That's Level 4.
These are the gatekeepers between levels. Use them every time you evaluate a cross-domain claim. Each test is a single question that determines whether a claim has earned promotion.
This is the master status board. Every ISO in the registry has a current level, the strongest feature supporting that placement, and an implicit kill condition. Levels are not permanent — evidence moves claims up and down.
Three things would destroy this classification system. We've been looking for all three. None have appeared. But they remain live possibilities — and naming them is how the framework protects its own integrity.
If the four isomorphism tests (structural preservation, non-arbitrariness, constraint, bidirectionality) presuppose the framework's conclusions, the whole hierarchy collapses. If the tests are circular — if they can only be applied by someone who already believes the framework — then they are not tests but confessions. We don't believe they are circular; they are domain-general tests from mathematics and philosophy of science. But if someone demonstrates circularity, we'll rebuild.
If a Level 3 claim is shown to fail its own tests — if ISO-027's SIR mapping doesn't actually predict sin propagation dynamics, or if ISO-032's attachment mapping doesn't replicate empirically — those demotions damage the framework's credibility in proportion to how central those claims were. The flagship ISOs are the framework's most exposed positions. Their fall would be serious.
If a Level 1 claim passes Level 3 tests — if ISO-029 (Music/Trinity) turns out to have genuine structure-preserving dynamics we missed — that means our screening is too conservative and needs recalibration. False demotions are as serious as false promotions: they mean we're discarding real evidence because our tests are poorly calibrated.
This document is the registry's immune system. Use it as a checklist, not a belief statement. The levels are measurement results, not endorsements.